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s
f
or
me
d“
l
i
k
eav
e
c
t
or
”
.Al
t
h
ou
g
ht
h
e
r
ea
r
emu
l
t
i
p
l
ec
oor
d
i
n
a
t
es
y
s
t
e
msi
nwh
i
c
h
v
e
c
t
or
sc
a
nb
ee
x
p
r
e
s
s
e
d
,t
h
e“
s
i
mp
l
e
s
t
”on
ei
sCa
r
t
e
s
i
a
n
,wh
e
r
eav
e
c
t
orc
a
nt
y
p
i
c
a
l
l
yb
e
wr
i
t
t
e
n
:
A =Axx
ˆ+Ayy
ˆ+Azz
ˆ
i
nt
e
r
msofc
omp
on
e
n
ts
c
a
l
a
ra
mp
l
i
t
u
d
e
s(
Ax,
Ay,
Az)a
n
du
n
i
tv
e
c
t
or
si
nt
h
e
or
t
h
og
on
a
l
d
i
r
e
c
t
i
on
s(
x
ˆ
,
y
ˆ
,
z
ˆ
)
.
Toa
ddv
e
c
t
or
s(
i
nCa
r
t
e
s
i
a
nc
oor
d
i
n
a
t
e
s
)wea
d
dc
omp
on
e
n
t
s
:
C =A+B=(
Ax+Bx)
x
ˆ+(
Ay+By)
y
ˆ+(
Az+Bz)
z
ˆ
Th
er
e
s
u
l
t
a
n
ti
sa
l
s
ot
h
er
e
s
u
l
tofag
e
ome
t
r
i
ct
r
i
a
n
g
l
eo
rp
a
r
a
l
l
e
l
og
r
a
mr
u
l
e
:
(
B)
A
C=
A+B
B
(
A)
Su
b
t
r
a
c
t
i
oni
sj
u
s
ta
dd
i
t
i
onofan
e
g
a
t
i
v
e
:
C=A−B=(
Ax−Bx)
x
ˆ+(
Ay−By)
y
ˆ+(
Az−Bz)
z
ˆ
1
5
I
tc
a
na
l
s
ob
ev
i
s
u
a
l
i
z
e
db
yme
a
n
sofag
e
ome
t
r
i
ct
r
i
a
n
g
l
es
ot
h
a
t(
A−B)+
B =A.
−B
C=
A−B
=
A+(
−B)
A
B
3.
1 Sc
a
l
a
r
sa
n
dVe
c
t
or
s
Anor
d
i
n
a
r
yn
u
mb
e
rt
h
a
td
oe
sn
otc
h
a
n
g
ewh
e
nt
h
ec
oor
d
i
n
a
t
ef
r
a
mec
h
a
n
g
e
si
sc
a
l
l
e
d
as
c
a
l
a
r
.Mu
l
t
i
p
l
i
c
a
t
i
onofav
e
c
t
orb
yas
c
a
l
a
rr
e
s
c
a
l
e
st
h
ev
e
c
t
orb
ymu
l
t
i
p
l
y
i
n
ge
a
c
h
ofi
t
sc
omp
on
e
n
t
sa
sas
p
e
c
i
a
l
c
a
s
eoft
h
i
sr
u
l
e
:
a
A=a
(
Axx
ˆ+Ayy
ˆ+Azz
ˆ
)=(
a
Ax)
x
ˆ+(
a
Ay)
y
ˆ+(
a
Az)
z
ˆ
Not
ewe
l
lt
h
a
tt
h
ev
e
c
t
orc
omp
on
e
n
t
sAx,Ay,Aza
r
et
h
e
ms
e
l
v
e
ss
c
a
l
a
r
s
.I
n
d
e
e
d
,
web
u
i
l
dav
e
c
t
ori
nt
h
ef
i
r
s
tp
l
a
c
eb
yt
a
k
i
n
gau
n
i
tv
e
c
t
or(
ofl
e
n
g
t
hon
e
,“
p
u
r
e
d
i
r
e
c
t
i
on
”
)a
n
ds
c
a
l
i
n
gi
tb
yi
t
sc
omp
on
e
n
tl
e
n
g
t
h
,e
.
g
.Axx
ˆ
,a
n
dt
h
e
ns
u
mmi
n
gt
h
e
v
e
c
t
or
st
h
a
tma
k
eu
pi
t
sc
omp
on
e
n
t
s
!
Th
emu
l
t
i
p
l
i
c
a
t
i
onofav
e
c
t
orb
yas
c
a
l
a
ri
sc
ommu
t
a
t
i
v
e
:
a
A=Aa
a
n
dd
i
s
t
r
i
b
u
t
i
v
e
.
a
(
A+B)=a
A+a
B
3.
2 Th
eSc
a
l
a
r
,
o
rDo
tPr
o
d
u
c
t
I
ti
sa
l
s
op
os
s
i
b
l
ef
or
ms
e
v
e
r
a
l“
mu
l
t
i
p
l
i
c
a
t
i
on
l
i
k
e
”p
r
od
u
c
t
soft
wo(
ormor
e
)v
e
c
t
or
s
.
Wec
a
nt
a
k
et
wov
e
c
t
or
sa
n
dma
k
eas
c
a
l
a
r
,a
n
ot
h
e
rv
e
c
t
or
,ora“
b
i
v
e
c
t
or
”(
t
e
n
s
or
)
.
Someoft
h
e
s
emi
g
h
tb
er
e
g
u
l
a
rv
e
r
s
i
onoft
h
eob
j
e
c
t
s
,s
omemi
g
h
tb
e“
p
s
e
u
d
o”
v
e
r
s
i
on
st
h
a
twewi
l
lc
omet
ou
n
de
r
s
t
a
n
d.Howe
v
e
r
,weh
a
v
et
ob
ec
a
r
e
f
u
ln
ott
og
e
t
s
we
p
toffofou
rf
e
e
tb
yt
h
ed
a
z
z
l
i
n
ga
r
r
a
yofp
os
s
i
b
i
l
i
t
i
e
sr
i
g
h
ta
tt
h
eb
e
g
i
n
n
i
n
g
.
Wewi
l
l
t
h
e
r
e
f
or
es
t
a
r
twi
t
ht
h
ea
r
g
u
a
b
l
ys
i
mp
l
e
s
tf
or
mofv
e
c
t
ormu
l
t
i
p
l
i
c
a
t
i
on
:
t
h
e
s
c
a
l
a
rord
otp
r
od
u
c
t
:
C=A·B.
Not
et
h
a
tt
h
edotp
r
od
u
c
tt
u
r
n
s
t
wov
e
c
t
or
si
n
t
oas
c
a
l
a
r
.I
ti
sa
l
s
oof
t
e
nc
a
l
l
e
da
ni
n
n
e
rp
r
od
u
c
t
,
a
l
t
h
ou
g
ht
h
el
a
t
t
e
ri
s
s
ome
wh
a
tmor
eg
e
n
e
r
a
l
t
h
a
nt
h
ed
otp
r
od
u
c
ti
naEu
c
l
i
d
e
a
n(
e
.
g
.
Ca
r
t
e
s
i
a
n
)s
p
a
c
e
.
Th
edotp
r
od
u
c
ti
sc
ommu
t
a
t
i
v
e
:
A·
B=
B·
A
I
ti
sdi
s
t
r
i
b
u
t
i
v
e
:
A·
(
B+
C)
=
A·
B+
A·
C
I
tc
a
nb
ee
v
a
l
u
a
t
e
dt
wo(
i
mp
or
t
a
n
t
)wa
y
s
:
C=A·B=ABc
os
(
θ
)=AxBx+AyBy+AzBz
1
wh
e
r
eAa
n
dBa
r
et
h
es
c
a
l
a
rma
gn
i
t
u
de
soft
h
ev
e
c
t
or
sAa
n
dBr
e
s
p
e
c
t
i
v
e
l
ya
n
dθi
s
t
h
ea
n
g
l
ei
nb
e
t
we
e
nt
h
e
m:
B
B
θ
B
A
F
r
omt
h
ef
i
r
s
toft
h
e
s
ef
or
ms
,
wes
e
et
h
a
tt
h
ed
otp
r
odu
c
tc
a
nb
et
h
ou
g
h
tofa
st
h
e
ma
g
n
i
t
u
deoft
h
ev
e
c
t
orAt
i
me
st
h
ema
g
n
i
t
u
d
eoft
h
ec
o
mp
on
e
n
toft
h
ev
e
c
t
orBi
nt
h
e
s
a
med
i
r
e
c
t
i
ona
sA,
i
n
d
i
c
a
t
e
da
sBi
nt
h
ef
i
g
u
r
ea
b
ov
e
.
I
n
d
e
e
d
:
A·
B=
AB=
AB
(
Th
el
a
t
t
e
rt
h
ema
g
n
i
t
u
d
eofBt
i
me
st
h
ec
omp
on
e
n
tofAp
a
r
a
l
l
e
lt
oB.
)Th
es
e
c
on
d
f
ol
l
owsf
r
omt
h
ef
ol
l
owi
n
gmu
l
t
i
p
l
i
c
a
t
i
ont
a
b
l
eofu
n
i
tv
e
c
t
or
s
,wh
i
c
hc
a
nb
et
h
ou
g
h
tofa
sd
e
f
i
n
i
n
gt
h
edotp
r
od
u
c
ta
n
dt
h
eu
n
i
tv
e
c
t
or
sof
“
or
t
h
on
or
ma
l
c
oor
d
i
n
a
t
e
s
”s
i
mu
l
t
a
n
e
ou
s
l
y
:
x
ˆ·x
ˆ=y
ˆ·y
ˆ=z
ˆ·z
ˆ=1
x
ˆ·y
ˆ=y
ˆ·z
ˆ=z
ˆ·x
ˆ=0
(
p
l
u
st
h
ec
ommu
t
a
t
e
df
o
r
msoft
h
el
a
s
tr
ow,
e
.
g
.
y
ˆ·x
ˆ=0a
swe
l
l
)
.
Twov
e
c
t
or
st
h
a
ta
r
ep
e
r
p
e
n
d
i
c
u
l
a
r(
or
t
h
og
on
a
l
)h
a
v
ead
otp
r
odu
c
tofz
e
r
oa
n
d
v
i
c
e
v
e
r
s
a
.I
fa
n
don
l
yi
f(
wr
i
t
t
e
nh
e
n
c
e
f
or
t
ha
s“
i
ff”
)A·B=0t
h
e
nA⊥B.
Wemi
g
h
ts
a
y
t
h
a
tAi
sn
or
ma
l
t
o,
p
e
r
p
e
n
d
i
c
u
l
a
rt
o,
a
tr
i
g
h
ta
n
g
l
e
st
o,
oror
t
h
og
on
a
l
t
oB.Al
l
oft
h
e
s
e
me
a
nt
h
es
a
met
h
i
n
g
.
1
Not
et
h
a
twed
e
f
i
n
et
h
ema
g
n
i
t
u
deoft
h
ev
e
c
t
orA(
wr
i
t
t
e
ne
i
t
h
e
rAor|
A|
)i
nt
e
r
msoft
h
ei
n
n
e
rp
r
od
u
c
t
:
2
2
2 1
A=|
A|
=+
A·
A=(
A x+A y+A z)2
p
3.
2.
1 Th
eL
a
wo
fCo
s
i
n
e
s
Th
el
a
wo
fc
o
s
i
n
e
si
se
a
s
i
l
yd
e
r
i
v
e
d(
on
eofs
e
v
e
r
a
lwa
y
s
)b
yf
i
n
di
n
gt
h
es
c
a
l
a
rl
e
n
g
t
h
oft
h
ed
i
ffe
r
e
n
c
ev
e
c
t
orA−B.
A B
B
θ
A
2
|
A−
B|=
(
A−
B)
·
(
A−
B)
=
A·
A−
A·
B−
B·
A+
B·
B
or(
c
ol
l
e
c
t
i
n
gt
e
r
msa
n
du
s
i
n
gr
u
l
e
sf
r
oma
b
ov
e
)
:
|
A−B|
=
2
2
A +B −2
ABc
osθ
Not
et
h
a
tt
h
ePy
t
h
a
g
or
e
a
nTh
e
or
e
mi
sas
p
e
c
i
a
l
c
a
s
eoft
h
i
sr
u
l
ewi
t
hθ=π/
2
.
3.
3 Th
eVe
c
t
o
r
,
o
rCr
o
s
sPr
od
u
c
t
Th
e
r
ei
sas
e
c
on
dwa
yt
omu
l
t
i
p
l
yt
wov
e
c
t
or
s
.Th
i
sp
r
od
u
c
toft
wov
e
c
t
or
sp
r
od
u
c
e
sa
t
h
i
r
dv
e
c
t
or
,wh
i
c
hi
swh
yi
ti
sof
t
e
nr
e
f
e
r
r
e
dt
oa
s“
t
h
e
”v
e
c
t
orp
r
od
u
c
t(
e
v
e
nt
h
ou
g
h
t
h
e
r
ea
r
ean
u
mb
e
rofp
r
odu
c
t
si
n
v
ol
v
i
n
gv
e
c
t
or
s
)
.I
ti
ss
y
mb
ol
i
c
a
l
l
yd
i
ffe
r
e
n
t
i
a
t
e
db
y
t
h
emu
l
t
i
p
l
i
c
a
t
i
ons
y
mb
olu
s
e
d
,
wh
i
c
hi
sal
a
r
g
e×s
i
g
n
,
h
e
n
c
ei
ti
sof
t
e
nr
e
f
e
r
r
e
dt
oa
s
t
h
ec
r
os
sp
r
od
u
c
tb
ot
hf
ort
h
e(
c
r
os
s
l
i
k
e
)s
h
a
p
eoft
h
i
ss
i
g
na
n
db
e
c
a
u
s
eoft
h
e
p
a
t
t
e
r
nofmu
l
t
i
p
l
i
c
a
t
i
onofc
omp
on
e
n
t
s
.Wewr
i
t
et
h
ec
r
os
sp
r
od
u
c
toft
wov
e
c
t
or
sa
s
e
.
g
.
C=A×B.
Th
ec
r
os
sp
r
od
u
c
ta
n
t
i
c
ommu
t
e
s
:
A×
B=
−
B×
A
I
ti
sdi
s
t
r
i
b
u
t
i
v
e
:
A×
(
B+C)
=
A×
B+
A×
C
(
a
l
t
h
ou
g
ht
h
eor
de
roft
h
ep
r
odu
c
tmu
s
tb
ema
i
n
t
a
i
n
e
d
!
)
I
ta
sn
ot
e
da
b
ov
ep
r
od
u
c
e
sav
e
c
t
o
r(
r
e
a
l
l
yap
s
e
u
d
ov
e
c
t
or
,e
x
p
l
a
i
n
e
dl
a
t
e
r
)f
r
om
t
wov
e
c
t
or
s
.
Th
ema
g
n
i
t
u
d
eoft
h
ec
r
os
sp
r
odu
c
toft
wov
e
c
t
or
si
sd
e
f
i
n
e
db
y
:
|
A×B|
=ABs
i
nθ=AB⊥=A⊥B
u
s
i
n
gt
e
r
mss
i
mi
l
a
rt
ot
h
os
eu
s
e
da
b
ov
ei
nou
rd
i
s
c
u
s
s
i
onofdotp
r
od
u
c
t
s
.
Not
ewe
l
l
!
I
f
t
h
ev
e
c
t
or
sb
ot
hh
a
v
ed
i
me
n
s
i
on
sofl
e
n
g
t
h
,
t
h
ec
r
os
sp
r
od
u
c
t
i
st
h
ea
r
e
aoft
h
ep
a
r
a
l
l
e
l
og
r
a
mf
or
me
db
yt
h
ev
e
c
t
or
sa
si
l
l
u
s
t
r
a
t
e
di
nf
i
g
u
r
e3
.
1
.I
ti
s
s
ome
t
i
me
sc
a
l
l
e
dt
h
ea
r
e
a
l
p
r
od
u
c
tf
ort
h
i
sr
e
a
s
on
,
a
l
t
h
ou
g
hon
e
B
Ar
e
a=|
AxB|
B
θ
B
A
AxBdi
r
e
c
t
i
oni
sou
tofpa
ge
F
i
g
u
r
e3
.
1
:Th
ea
r
e
ab
e
t
we
e
nt
wov
e
c
t
or
si
nap
l
a
n
ei
st
h
ema
g
n
i
t
u
deoft
h
ec
r
os
s
p
r
od
u
c
toft
h
os
ev
e
c
t
or
s
.
wou
l
dt
h
i
n
kt
won
a
me
si
se
n
ou
g
h(
a
n
di
nma
n
yc
on
t
e
x
t
s
,a
r
e
a
lp
r
od
u
c
tma
k
e
sn
o
s
e
n
s
e
)
.
Th
edi
r
e
c
t
i
onofA×Bi
sg
i
v
e
nb
yt
h
er
i
g
h
t
h
a
n
dr
u
l
e
.Th
ed
i
r
e
c
t
i
oni
sa
l
wa
y
s
p
e
r
p
e
n
di
c
u
l
a
rorn
or
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lt
ot
h
ep
l
a
n
ed
e
f
i
n
e
db
yt
h
et
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on
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ol
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n
e
a
rv
e
c
t
or
si
nt
h
e
c
r
os
sp
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od
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c
t
.Th
a
tl
e
a
v
e
st
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os
s
i
b
i
l
i
t
i
e
s
.
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fy
oul
e
tt
h
ef
i
n
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e
r
sofy
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rr
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g
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th
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n
dl
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n
e
u
pwi
t
hA(
t
h
ef
i
r
s
t
)s
ot
h
a
tt
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e
yc
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r
lt
h
r
ou
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ht
h
es
ma
l
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n
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e(
t
h
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el
e
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st
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t
h
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l
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oth
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r
ty
ou
rwr
i
s
t
)i
n
t
oBt
h
e
nt
h
et
h
u
mbofy
ou
rr
i
g
h
th
a
n
dwi
l
lp
i
c
kou
tt
h
e
p
e
r
p
e
n
d
i
c
u
l
a
rdi
r
e
c
t
i
onoft
h
ec
r
os
sp
r
od
u
c
t
.
I
nt
h
ef
i
g
u
r
ea
b
ov
e
,
i
ti
sou
toft
h
ep
a
g
e
.
F
i
n
a
l
l
y
:
A×
A=
−
(
A×
A)
=
0
Tog
e
t
h
e
rwi
t
ht
h
er
u
l
ef
orr
e
s
c
a
l
i
n
gv
e
c
t
or
st
h
i
sp
r
ov
e
st
h
a
tt
h
ec
r
os
sp
r
od
u
c
tofa
n
y
v
e
c
t
orwi
t
hi
t
s
e
l
fora
n
yv
e
c
t
orp
a
r
a
l
l
e
l
o
ra
n
t
i
p
a
r
a
l
l
e
l
t
oi
t
s
e
l
fi
sz
e
r
o.Th
i
sa
l
s
of
ol
l
ows
f
r
omt
h
ee
x
p
r
e
s
s
i
onf
ort
h
ema
g
n
i
t
u
d
eABs
i
nθwi
t
hθ=0or
π.
L
e
tu
sf
or
mt
h
eCa
r
t
e
s
i
a
nr
e
p
r
e
s
e
n
t
a
t
i
onofac
r
os
sp
r
od
u
c
toft
wov
e
c
t
or
s
.We
b
e
g
i
nb
yn
ot
i
n
gt
h
a
tar
i
g
h
th
a
n
d
e
dc
oor
d
i
n
a
t
es
y
s
t
e
mi
sd
e
f
i
n
e
db
yt
h
er
e
qu
i
r
e
me
n
t
t
h
a
tt
h
eu
n
i
tv
e
c
t
or
ss
a
t
i
s
f
y
:
x
ˆ×y
ˆ=z
ˆ
Th
i
si
si
l
l
u
s
t
r
a
t
e
dh
e
r
e
:
y
y
z
x
x
z
Youc
a
ne
a
s
i
l
yc
h
e
c
kt
h
a
ti
ti
sa
l
s
ot
r
u
et
h
a
t
:
x
ˆ×y
ˆ=z
ˆ
y
ˆ×z
ˆ=z
ˆ
z
ˆ×x
ˆ=y
ˆ
Weu
s
et
h
ea
n
t
i
c
ommu
t
i
onr
u
l
eont
h
e
s
et
h
r
e
ee
qu
a
t
i
on
s
:
y
ˆ×x
ˆ=−
z
ˆ
z
ˆ×y
ˆ=−
z
ˆ
x
ˆ×z
ˆ=−
y
ˆ
An
dn
ot
et
h
a
t
:
x
ˆ×x
ˆ=y
ˆ×y
ˆ=z
ˆ×z
ˆ=0
Th
i
sf
or
mst
h
ef
u
l
lmu
l
t
i
p
l
i
c
a
t
i
ont
a
b
l
eoft
h
eor
t
h
on
or
ma
lu
n
i
tv
e
c
t
or
sofas
t
a
n
d
a
r
d
r
i
g
h
t
h
a
n
d
e
dCa
r
t
e
s
i
a
nc
oor
d
i
n
a
t
es
y
s
t
e
m,a
n
dt
h
eCa
r
t
e
s
i
a
n(
a
n
dv
a
r
i
ou
sot
h
e
r
or
t
h
on
or
ma
l
)c
oor
di
n
a
t
ec
r
os
sp
r
od
u
c
tn
owf
ol
l
ows
.
Ap
p
l
y
i
n
gt
h
ed
i
s
t
r
i
b
u
t
i
v
er
u
l
ea
n
dt
h
es
c
a
l
a
rmu
l
t
i
p
l
i
c
a
t
i
onr
u
l
e
,mu
l
t
i
p
l
you
ta
l
lof
t
h
et
e
r
msi
nA×B:
(
Axx
ˆ+Ayy
ˆ+Azz
ˆ
)× (
Bxx
ˆ+Byy
ˆ+Bzz
ˆ
)=
AxBxx
ˆ×x
ˆ+AxByx
ˆ×y
ˆ+AxBzx
ˆ×z
ˆ
+
AyBxy
ˆ×x
ˆ+AyByy
ˆ×y
ˆ+AyBzy
ˆ×z
ˆ
+
AzBxz
ˆ×x
ˆ+AzByz
ˆ×y
ˆ+AzBzz
ˆ×z
ˆ
Th
edi
a
g
n
on
a
l
t
e
r
msv
a
n
i
s
h
.
Th
eot
h
e
rt
e
r
msc
a
na
l
l
b
es
i
mp
l
i
f
i
e
dwi
t
ht
h
eu
n
i
t
v
e
c
t
orr
u
l
e
sa
b
ov
e
.
Th
er
e
s
u
l
ti
s
:
A×B=(
AyBz−AzBy)
x
ˆ+(
AzBx−AxBz)
y
ˆ+(
AxBy−AyBx)
z
ˆ
Th
i
sf
or
mi
se
a
s
yt
or
e
me
mb
e
ri
fy
oun
ot
et
h
a
te
a
c
hl
e
a
d
i
n
gt
e
r
mi
s
ac
y
c
l
i
cp
e
r
mu
t
a
t
i
onofx
y
z
.Th
a
ti
s
,
AyBzx
ˆ
,
AzBxy
ˆa
n
dAxByz
ˆa
r
ey
z
x
,
z
x
y
,
a
n
dx
y
z
.
Th
es
e
c
on
dt
e
r
mi
ne
a
c
hp
a
r
e
n
t
h
e
s
e
si
st
h
es
a
mea
st
h
ef
i
r
s
tb
u
ti
nt
h
eop
p
os
i
t
eor
d
e
r
,
wi
t
ht
h
ea
t
t
e
n
da
n
tmi
n
u
ss
i
g
n
,
f
r
omt
h
ec
y
c
l
i
cp
e
r
mu
t
a
t
i
on
sofz
y
x
.
3.
4 Tr
i
p
l
ePr
o
d
u
c
t
sofVe
c
t
o
r
s
Th
e
r
ea
r
et
wot
r
i
p
l
ep
r
odu
c
t
sofv
e
c
t
or
s
.
Th
ef
i
r
s
ti
st
h
es
c
a
l
a
rt
r
i
p
l
ep
r
od
u
c
t
:
A·
(
B×
C)
I
fA,Ba
n
dCa
r
ea
l
ll
e
n
gt
hv
e
c
t
or
s
,t
h
i
sr
e
p
r
e
s
e
n
t
st
h
ev
ol
u
meofp
a
r
a
l
l
e
l
op
i
p
e
d
f
or
me
db
yt
h
ev
e
c
t
or
s
.
Th
es
e
c
on
di
st
h
ev
e
c
t
ort
r
i
p
l
ep
r
odu
c
t
:
A×
(
B×
C)
=
B(
A·
C)
−
C(
A·
B)
Th
i
sl
a
s
ti
d
e
n
t
i
t
yi
sc
a
l
l
e
dt
h
eBACCAB(
p
a
l
i
n
d
r
omi
c
)r
u
l
e
.I
ti
st
e
di
ou
sb
u
t
s
t
r
a
i
g
h
t
f
or
wa
r
dt
op
r
ov
ei
tf
orCa
r
t
e
s
i
a
nv
e
c
t
orc
omp
on
e
n
t
s
.F
i
r
s
t
,h
owe
v
e
r
,
wewou
l
d
l
i
k
et
oi
n
t
r
odu
c
et
wos
p
e
c
i
a
lt
e
n
s
orf
or
mst
h
a
tg
r
e
a
t
l
ys
i
mp
l
i
f
yt
h
ea
l
g
e
b
r
aofb
ot
hd
ot
a
n
dc
r
os
sp
r
odu
c
t
sa
n
de
n
a
b
l
eu
st
op
r
ov
ev
a
r
i
ou
sv
e
c
t
ori
d
e
n
t
i
t
i
e
su
s
i
n
ga
l
g
e
b
r
a
i
n
s
t
e
a
dofat
e
d
i
ou
se
n
u
me
r
a
t
i
onoft
e
r
ms
.
3.
5 δ
n
dǫ
i
ja
i
j
k
Asn
ot
e
da
b
ov
e
,
wewou
l
dl
i
k
et
ob
ea
b
l
et
os
i
mp
l
i
f
yv
e
c
t
ora
l
g
e
b
r
ai
nor
d
e
rt
op
r
ov
et
h
e
t
r
i
p
l
ep
r
od
u
c
tr
u
l
ea
n
dv
a
r
i
ou
sot
h
e
rv
e
c
t
ori
d
e
n
t
i
t
i
e
swi
t
h
ou
th
a
v
i
n
gt
oe
n
u
me
r
a
t
ewh
a
t
ma
yt
u
r
nou
tt
ob
eal
a
r
g
en
u
mb
e
roft
e
r
ms
.Ag
r
e
a
td
e
a
lofs
i
mp
l
i
f
i
c
a
t
i
oni
sp
os
s
i
b
l
e
u
s
i
n
gt
wo“
s
p
e
c
i
a
l
”t
e
n
s
or
st
h
a
ta
p
p
e
a
ri
nt
h
ema
n
ys
u
mma
t
i
on
st
h
a
toc
c
u
ri
nt
h
e
e
x
p
r
e
s
s
i
on
sa
b
ov
e
,
a
swe
l
la
sas
p
e
c
i
a
lr
u
l
et
h
a
ta
l
l
owsu
st
o“
c
omp
r
e
s
s
”t
h
ea
l
g
e
b
r
a
b
ye
l
i
mi
n
a
t
i
n
gar
e
d
u
n
d
a
n
ts
u
mma
t
i
o
ns
y
mb
ol
.
3.
5.
1 Th
eKr
on
e
c
k
e
rDe
l
t
aF
u
n
c
t
i
o
na
n
dt
h
eE
i
n
s
t
e
i
nSu
mma
t
i
on
Co
n
v
e
n
t
i
o
n
Th
eKr
on
e
c
k
e
rd
e
l
t
af
u
n
c
t
i
oni
sd
e
f
i
n
e
db
yt
h
er
u
l
e
s
:
1i
fi
=j
δi
j =
0i
fi
=j
Us
i
n
gt
h
i
swec
a
nr
e
d
u
c
et
h
ed
otp
r
od
u
c
tt
ot
h
ef
ol
l
owi
n
gt
e
n
s
orc
on
t
r
a
c
t
i
on
,
u
s
i
n
gt
h
e
Ei
n
s
t
e
i
ns
u
mma
t
i
onc
on
v
e
n
t
i
on
:
3
A·B=
Ai
Bi=Ai
δ
Bi
i
jB
j=Ai
i
=
1
wh
e
r
ewes
u
mr
e
p
e
a
t
e
di
n
d
i
c
e
sov
e
ra
l
l
oft
h
eor
t
h
og
on
a
l
c
a
r
t
e
s
i
a
n
3
c
oor
d
i
n
a
t
ei
n
d
i
c
e
swi
t
h
ou
th
a
v
i
n
gt
owr
i
t
ea
ne
x
p
l
i
c
i
ti
.Wewi
l
lh
e
n
c
e
f
or
t
hu
s
et
h
i
s
=
1
c
on
v
e
n
t
i
ona
l
mos
ta
l
lt
h
et
i
met
os
t
r
e
a
ml
i
n
et
h
en
ot
a
t
i
onofc
e
r
t
a
i
nk
i
n
d
sof(
v
e
c
t
or
a
n
dt
e
n
s
or
)a
l
g
e
b
r
a
.
Th
eKr
on
e
c
k
e
rd
e
l
t
af
u
n
c
t
i
oni
sob
v
i
ou
s
l
yu
s
e
f
u
lf
orr
e
p
r
e
s
e
n
t
i
n
gt
h
edotp
r
odu
c
ti
na
c
omp
a
c
twa
y
.Wec
a
ns
i
mi
l
a
r
l
yi
n
v
e
n
tas
y
mb
olt
h
a
ti
n
c
or
p
or
a
t
e
sa
l
loft
h
ed
e
t
a
i
l
soft
h
e
wa
y
st
h
eu
n
i
tv
e
c
t
or
smu
l
t
i
p
l
yi
nt
h
ec
r
os
sp
r
od
u
c
t
,
n
e
x
t
.
3.
5.
2 Th
eL
e
v
i
Ci
v
i
t
aTe
n
s
o
r
Th
eL
e
v
i
Ci
v
i
t
at
e
n
s
ori
sa
l
s
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n
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st
h
et
h
i
r
dr
a
n
kf
u
l
l
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n
t
i
s
y
mme
t
r
i
cu
n
i
tt
e
n
s
or
a
n
di
sd
e
f
i
n
e
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y
:
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fi
j
ka
r
ea
n
yc
y
c
l
i
cp
e
r
mu
t
a
t
i
onof1
2
3ǫi
j
k=
−
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fi
j
ka
r
ea
n
yc
y
c
l
i
cp
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r
mu
t
a
t
i
onof3
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r
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i
fa
n
yp
a
i
rofi
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c
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os
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c
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n
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n
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c
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on
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n
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t
e
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i
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e
n
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i
on
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(
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k=
3
Ai
Bjǫi
Bjǫi
j
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j
k
i
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=
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e(
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ora
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.
3.
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3 Th
eE
p
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l
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l
t
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c
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rt
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r
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e
de
p
s
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l
on
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e
l
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ai
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e
n
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t
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j
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n
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st
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td
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l
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h
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yd
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n
e
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l
u
a
t
e
:
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(
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k
m)
b
yu
s
i
n
gǫmni=ǫi
n
dǫj
ǫi
.
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i
k=−
j
k
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x
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i
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p
p
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s
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r
ov
et
h
a
t
:
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(
B×
C)
=
B·
(
C×
A)
=
C·
(
A×
B)
L
e
t
’
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t
et
h
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r
s
tt
e
r
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on
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h
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omp
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t
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nf
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v
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h
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:
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B×C)=ǫ
Ai
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CnAi=Bmǫni
Ai
mn
i
mCn
2
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r
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h
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e
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)=B·(
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jCn
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e
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t
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i
r
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t
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h
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e
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n
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e
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(
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ǫi
j
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e
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et
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mi
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h
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omp
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h
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r
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ǫj
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d
e
n
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ov
e
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mi
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l
t
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n
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t
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on
si
n
f
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v
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a
n
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:
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B×C)=Ai
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Ck=Bk(
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k
wh
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e
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n
d
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e
s
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x
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.
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g
.
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(
4
.
7
)
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0
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4
.
8
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4
.
9
)
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0 0 0
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or
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p
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t
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ong
r
ou
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st
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r
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rt
or:
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= Ryx
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t
i
on
sd
i
s
c
u
s
s
e
d)f
u
n
c
t
i
o
n
f(
t
)
:
d
f =l
t+
i
m f(
t
→0
dx
t
)−f(
t
)
t
Not
emye
x
p
l
i
c
i
ta
n
dde
l
i
b
e
r
a
t
eu
s
eofta
st
h
ei
n
d
e
p
e
n
de
n
tv
a
r
i
a
b
l
eu
p
onwh
i
c
hf
d
e
p
e
n
d
s
.Th
i
si
n
v
i
t
e
su
st
ot
h
i
n
koft
h
i
sa
sar
a
t
eo
fc
h
a
n
gei
np
h
y
s
i
c
swh
e
r
efi
ss
ome
p
h
y
s
i
c
a
l
qu
a
n
t
i
t
ya
saf
u
n
c
t
i
onoftt
h
et
i
me
.
F
r
omt
h
i
son
ec
a
ne
a
s
i
l
yd
e
r
i
v
ea
l
l
s
or
t
sofa
s
s
oc
i
a
t
e
dr
u
l
e
s
,
t
h
emos
ti
mp
or
t
a
n
tof
wh
i
c
ha
r
e
:
•Th
eCh
a
i
nr
u
l
e
.
Su
p
p
os
eweh
a
v
eaf
u
n
c
t
i
onf(
x
)wh
e
r
ex
(
t
)i
si
t
s
e
l
faf
u
n
c
t
i
onof
t(
a
n
dt
h
e
r
ei
sn
o“
s
e
p
a
r
a
t
e
”t
i
mede
p
e
n
d
e
n
c
ei
nf)
.
Th
e
n
:
d
f =d
fdx
xd
t
dt d
•Th
eSu
mr
u
l
e
.
Su
p
p
os
eweh
a
v
et
wof
u
n
c
t
i
on
s
,
f(
t
)a
n
dg
(
t
)
.
Th
e
n
:
d
(
f+g
) df dg
= +
dt
dt dt
•Th
ePr
od
u
c
tr
u
l
e
.
Su
p
p
os
eweh
a
v
et
wof
u
n
c
t
i
on
s
,
f(
t
)a
n
dg
(
t
)
.
Th
e
n
:
d(
d
f d
g
+f d
d
tfg)=g d
t
t
Wewi
l
l
of
t
e
ne
x
p
r
e
s
st
h
e
s
er
u
l
e
si
nt
e
r
msofd
i
ffe
r
e
n
t
i
a
l
s
,
n
otd
e
r
i
v
a
t
i
v
e
swi
t
h
r
e
s
p
e
c
tt
os
p
e
c
i
f
i
cc
o
or
d
i
n
a
t
e
s
.
F
ore
x
a
mp
l
e
:
d
f d
f
d
t
d
t
d
f=d
x dx=
d
(
fg
)=gd
f+fdg
Mos
toft
h
e
s
es
i
mp
l
es
c
a
l
a
rr
u
l
e
sh
a
v
ec
ou
n
t
e
r
p
a
r
t
swh
e
nwec
on
s
i
d
e
rd
i
ffe
r
e
n
t
k
i
n
d
sofv
e
c
t
ord
i
ffe
r
e
n
t
i
a
t
i
on
.
6.
2 Ve
c
t
o
rDi
ffe
r
e
n
t
i
a
t
i
o
n
Wh
e
nwec
on
s
i
d
e
rv
e
c
t
orf
u
n
c
t
i
on
sofc
oor
d
i
n
a
t
e
s
,weh
a
v
ead
ou
b
l
eh
e
l
p
i
n
gof
c
omp
l
e
x
i
t
y
.F
i
r
s
t
,
t
h
e
r
ea
r
et
y
p
i
c
a
l
l
ys
e
v
e
r
a
lc
oor
di
n
a
t
e
s–(
x
,
y
,
z
,
t
)f
ore
x
a
mp
l
e–t
h
a
t
t
h
e
ms
e
l
v
e
sma
yf
or
mav
e
c
t
or
.Se
c
on
d
,
t
h
ef
u
n
c
t
i
on(
p
h
y
s
i
c
a
l
qu
a
n
t
i
t
yofi
n
t
e
r
e
s
t
)ma
y
b
eav
e
c
t
or
,
ore
v
e
nat
e
n
s
or
.Th
i
sme
a
n
st
h
a
twec
a
nt
a
k
eav
e
c
t
or
l
i
k
ede
r
i
v
a
t
i
v
eofa
s
c
a
l
a
rf
u
n
c
t
i
onofv
e
c
t
orc
oor
d
i
n
a
t
e
sa
n
dp
r
od
u
c
eav
e
c
t
or
!Al
t
e
r
n
a
t
i
v
e
l
y
,wec
a
nt
a
k
e
d
e
r
i
v
a
t
i
v
e
st
h
a
tb
o
t
ha
c
tont
h
eu
n
d
e
r
l
y
i
n
gv
e
c
t
orc
oor
d
i
n
a
t
e
sa
n
ds
e
l
e
c
tou
ta
n
d
t
r
a
n
s
f
or
ms
p
e
c
i
f
i
cc
omp
on
e
n
t
soft
h
ev
e
c
t
orqu
a
n
t
i
t
yi
t
s
e
l
fi
ns
p
e
c
i
f
i
cwa
y
s
.Aswa
s
t
h
ec
a
s
ef
ormu
l
t
i
p
l
i
c
a
t
i
onofs
c
a
l
a
r
sa
n
dv
e
c
t
or
s
,wewon
’
th
a
v
ej
u
s
ton
ek
i
n
d–we
ma
ye
n
du
pwi
t
ht
h
r
e
e
,orf
ou
r
,
ormor
e
!I
n
d
e
e
d
,
s
omeofou
rd
e
r
i
v
a
t
i
v
e
swi
l
le
c
h
ot
h
e
mu
l
t
i
p
l
i
c
a
t
i
onr
u
l
e
sa
l
g
e
b
r
a
i
c
a
l
l
ys
p
e
c
i
f
i
e
da
b
ov
e
.
6.
2.
1 Th
ePa
r
t
i
a
l
De
r
i
v
a
t
i
v
e
Th
ep
a
r
t
i
a
ld
e
r
i
v
a
t
i
v
ei
swh
a
twet
y
p
i
c
a
l
l
yu
s
ewh
e
nweh
a
v
eaf
u
n
c
t
i
onofmu
l
t
i
p
l
e
c
oo
r
d
i
n
a
t
e
s
.Su
p
p
os
eweh
a
v
ef(
x
,
y
,
z
)
,
b
u
twi
s
ht
os
e
eh
owt
h
i
sf
u
n
c
t
i
onv
a
r
i
e
swh
e
n
wev
a
r
yon
l
yx
,
h
ol
d
i
n
gt
h
eot
h
e
rv
a
r
i
a
b
l
e
sc
on
s
t
a
n
t
.
Th
i
sd
e
f
i
n
e
st
h
ep
a
r
t
i
a
l
de
r
i
v
a
t
i
v
e
:
∂f =l
x+
i
m f(
t
→0
∂x
x
,
y
,
z
)−f(
x
,
y
,
z
)
x
Not
et
h
a
tt
h
i
si
sj
u
s
tt
a
k
i
n
gt
h
eo
r
d
i
n
a
r
ys
c
a
l
a
rd
e
r
i
v
a
t
i
v
e
,
wh
i
l
et
r
e
a
t
i
n
gt
h
eot
h
e
r
v
a
r
i
a
b
l
e
sa
sc
on
s
t
a
n
t
s
.
I
n
de
e
d
,
ou
rs
c
a
l
a
rd
e
r
i
v
a
t
i
v
ea
b
ov
ei
sa
l
s
oap
a
r
t
i
a
l
d
e
r
i
v
a
t
i
v
ei
n
t
h
ec
a
s
ewh
e
r
et
h
e
r
ea
r
en
oot
h
e
rv
a
r
i
a
b
l
e
s
!
F
or
mi
n
gt
h
et
ot
a
ldi
ffe
r
e
n
t
i
a
l
,h
o
we
v
e
r
,n
owr
e
qu
i
r
e
su
st
oc
on
s
i
de
rwh
a
th
a
p
p
e
n
s
wh
e
nwev
a
r
ya
l
l
t
h
r
e
ec
oor
d
i
n
a
t
e
s
:
d
f=
∂f
∂f
x+ ∂y d
y+
∂x d
∂f
∂z dz
Th
e
s
ea
r
en
otn
e
c
e
s
s
a
r
i
l
ys
p
a
t
i
a
l
v
a
r
i
a
t
i
on
s–wec
o
u
l
dt
h
r
owt
i
mei
nt
h
e
r
ea
swe
l
l
,
b
u
tf
ort
h
emome
n
twewi
l
lc
on
s
i
d
e
rt
i
mea
ni
n
de
p
e
n
d
e
n
tv
a
r
i
a
b
l
et
h
a
twen
e
e
d
c
on
s
i
d
e
ron
l
yv
i
at
h
ec
h
a
i
nr
u
l
e
.
Wec
a
nwr
i
t
et
h
i
sa
sad
otp
r
od
u
c
t
:
∂f
d
f=
∂f
∂f
∂x x
ˆ+ ∂y y
ˆ+ ∂z z
ˆ·{
dx
x
ˆ+dy
y
ˆ+dz
z
ˆ
}
wh
i
c
hwewr
i
t
ea
s
:
d
f=(
∇f)·dℓ
wh
e
r
eweh
a
v
ei
mp
l
i
c
i
t
l
yd
e
f
i
n
e
d∇fa
n
ddℓ
.
6.
3 Th
eGr
a
d
i
e
n
t
Th
eg
r
a
d
i
e
n
tofaf
u
n
c
t
i
on
:
∂f
∇f=
∂f
∂f
ˆ+ ∂y y
ˆ+ ∂z zˆ
∂x x
i
sav
e
c
t
orwh
os
ema
g
n
i
t
u
d
ei
st
h
ema
x
i
mu
ms
l
op
e(
r
a
t
eofc
h
a
n
g
ewi
t
hr
e
s
p
e
c
tt
ot
h
e
u
n
d
e
r
l
y
i
n
gc
oor
d
i
n
a
t
e
s
)oft
h
ef
u
n
c
t
i
oni
na
n
yd
i
r
e
c
t
i
o
n
,
wh
i
c
hp
o
i
n
t
si
nt
h
ed
i
r
e
c
t
i
on
i
nwh
i
c
ht
h
ema
x
i
mu
ms
l
o
peoc
c
u
r
s
.
Weu
s
u
a
l
l
ye
x
p
r
e
s
s∇a
sad
i
ffe
r
e
n
t
i
a
l
op
e
r
a
t
or:
∇=
∂
∂
∂
∂x x
ˆ+ ∂y y
ˆ+ ∂z zˆ
t
h
a
ta
c
t
sona
nob
j
e
c
tont
h
er
i
g
h
t
,
a
n
dwh
i
c
hf
ol
l
owst
h
eu
s
u
a
lp
a
r
e
n
t
h
e
s
e
sr
u
l
e
s
t
h
a
tc
a
nl
i
mi
tt
h
es
c
op
eoft
h
i
sr
i
g
h
ta
c
t
i
on
:
(
∇f)
g=g
(
∇f)=g
∇f
Nowweg
e
tt
ot
h
ei
n
t
e
r
e
s
t
i
n
gs
t
u
ff.
6.
4 Ve
c
t
o
rDe
r
i
v
a
t
i
v
e
s
Re
c
a
l
lt
h
a
tweh
a
v
et
h
r
e
er
u
l
e
sf
orv
e
c
t
ormu
l
t
i
p
l
i
c
a
t
i
on(
n
oti
n
c
l
u
d
i
n
gt
h
eou
t
e
r
p
r
od
u
c
t
)
:
Ab
,
A·B,
A×B
wh
e
r
ebi
sas
c
a
l
a
r
,
a
n
dAa
n
dBa
r
ev
e
c
t
or
sa
su
s
u
a
l
.Wee
v
i
d
e
n
t
l
ymu
s
th
a
v
et
h
r
e
es
i
mi
l
a
r
r
u
l
e
sf
ort
h
eg
r
a
d
i
e
n
top
e
r
a
t
ort
r
e
a
t
e
da
si
fi
ti
sav
e
c
t
or(
op
e
r
a
t
or
)
:
∇f
,
∇·A,
∇×A
wh
e
r
efi
samu
l
t
i
v
a
r
i
a
t
es
c
a
l
a
rf
u
n
c
t
i
on
,a
n
dAi
samu
l
t
i
v
a
r
i
a
t
ev
e
c
t
orf
u
n
c
t
i
on
.We
c
a
l
lt
h
e
s
e
,r
e
s
p
e
c
t
i
v
e
l
y
,t
h
eg
r
a
d
i
e
n
tofas
c
a
l
a
rf
u
n
c
t
i
on
,t
h
ed
i
v
e
r
g
e
n
c
eofav
e
c
t
or
f
u
n
c
t
i
on
,
a
n
dt
h
ec
u
r
l
ofav
e
c
t
orf
u
n
c
t
i
on
.
Th
eg
r
a
d
i
e
n
ti
st
h
edi
r
e
c
t
e
ds
l
op
eoffa
tap
oi
n
t
.Th
edi
v
e
r
g
e
n
c
ei
same
a
s
u
r
eof
t
h
ei
n
/
ou
t
f
l
owofav
e
c
t
orf
i
e
l
dAr
e
l
a
t
i
v
et
oap
oi
n
t
.Th
ec
u
r
li
same
a
s
u
r
eoft
h
e
r
ot
a
t
i
onofav
e
c
t
orf
i
e
l
dAa
b
ou
tap
oi
n
t
.Al
lt
h
r
e
ea
r
ed
e
f
i
n
e
da
t(
i
nt
h
en
e
i
g
h
b
or
h
ood
of
)ap
oi
n
ti
ns
p
a
c
eb
yme
a
n
soft
h
el
i
mi
t
i
n
gp
r
oc
e
s
si
n
d
i
c
a
t
e
da
b
ov
ea
n
dp
r
e
s
u
met
h
a
t
t
h
eob
j
e
c
t
st
h
e
ya
c
tona
r
ewe
l
l
b
e
h
a
v
e
de
n
ou
g
ht
op
e
r
mi
tl
i
mi
t
st
ob
et
a
k
e
n
.
I
nCa
r
t
e
s
i
a
nc
omp
on
e
n
t
s
,
t
h
eg
r
a
d
i
e
n
tofav
e
c
t
orVi
s
:
∇·
V=∂Vx+∂Vy+∂Vz∂x
∂y ∂
z
a
n
dt
h
ec
u
r
l
i
s
:
∂Vz
∇×
V=
∂Vy
∂Vx
∂Vz
∂Vy
∂Vx
∂y − ∂z x
ˆ+ ∂z − ∂x y
ˆ+ ∂x − ∂y zˆ
Wh
a
ta
r
et
h
ea
n
a
l
og
u
e
soft
h
es
c
a
l
a
rr
u
l
e
swel
i
s
t
e
da
b
ov
e
?Wen
owh
a
v
et
h
r
e
e
v
e
r
s
i
on
sofe
a
c
hoft
h
e
m.Th
ec
h
a
i
nr
u
l
ei
sf
or
me
db
yc
omp
os
i
t
i
onoft
h
er
u
l
ef
ort
h
e
t
ot
a
l
d
i
ffe
r
e
n
t
i
a
l
wi
t
hr
u
l
e
sf
ort
h
ec
omp
on
e
n
td
i
ffe
r
e
n
t
i
a
l
sa
n
dwewon
’
th
a
v
emu
c
hu
s
e
f
ori
t
.
Th
es
u
mr
u
l
e
,
h
owe
v
e
r
,
i
si
mp
or
t
a
n
t(
a
l
l
t
h
r
e
ewa
y
s
)i
fob
v
i
ou
s
.
6.
4.
1 Th
eSu
mRu
l
e
s
Su
p
p
os
efa
n
dga
r
es
c
a
l
a
rf
u
n
c
t
i
on
sa
n
dAa
n
dBa
r
ev
e
c
t
orf
u
n
c
t
i
on
s
.
Th
e
n
:
∇(
f+g
)=∇f+∇g
∇·
(
A+
B)
=
∇·
A+
∇·
B
∇×
(
A+B)
=
∇×
A+
∇×
B
6.
4.
2 Th
ePr
o
d
u
c
tRu
l
e
s
Th
ep
r
odu
c
tr
u
l
e
sa
r
emu
c
hmor
ed
i
ffic
u
l
t
.Weh
a
v
et
wowa
y
sofma
k
i
n
gas
c
a
l
a
r
p
r
od
u
c
t–fga
n
dA·B.Wec
a
nma
k
et
wov
e
c
t
orp
r
od
u
c
t
sa
swe
l
l
–fAa
n
dA×B(
n
ot
e
t
h
a
twewi
l
ln
otwor
r
ya
b
ou
tt
h
e“
p
s
e
u
do”c
h
a
r
a
c
t
e
roft
h
ec
r
os
sp
r
od
u
c
tu
n
l
e
s
si
t
ma
t
t
e
r
st
ot
h
ep
oi
n
twea
r
et
r
y
i
n
gt
oma
k
e
)
.Th
e
r
ea
r
ea
si
tt
u
r
n
sou
ts
i
xd
i
ffe
r
e
n
t
p
r
od
u
c
tr
u
l
e
s
!
∇(
fg
)=f∇g+g
∇f
∇(
A·
B)
=
A×
(
∇×
B)
+
B×
(
∇×
A)
+
(
A·
∇)
B+(
B·
∇)
A
Th
ef
i
r
s
ti
sob
v
i
ou
sa
n
ds
i
mp
l
e
,t
h
es
e
c
on
di
sdi
ffic
u
l
tt
op
r
ov
eb
u
ti
mp
or
t
a
n
tt
o
p
r
ov
ea
sweu
s
et
h
i
si
de
n
t
i
t
yaf
a
i
rb
i
t
.
Not
ewe
l
l
t
h
a
t
:
(
A·∇)=Ax
∂
∂
∂
∂x+Ay ∂y+Az ∂z
Weh
a
v
et
wodi
v
e
r
g
e
n
c
er
u
l
e
s
:
∇·(
fA)=f(
∇·A)+(
A·∇)
f
∇·
(
A×
B)
=
B·
(
∇×
A)
−
A·
(
∇×
B)
Th
ef
i
r
s
ti
sa
g
a
i
nf
a
i
r
l
yob
v
i
ou
s
.Th
es
e
c
on
don
ec
a
ne
a
s
i
l
yb
ep
r
ov
e
nb
yd
i
s
t
r
i
b
u
t
i
n
gt
h
e
d
i
v
e
r
g
e
n
c
ea
g
a
i
n
s
tt
h
ec
r
os
sp
r
od
u
c
ta
n
dl
ook
i
n
gf
ort
e
r
mst
h
a
ts
h
a
r
ea
nu
n
di
ffe
r
e
n
t
i
a
t
e
d
c
omp
on
e
n
t
,t
h
e
nc
ol
l
e
c
t
i
n
gt
h
os
et
e
r
mst
of
or
mt
h
et
woc
r
os
sp
r
od
u
c
t
s
.I
tc
a
na
l
mos
tb
e
i
n
t
e
r
p
r
e
t
e
da
sa
nor
d
i
n
a
r
yp
r
od
u
c
tr
u
l
ei
fy
oun
ot
et
h
a
twh
e
ny
oup
u
l
l
∇“
t
h
r
ou
g
h
”Ay
oua
r
e
e
ffe
c
t
i
v
e
l
yc
h
a
n
g
i
n
gt
h
eor
de
roft
h
ec
r
os
sp
r
odu
c
ta
n
dh
e
n
c
en
e
e
dami
n
u
ss
i
g
n
.Th
e
p
r
od
u
c
th
a
st
ob
ea
n
t
i
s
y
mme
t
r
i
ci
nt
h
ei
n
t
e
r
c
h
a
n
g
eofAa
n
dB,s
ot
h
e
r
eh
a
st
ob
eas
i
g
n
d
i
ffe
r
e
n
c
eb
e
t
we
e
nt
h
eot
h
e
r
wi
s
es
y
mme
t
r
i
ct
e
r
msf
r
omd
i
s
t
r
i
b
u
t
i
n
gt
h
ede
r
i
v
a
t
i
v
e
s
.
F
i
n
a
l
l
y
,
weh
a
v
et
woc
u
r
l
r
u
l
e
s
:
∇×(
fA)=f(
∇×A)−(
A×∇)
f
∇×
(
A×
B)
=
(
B·
∇)
A−
(
A·
∇)
B+
A(
∇·
B)
−
B(
∇·
A)
Th
ef
i
r
s
ti
sa
g
a
i
nr
e
me
mb
e
r
a
b
l
ea
st
h
eu
s
u
a
lp
r
od
u
c
tr
u
l
eb
u
twi
t
hami
n
u
ss
i
g
n
wh
e
nwep
u
l
l
At
ot
h
eo
t
h
e
rs
i
d
eof∇.Th
es
e
c
on
don
ei
sn
a
s
t
yt
op
r
ov
eb
e
c
a
u
s
et
h
e
r
e
a
r
es
ov
e
r
yma
n
yt
e
r
msi
nt
h
ef
u
l
l
ye
x
p
a
n
d
e
dc
u
r
loft
h
ec
r
os
s
p
r
od
u
c
tt
h
a
tmu
s
tb
e
c
ol
l
e
c
t
e
da
n
dr
e
a
r
r
a
n
g
e
d
,
b
u
ti
sv
e
r
yu
s
e
f
u
l
.Not
et
h
a
ti
ne
l
e
c
t
r
od
y
n
a
mi
c
swewi
l
l
of
t
e
n
b
ema
n
i
p
u
l
a
t
i
n
gors
ol
v
i
n
gv
e
c
t
orp
a
r
t
i
a
l
d
i
ffe
r
e
n
t
i
a
l
e
qu
a
t
i
on
si
nc
on
t
e
x
t
swh
e
r
ee
.
g
.∇
·E=0or∇·E=0
,
s
os
e
v
e
r
a
l
oft
h
e
s
et
e
r
msmi
g
h
tb
ez
e
r
o.
6.
5 Se
c
o
n
dDe
r
i
v
a
t
i
v
e
s
Th
e
r
ea
r
ef
i
v
es
e
c
on
dd
e
r
i
v
a
t
i
v
e
s
.Twoa
r
ei
mp
or
t
a
n
t
,
a
n
dat
h
i
r
dc
ou
l
dc
on
c
e
i
v
a
b
l
yb
e
i
mp
or
t
a
n
tb
u
twi
l
l
of
t
e
nv
a
n
i
s
hf
ort
h
es
a
mer
e
a
s
on
.Th
ef
i
r
s
tr
u
l
ed
e
f
i
n
e
sa
n
dop
e
r
a
t
or
t
h
a
ti
sa
r
g
u
a
b
l
yt
h
emos
ti
mp
or
t
a
n
ts
e
c
on
dd
e
r
i
v
a
t
i
v
ei
np
h
y
s
i
c
s
:
2
∇·∇f=∇f
2
Th
e∇ op
e
r
a
t
ori
sc
a
l
l
e
dt
h
eL
a
p
l
a
c
i
a
na
n
di
te
n
or
mou
s
l
yi
mp
or
t
a
n
ti
nb
ot
h
d
e
l
e
c
t
r
od
y
n
a
mi
c
sa
n
dq
u
a
n
t
u
m me
c
h
a
n
i
c
s
.I
ti
st
h
e3
de
qu
i
v
a
l
e
n
tofdx22 ,g
i
v
e
n
e
x
p
l
i
c
i
t
l
yb
y
:
∇2=
∂
2∂
2∂
2
++
2
∂x
2
∂
y
2
∂
z
Ne
x
tweh
a
v
e
:
∇×(
∇f)=0
(
n
otp
r
e
c
i
s
e
l
yt
r
i
v
i
a
l
t
op
r
ov
eb
u
ti
mp
or
t
a
n
t
)
.
Al
s
o:
∇(
∇·A)
wh
i
c
hh
a
sn
os
i
mp
l
e
rf
or
mb
u
twh
i
c
hi
sof
t
e
nz
e
r
of
orA=E
,
Bi
ne
l
e
c
t
r
od
y
n
a
mi
c
s
.
Ne
x
t
:
∇·
(
∇×
A)
=
0
(
n
otp
r
e
c
i
s
e
l
yt
r
i
v
i
a
l
t
op
r
ov
eb
u
ti
mp
or
t
a
n
t
)
.
F
i
n
a
l
l
y
:
2
∇×
(
∇×
A)
=
∇(
∇·
A)
−
∇A
wh
i
c
hi
sv
e
r
yi
mp
or
t
a
n
t–ak
e
ys
t
e
pi
nt
h
ed
e
r
i
v
a
t
i
onoft
h
e3
dwa
v
ee
qu
a
t
i
onf
r
om
Ma
x
we
l
l
’
se
qu
a
t
i
on
si
ndi
ffe
r
e
n
t
i
a
l
f
or
m!
6.
6 Sc
a
l
a
rI
n
t
e
gr
a
t
i
o
n
I
n
t
e
g
r
a
t
i
oni
sb
a
s
e
dond
i
ffe
r
e
n
t
i
a
t
i
on
,b
u
tr
u
n
st
h
ep
r
oc
e
s
sb
a
c
k
wa
r
d
s
.Th
i
si
st
h
e
b
a
s
i
sf
ort
h
ef
u
n
d
a
me
n
t
a
l
t
h
e
or
e
mofc
a
l
c
u
l
u
s
.
6.
6.
1 Th
eF
u
n
d
a
me
n
t
a
l
Th
e
o
r
e
mofCa
l
c
u
l
u
s
Re
c
a
l
l
t
h
a
tt
h
ef
u
n
d
a
me
n
t
a
l
t
h
e
or
e
mofc
a
l
c
u
l
u
sb
a
s
i
c
a
l
l
yde
f
i
n
e
st
h
ei
n
t
e
g
r
a
l
:
b
d
f
b
a
d
f=
dxd
x=f(
b
)−f(
a
)
a
d
f
Top
u
ti
ta
n
ot
h
e
rwa
y
,
i
fF=dx :
b
a
Fd
x=f(
b
)−f(
a
)
Th
i
sj
u
s
t
i
f
i
e
sr
e
f
e
r
r
i
n
gt
oi
n
t
e
g
r
a
t
i
ona
s“
a
n
t
i
d
i
ffe
r
e
n
t
i
a
t
i
on
”– d
i
ffe
r
e
n
t
i
a
t
i
onr
u
n
b
a
c
k
wa
r
ds
.I
n
t
e
g
r
a
t
i
onc
on
s
i
s
t
soff
i
n
d
i
n
gaf
u
n
c
t
i
onwh
os
ed
e
r
i
v
a
t
i
v
ei
st
h
ef
u
n
c
t
i
on
b
e
i
n
gi
n
t
e
g
r
a
t
e
d
.
Asb
e
f
or
e
,
wh
a
twec
a
nd
owi
t
hs
c
a
l
a
r
s
,
wec
a
nd
owi
t
hv
e
c
t
or
s–wi
t
hb
e
l
l
son
,
t
wo
ort
h
r
e
edi
ffe
r
e
n
twa
y
s
.
6.
7 Ve
c
t
o
rI
n
t
e
gr
a
t
i
on
Wen
e
e
dt
og
e
n
e
r
a
l
i
z
et
h
es
c
a
l
a
rt
h
e
or
e
mt
oaf
u
n
d
a
me
n
t
a
lt
h
e
or
e
mf
orv
e
c
t
or
de
r
i
v
a
t
i
v
e
s
.Howe
v
e
r
,wema
ye
n
du
ph
a
v
i
n
gmor
et
h
a
non
e
!Th
a
ti
sb
e
c
a
u
s
ewec
a
n
i
n
t
e
g
r
a
t
eov
e
r1
,2ora
l
lt
h
r
e
ed
i
me
n
s
i
on
a
ld
oma
i
n
sf
ors
c
a
l
a
ra
n
dv
e
c
t
orf
u
n
c
t
i
on
s
d
e
f
i
n
e
di
n3
dEu
c
l
i
d
e
a
ns
p
a
c
e
.He
r
ei
san
on
e
x
h
a
u
s
t
i
v
el
i
s
tofi
mp
or
t
a
n
ti
n
t
e
g
r
a
l
t
y
p
e
s
(
s
omeofwh
i
c
hy
ouh
a
v
ee
n
c
ou
n
t
e
r
e
di
ni
n
t
r
od
u
c
t
or
yp
h
y
s
i
c
sc
ou
r
s
e
s
)
:
Al
i
n
ei
n
t
e
g
r
a
l
a
l
on
gs
omes
p
e
c
i
f
i
e
dc
u
r
v
i
l
i
n
e
a
rp
a
t
hora
r
ou
n
ds
omes
p
e
c
i
f
i
e
dl
oop
C:
C
V·d
ℓ or
C
V·dℓ
Yous
h
ou
l
dr
e
c
og
n
i
z
et
h
i
st
y
p
eofi
n
t
e
g
r
a
lf
r
om wh
a
ty
ouh
a
v
el
e
a
r
n
e
da
b
ou
t
p
ot
e
n
t
i
a
l
orp
ot
e
n
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i
a
l
e
n
e
r
g
yorc
e
r
t
a
i
nf
i
e
l
di
n
t
e
g
r
a
l
si
nMa
x
we
l
l
’
sE
qu
a
t
i
on
sl
e
a
r
n
e
di
n
i
n
t
r
od
u
c
t
or
ye
l
e
c
t
r
i
c
i
t
ya
n
dma
g
n
e
t
i
s
m.
Ne
x
tweh
a
v
es
u
r
f
a
c
ei
n
t
e
g
r
a
l
s(
oft
h
ep
a
r
t
i
c
u
l
a
rk
i
n
da
s
s
oc
i
a
t
e
dwi
t
ht
h
ef
l
u
xofa
v
e
c
t
orf
i
e
l
d
)
:
S
V·n
ˆ
d
A=
V·a
d
S
or
V·a
d
S
f
ort
woc
ommonn
ot
a
t
i
on
s
,
t
h
es
e
c
on
don
ef
a
v
or
e
db
ye
.
g
.
Gr
i
ffit
h
sa
l
t
h
ou
g
hI
p
e
r
s
on
a
l
l
yp
r
e
f
e
rt
h
ef
i
r
s
ton
ea
n
di
ti
smor
ec
ommoni
np
h
y
s
i
c
st
e
x
t
b
ook
s
.
I
n
t
h
ef
i
r
s
tc
a
s
e
,
Si
sa
nop
e
ns
u
r
f
a
c
e
,
wh
i
c
hme
a
n
si
ti
sa
)(
p
i
e
c
e
wi
s
e
)b
ou
n
d
e
db
yac
l
os
e
d
c
u
r
v
eCa
n
dt
h
ed
i
r
e
c
t
i
onoft
h
en
or
ma
lt
ot
h
es
u
r
f
a
c
ei
sa
r
b
i
t
r
a
r
y
.I
nt
h
es
e
c
on
d
,Si
sa
c
l
os
e
ds
u
r
f
a
c
e–as
u
r
f
a
c
et
h
a
ti
st
op
ol
og
i
c
a
l
l
ye
qu
i
v
a
l
e
n
tt
os
oa
pb
u
b
b
l
e–i
nwh
i
c
hc
a
s
e
i
te
n
c
l
os
e
sav
ol
u
me
.F
ore
x
a
mp
l
ei
fwel
e
tSb
eas
qu
a
r
eont
h
ex
y
p
l
a
n
e
,
wemi
g
h
tc
h
os
e
t
oma
k
en
ˆ
dA=a
d=z
ˆ
d
x
d
y
,s
oy
ouc
a
ns
e
et
h
a
ti
na
l
mos
ta
l
lc
a
s
e
sy
ouwi
l
lh
a
v
et
oa
t
l
e
a
s
tme
n
t
a
l
l
ye
x
p
r
e
s
sn
ˆe
x
p
l
i
c
i
t
l
yi
nor
d
e
rt
oe
v
a
l
u
a
t
ea
da
n
y
wa
y
.
[
As
i
d
e
:Ac
l
os
e
dl
i
n
eb
ou
n
dsa
nop
e
ns
u
r
f
a
c
e
.Ac
l
os
e
ds
u
r
f
a
c
eb
ou
n
d
sa
nop
e
n
v
ol
u
me
.I
fy
ouwa
n
tt
oma
k
ey
ou
rh
e
a
dh
u
r
t(
i
nc
on
s
t
r
u
c
t
i
v
ewa
y
s–wewi
l
ln
e
e
dt
o
t
h
i
n
ka
b
ou
tt
h
i
n
g
sl
i
k
et
h
i
si
nr
e
l
a
t
i
v
i
t
yt
h
e
or
y
)t
h
i
n
ka
b
ou
twh
a
tac
l
os
e
dv
ol
u
memi
g
h
t
b
ou
n
d.
.
.
]
F
i
n
a
l
l
y
,
weh
a
v
ei
n
t
e
g
r
a
t
i
onov
e
rav
ol
u
me
:
3
Fd
V=
Fdr=
V
V
Fdτ
V
wh
e
r
eVi
st
h
e(
op
e
n
)v
ol
u
met
h
a
tmi
g
h
th
a
v
eb
e
e
nb
ou
n
d
e
db
yac
l
os
e
dS,a
n
dI
’
v
e
i
n
d
i
c
a
t
e
dt
h
r
e
ed
i
ffe
r
e
n
twa
y
sp
e
op
l
ewr
i
t
et
h
ev
ol
u
mee
l
e
me
n
t
.Gr
i
ffit
h
sf
a
v
or
se
.
g
.dτ
=d
xd
yd
z
.
On
edo
e
s
n
’
th
a
v
et
oi
n
t
e
g
r
a
t
eon
l
ys
c
a
l
a
rf
u
n
c
t
i
on
s
,
a
n
dt
h
e
r
ea
r
eot
h
e
rl
i
n
ea
n
ds
u
r
f
a
c
e
i
n
t
e
g
r
a
l
son
ec
a
nd
e
f
i
n
eors
e
n
s
i
b
l
ye
v
a
l
u
a
t
e
.
F
ore
x
a
mp
l
ea
l
l
of
:
Vd
ℓ or
fd
a or
C
S
Fdτ
V
mi
g
h
tma
k
es
e
n
s
ei
ns
o
mec
on
t
e
x
t
.
6.
8 Th
eF
u
n
d
a
me
n
t
a
l
Th
e
o
r
e
m(
s
)o
fVe
c
t
o
rCa
l
c
u
l
u
s
6.
8.
1 ASc
a
l
a
rF
u
n
c
t
i
ono
fVe
c
t
o
rCo
or
d
i
n
a
t
e
s
L
e
t
’
sr
e
t
u
r
nt
oou
re
x
p
r
e
s
s
i
onf
orat
ot
a
l
di
ffe
r
e
n
t
i
a
l
ofas
c
a
l
a
rf
u
n
c
t
i
on
,
g
i
v
e
na
b
ov
e
:
d
f=∇f·d
ℓ
Th
e
n
b
a
d
f=
b
a
∇f·d
ℓ=f(
b
)−f(
b
)
i
n
d
e
p
e
n
d
e
n
tofp
a
t
h
!Th
ei
n
t
e
g
r
a
ld
e
p
e
n
d
son
l
yont
h
ee
n
dp
oi
n
t
sf
ora
n
yt
ot
a
l
di
ffe
r
e
n
t
i
a
l
t
h
a
ti
si
n
t
e
g
r
a
t
e
d
!
He
n
c
ewek
n
owt
h
a
t
:
C
∇f·d
ℓ=0
Th
i
ss
h
ou
l
ds
e
e
mv
e
r
yf
a
mi
l
i
a
rt
oy
ou
.Su
p
p
os
eF=−
∇Uf
orawe
l
l
b
e
h
a
v
e
ds
c
a
l
a
r
f
u
n
c
t
i
onU.
Th
e
n
:
b
W(
a→ b
)=
a
b
F·dℓ=−
a
∇U·d
ℓ
i
n
d
e
p
e
n
d
e
n
tofp
a
t
h
.I
ni
n
t
r
od
u
c
t
or
yme
c
h
a
n
i
c
sy
oup
r
ob
a
b
l
ywe
n
tf
r
omt
h
ep
r
op
os
i
t
i
on
t
h
a
tt
h
ewor
ki
n
t
e
g
r
a
lwa
si
n
de
p
e
n
d
e
n
tofp
a
t
hf
orac
on
s
e
r
v
a
t
i
v
ef
or
c
et
oad
e
f
i
n
i
t
i
on
oft
h
ep
ot
e
n
t
i
a
le
n
e
r
g
y
,
b
u
ta
sf
a
ra
sv
e
c
t
orc
a
l
c
u
l
u
si
sc
on
c
e
r
n
e
d
,
t
h
eot
h
e
rd
i
r
e
c
t
i
on
i
sat
r
i
v
i
a
li
d
e
n
t
i
t
y
.An
yv
e
c
t
orf
or
c
et
h
a
tc
a
nb
ewr
i
t
t
e
na
st
h
e(
n
e
g
a
t
i
v
e
)g
r
a
d
i
e
n
tofa
s
moot
h
,
di
ffe
r
e
n
t
i
a
b
l
ep
ot
e
n
t
i
a
l
e
n
e
r
g
yf
u
n
c
t
i
oni
sac
on
s
e
r
v
a
t
i
v
ef
or
c
e
!
6.
8.
2 Th
eDi
v
e
r
ge
n
c
eTh
e
or
e
m
Th
i
si
sas
e
c
on
d
,
v
e
r
y
,
v
e
r
yi
mp
or
t
a
n
ts
t
a
t
e
me
n
toft
h
eF
u
n
d
a
me
n
t
a
l
Th
e
or
e
m:
V/
S
(
∇·V)
dτ=
V·n
ˆ
dA
S
I
nt
h
i
se
x
p
r
e
s
s
i
onV/
Ss
h
ou
l
db
er
e
a
di
ny
ou
rmi
n
da
s“
ov
e
rt
h
eop
e
nv
ol
u
meV
b
ou
n
d
e
db
yt
h
ec
l
os
e
ds
u
r
f
a
c
eS”
,a
n
dVi
sa
na
r
b
i
t
r
a
r
yv
e
c
t
orqu
a
n
t
i
t
y
,t
y
p
i
c
a
l
l
ya
v
e
c
t
orf
i
e
l
dl
i
k
eEorBorav
e
c
t
orc
u
r
r
e
n
tde
n
s
i
t
ys
u
c
ha
sJ
.Not
ewe
l
lt
h
a
tt
h
er
i
g
h
t
h
a
n
ds
i
dey
ous
h
ou
l
db
er
e
a
d
i
n
ga
s“
t
h
ef
l
u
xoft
h
ev
e
c
t
orf
u
n
c
t
i
onVou
tt
h
r
ou
g
ht
h
e
c
l
os
e
ds
u
r
f
a
c
eS”
.
Yo
umi
g
h
ta
l
s
os
e
et
h
i
swr
i
t
t
e
na
s
:
(
∇·V)
dτ=
V
V·n
ˆ
dA
∂V
wh
e
r
e∂Vi
sr
e
a
da
s“
t
h
es
u
r
f
a
c
eb
ou
n
d
i
n
gt
h
ev
ol
u
meV”
.Th
i
si
ss
l
i
g
h
t
l
ymor
e
c
omp
a
c
tn
ot
a
t
i
on
,
b
u
tas
t
u
d
e
n
tc
a
ne
a
s
i
l
yb
ec
on
f
u
s
e
db
ywh
a
ta
p
p
e
a
r
st
ob
eap
a
r
t
i
a
l
d
i
ffe
r
e
n
t
i
a
l
i
nt
h
es
u
r
f
a
c
el
i
mi
t
s
.
As
i
mp
l
ec
on
s
e
qu
e
n
c
eoft
h
ed
i
v
e
r
g
e
n
c
et
h
e
or
e
mi
s
:
∇fdτ=
V/
S
fn
ˆ
d
A=
S
fa
d
S
Pr
oof
:
As
s
u
me
A =fc
ˆ
t
h
e
n
∇·A=(
c
ˆ·∇)
f+f(
∇·c
ˆ
)=(
c
ˆ·∇)
f
s
ot
h
a
t
V/
S
∇·Ad
τ=
V/
S
(
c
ˆ·∇)
fd
τ=
A·n
ˆ
d
A=
s
c
ˆ
f·n
ˆ
d
A
s
Si
n
c
ec
ˆi
sc
on
s
t
a
n
ta
n
da
r
b
i
t
r
a
r
y
,
wec
a
nf
a
c
t
ori
tou
tf
r
omt
h
ei
n
t
e
g
r
a
l
:
c
ˆ·
V/
S
∇fd
τ=c
ˆ·
fn
ˆ
d
A
s
Si
n
c
et
h
i
sh
a
st
ob
et
r
u
ef
ora
n
yn
on
z
e
r
oc
ˆ,wec
a
ne
s
s
e
n
t
i
a
l
l
yd
i
v
i
deou
tt
h
e
c
on
s
t
a
n
ta
n
dc
on
c
l
u
det
h
a
t
:
∇fdτ=
V/
S
fn
ˆ
dA
s
Q.
E
.
D.
Yous
h
ou
l
dp
r
ov
eony
ou
rown(
u
s
i
n
ge
x
a
c
t
l
yt
h
es
a
mes
or
tofr
e
a
s
on
i
n
g
)
t
h
a
t
:
V/
S
∇×Adτ=
n
ˆ×Ad
A
s
Th
e
r
et
h
u
si
son
es
u
c
ht
h
e
or
e
mf
or∇(
a
c
t
i
n
gona
n
ys
c
a
l
a
rf)
,
∇·A(
a
c
t
i
n
gona
n
y
v
e
c
t
orf
u
n
c
t
i
onA)or∇×Aa
c
t
i
n
gona
n
yv
e
c
t
orf
u
n
c
t
i
onA.Wec
a
nu
s
ea
l
loft
h
e
s
e
f
or
msi
ni
n
t
e
g
r
a
t
i
onb
yp
a
r
t
s
.
6.
8.
3 St
o
k
e
s
’
Th
e
o
r
e
m
St
ok
e
s
’
t
h
e
or
e
m(
wh
i
c
hmi
g
h
twe
l
lb
ec
a
l
l
e
dt
h
ec
u
r
lt
h
e
or
e
mi
fwewa
n
t
e
dt
ob
emor
e
c
on
s
i
s
t
e
n
ti
nou
rt
e
r
mi
n
ol
og
y
)i
se
qu
a
l
l
yc
r
i
t
i
c
a
l
t
oou
rf
u
t
u
r
ewor
k
:
S/
C
(
∇×V)·n
ˆ
dA=
C
V·d
ℓ
Ag
a
i
n
,r
e
a
dS/
Ca
s“
t
h
eop
e
ns
u
r
f
a
c
eSb
ou
n
d
e
db
yt
h
ec
l
os
e
dc
u
r
v
eC,a
n
dn
ot
e
t
h
a
tt
h
e
r
ei
sa
ni
mp
l
i
c
i
td
i
r
e
c
t
i
o
ni
nt
h
i
se
q
u
a
t
i
on
.I
np
a
r
t
i
c
u
l
a
r
,
y
oumu
s
tc
h
oos
e(
f
r
om
t
h
et
wop
os
s
i
b
l
ec
h
oi
c
e
s
)t
h
ed
i
r
e
c
t
i
onf
orn
ˆt
h
a
tc
or
r
e
s
p
on
d
st
ot
h
er
i
g
h
t
h
a
n
de
d
d
i
r
e
c
t
i
ona
r
ou
n
dt
h
el
oopC.I
nwor
d
s
,
i
fy
ouc
u
r
l
t
h
ef
i
n
g
e
r
sofy
ou
rr
i
g
h
th
a
n
da
r
ou
n
dC
i
nt
h
ed
i
r
e
c
t
i
oni
nwh
i
c
hy
ouwi
s
ht
od
ot
h
ei
n
t
e
g
r
a
l
,
y
ou
rt
h
u
mbs
h
ou
l
dp
oi
n
t“
t
h
r
ou
g
h
”
t
h
el
oopCi
nt
h
ed
i
r
e
c
t
i
ony
oumu
s
ts
e
l
e
c
tf
ort
h
en
or
ma
l
.
Wec
a
non
c
ea
g
a
i
nd
e
r
i
v
ea
na
d
di
t
i
on
a
l
f
or
moft
h
ec
u
r
l
t
h
e
or
e
m/
St
ok
e
s
’
t
h
e
or
e
m:
S/
C
(
n
ˆ×∇f)·d
A=
Not
ewe
l
l
t
h
a
tt
h
en
ˆh
a
sb
e
e
nmov
e
dt
ot
h
ef
r
on
t
!
C
fdℓ
6.
9 I
n
t
e
gr
a
t
i
o
nb
yPa
r
t
s
I
n
t
e
g
r
a
t
i
onb
yp
a
r
t
si
son
eoft
h
emos
ti
mp
or
t
a
n
tt
op
i
c
si
nt
h
i
sc
h
a
p
t
e
r
.I
n
d
e
e
d
,y
oumi
g
h
t
h
a
v
eb
e
e
nab
i
tb
or
e
db
yt
h
er
e
c
i
t
a
t
i
onoft
h
i
n
g
st
h
a
tp
r
ob
a
b
l
ywe
r
ec
ov
e
r
e
di
ny
ou
r
mu
l
t
i
v
a
r
i
a
t
ec
a
l
c
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l
u
sc
l
a
s
s
e
s
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i
smi
g
h
th
a
v
eb
e
e
na
swe
l
l
,
b
u
tc
h
a
n
c
e
sa
r
ev
e
r
yg
oodt
h
a
t
y
oud
i
d
n
’
tf
i
n
i
s
hl
e
a
r
n
i
n
gh
ow t
oma
k
ei
twor
ki
nt
h
eg
e
n
e
r
a
lc
on
t
e
x
toft
h
ev
a
r
i
ou
s
f
u
n
da
me
n
t
a
l
t
h
e
or
e
msl
i
s
t
e
da
b
ov
e
.
6.
9.
1 Sc
a
l
a
rI
n
t
e
g
r
a
t
i
onb
yPa
r
t
s
Weh
a
v
ea
l
r
e
a
d
ydon
ea
l
mos
ta
l
loft
h
ewor
kh
e
r
e
.St
a
r
twi
t
ht
h
ep
r
od
u
c
tr
u
l
ef
ort
h
e
d
i
ffe
r
e
n
t
i
a
l
:
d
(
fg
)=fd
g+gd
f
I
n
t
e
g
r
a
t
eb
ot
hs
i
d
e
s
.
b
b
a
b
b
d
(
fg
)=fg
=
a
fdg+
a
gd
f
a
a
n
dr
e
a
r
r
a
n
g
e
:
b
b
fdg=fg
b
a
gd
f
−
a
a
Th
i
si
son
ewa
yofwr
i
t
i
n
gi
n
t
e
g
r
a
t
i
onb
yp
a
r
t
s
,
b
u
twea
r
e
n
’
tu
s
u
a
l
l
yg
i
v
e
nb
ot
h“
d
f”
a
n
d
/
or“
d
g
”
.Not
ewe
l
l
t
h
a
twec
a
ne
x
p
r
e
s
sd
fa
n
ddgi
nt
e
r
msoft
h
ec
h
a
i
nr
u
l
e
,
t
h
ou
g
h
,
wh
i
c
hi
se
x
a
c
t
l
ywh
a
twewi
l
lu
s
u
a
l
l
yb
ed
oi
n
gt
oe
x
p
r
e
s
st
h
ei
n
t
e
g
r
a
lofk
n
own
f
u
n
c
t
i
on
sf(
x
)a
n
dg
(
x
)
:
b
a
d
g
f dxd
x=fg
b
a
b
−
a
d
f
gdxdx
I
n
t
e
g
r
a
t
i
onb
yp
a
r
t
si
sa
ne
n
or
mou
s
l
yv
a
l
u
a
b
l
et
ooli
ns
c
a
l
a
r
/
on
edi
me
n
s
i
on
a
li
n
t
e
g
r
a
l
c
a
l
c
u
l
u
s
.
I
ti
sj
u
s
ta
si
mp
o
r
t
a
n
ti
nmu
l
t
i
v
a
r
i
a
t
ei
n
t
e
gr
a
l
c
a
l
c
u
l
u
s
!
6.
9.
2 Ve
c
t
orI
n
t
e
gr
a
t
i
o
nb
yPa
r
t
s
Th
e
r
ea
r
ema
n
ywa
y
st
oi
n
t
e
g
r
a
t
eb
yp
a
r
t
si
nv
e
c
t
orc
a
l
c
u
l
u
s
.Soma
n
yt
h
a
tI
c
a
n
’
ts
h
ow
y
oua
l
l
oft
h
e
m.Th
e
r
ea
r
e
,
a
f
t
e
ra
l
l
,
l
o
t
sofwa
y
st
op
u
tav
e
c
t
ord
i
ffe
r
e
n
t
i
a
l
f
or
mi
n
t
oa
n
e
qu
a
t
i
on
,a
n
d(
a
tl
e
a
s
t
)t
h
r
e
ed
i
me
n
s
i
on
a
l
i
t
i
e
sofi
n
t
e
g
r
a
ly
oumi
g
h
tb
et
r
y
i
n
gt
od
o!I
wi
l
lt
h
e
r
e
f
or
ed
e
mon
s
t
r
a
t
eh
owt
ot
h
i
n
ka
b
ou
ti
n
t
e
g
r
a
t
i
n
gb
yp
a
r
t
si
nv
e
c
t
orc
a
l
c
u
l
u
s
,
e
x
p
l
oi
t
i
n
gt
h
eg
r
a
di
e
n
tp
r
odu
c
tr
u
l
e
,t
h
ed
i
v
e
r
g
e
n
c
et
h
e
or
e
m,orSt
ok
e
s
’t
h
e
or
e
m.I
n
a
l
mo
s
ta
l
l
oft
h
e
s
ec
a
s
e
s
,
t
h
e
yr
e
s
u
l
tf
r
omi
n
t
e
g
r
a
t
i
n
gat
ot
a
l
d
e
r
i
v
a
t
i
v
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omes
or
tor
a
n
ot
h
e
rov
e
rs
omep
a
r
t
i
c
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l
a
rd
oma
i
n(
a
sy
ouc
a
ns
e
ef
r
omt
h
e
i
ri
n
t
e
r
n
a
ld
e
r
i
v
a
t
i
on
sor
p
r
oof
s
,
b
e
y
on
dt
h
es
c
op
eoft
h
i
sc
ou
r
s
e
)
.
I
ti
se
a
s
i
e
s
tt
ot
e
a
c
ht
h
i
sb
ye
x
a
mp
l
e
.
L
e
t
’
swr
i
t
eon
eofou
rp
r
od
u
c
tr
u
l
e
s
:
∇·(
fA)=f(
∇·A)+(
A·∇)
f
Not
et
h
a
tt
h
el
e
f
th
a
n
ds
i
d
ei
sap
u
r
ed
i
v
e
r
g
e
n
c
e
.L
e
t
’
si
n
t
e
g
r
a
t
ei
tov
e
rav
ol
u
me
b
ou
n
d
e
db
yac
l
os
e
ds
u
r
f
a
c
e
:
V/
S
∇·(
fA)
d
τ=
f(
∇·A)
dτ+
V/
S
V/
S
(
A·∇)
fd
τ
Nowwewi
l
la
p
p
l
yt
h
edi
v
e
r
g
e
n
c
et
h
e
or
e
m(
on
eofou
r“
f
u
n
d
a
me
n
t
a
lt
h
e
or
e
ms
”
a
b
ov
e
)t
ot
h
el
e
f
th
a
n
ds
i
d
eon
l
y
:
fA·n
ˆ
d
A=
f(
∇·A)
d
τ+
(
A·∇)
fd
τ
V/
S
S
V/
S
F
i
n
a
l
l
y
,
l
e
t
’
sr
e
a
r
r
a
n
g
e
:
V/
S
(
A·∇)
fd
τ=
fA·n
ˆ
d
A−
S
V/
S
f(
∇·A)
d
τ
3
I
ne
l
e
c
t
r
od
y
n
a
mi
c
s
,
i
ti
so
f
t
e
nt
h
ec
a
s
et
h
eV=R,
a
l
lofr
e
a
ls
p
a
c
e
,
a
n
de
i
t
h
e
rfor
Av
a
n
i
s
ha
ti
n
f
i
n
i
t
y
,
wh
e
r
ewewou
l
dg
e
t
:
V/
S
(
A·∇)
fd
τ=−
V/
S
f(
∇·A)
d
τ
or∇·A=0(
ad
i
v
e
r
g
e
n
c
e
l
e
s
sf
i
e
l
d
)i
nwh
i
c
hc
a
s
e
:
V/
S
(
A·∇)
fd
τ=
fA·n
ˆ
dA
S
Bot
hoft
h
e
s
ee
x
p
r
e
s
s
i
on
sc
a
nb
ea
l
g
e
b
r
a
i
c
a
l
l
yu
s
e
f
u
l
.
Th
i
si
sn
otb
ya
n
yme
a
n
st
h
eon
l
yp
os
s
i
b
i
l
i
t
y
.Wec
a
ndoa
l
mos
te
x
a
c
t
l
yt
h
es
a
me
t
h
i
n
gwi
t
h∇×(
fA)a
n
dt
h
ec
u
r
lt
h
e
or
e
m.Wec
a
nd
oi
twi
t
ht
h
edi
v
e
r
g
e
n
c
eofac
r
os
s
p
r
od
u
c
t
,
∇·(
A×B)
.Youc
a
ns
e
ewh
yt
h
e
r
ei
sl
i
t
t
l
ep
oi
n
ti
nt
e
d
i
ou
s
l
ye
n
u
me
r
a
t
i
n
ge
v
e
r
y
s
i
n
g
l
ec
a
s
et
h
a
ton
ec
a
nb
u
i
l
df
r
oma
p
p
l
y
i
n
gap
r
od
u
c
tr
u
l
ef
orat
ot
a
ld
i
ffe
r
e
n
t
i
a
lor
c
on
n
e
c
t
e
dt
oon
eoft
h
eot
h
e
rwa
y
sofb
u
i
l
d
i
n
gaf
u
n
d
a
me
n
t
a
l
t
h
e
or
e
m.
Th
ema
i
np
oi
n
ti
st
h
i
s
:I
fy
oun
e
e
dt
oi
n
t
e
gr
a
t
ea
ne
x
p
r
e
s
s
i
oni
nv
e
c
t
orc
a
l
c
u
l
u
s
c
o
n
t
a
i
n
i
n
gt
h
e∇o
p
e
r
a
t
or
,t
r
yt
of
i
n
dap
r
od
u
c
tr
u
l
ec
o
n
n
e
c
t
e
dt
oav
e
r
s
i
o
no
ft
h
e
f
u
n
d
a
me
n
t
a
l
t
h
e
o
r
e
mt
h
a
tp
r
o
d
u
c
e
st
h
ee
x
p
r
e
s
s
i
ona
son
eofi
t
st
wot
e
r
ms
.
Th
e
ng
ot
h
r
ou
g
ht
h
ec
o
n
c
e
p
t
u
a
lp
r
oc
e
s
sofwr
i
t
i
n
gou
tt
h
edi
ffe
r
e
n
t
i
a
lp
r
odu
c
t
e
x
p
r
e
s
s
i
on
,
i
n
t
e
g
r
a
t
i
n
gb
ot
hs
i
d
e
s
,
a
p
p
l
y
i
n
ge
.
g
.t
h
ed
i
v
e
r
g
e
n
c
et
h
e
or
e
m,
t
h
ec
u
r
lt
h
e
or
e
m,
org
e
n
e
r
a
l
i
z
a
t
i
on
sors
p
e
c
i
a
l
c
a
s
e
soft
h
e
mi
n
d
i
c
a
t
e
da
b
ov
e
:
Th
e
r
ea
r
et
womod
e
r
a
t
e
l
yi
mp
or
t
a
n
t(
a
n
df
a
i
r
l
ye
a
s
yt
od
e
r
i
v
e
,a
tt
h
i
sp
oi
n
t
)
c
on
s
e
qu
e
n
c
e
sofa
l
loft
h
ewa
y
sofmi
x
i
n
gt
h
ef
u
n
d
a
me
n
t
a
lt
h
e
or
e
msa
n
dt
h
ep
r
od
u
c
t
r
u
l
e
si
n
t
os
t
a
t
e
me
n
t
sofi
n
t
e
g
r
a
t
i
onb
yp
a
r
t
s
.On
ei
st
h
es
l
i
g
h
t
l
yl
e
s
su
s
e
f
u
lGr
e
e
n
’
s
F
i
r
s
tI
d
e
n
t
i
t
y(
ort
h
e
or
e
m)
.
Su
p
p
os
efa
n
dga
r
e
,
a
su
s
u
a
l
,
s
c
a
l
a
rf
u
n
c
t
i
on
s
.
Th
e
n
:
2
f∇g−∇g·∇f
V
d
τ=
f ∇g·n
ˆ
) dA
∂V
∂g
=n
ˆ·∇gi
st
h
er
a
t
eofc
h
a
n
g
eoft
h
ef
u
n
c
t
i
ongi
nt
h
ed
i
r
e
c
t
i
onoft
h
e
ou
t
g
oi
n
gn
or
ma
l
(
a
n
dd
i
t
t
of
ort
h
es
i
mi
l
a
re
x
p
r
e
s
s
i
onf
orf
)
.
Hi
n
tf
orp
r
oof
:
Con
s
i
d
e
ri
n
t
e
g
r
a
t
i
n
g∇·f(
∇g
)
.
On
eu
s
eoft
h
i
si
st
op
r
ov
et
h
ev
e
r
yu
s
e
f
u
l
Gr
e
e
n
’
sSe
c
on
dI
d
e
n
t
i
t
y(
ort
h
e
or
e
m)
:
∂
g
∂
f
2
2
f∇g−g
∇fd
τ=
f∂n −g∂n dA
wh
e
r
e∂n
V
∂V
Youc
a
nj
u
s
twr
i
t
et
h
ef
i
r
s
ti
d
e
n
t
i
t
yt
wi
c
ewi
t
hfa
n
dgr
e
v
e
r
s
e
da
n
ds
u
b
t
r
a
c
tt
h
e
m
t
h
e
mt
og
e
tt
h
i
sr
e
s
u
l
t
.
Att
h
i
sp
oi
n
ti
ti
si
mp
or
t
a
n
tt
oc
on
n
e
c
tt
h
i
s“
t
ooa
b
s
t
r
a
c
t
”r
e
v
i
e
wofr
u
l
e
sa
n
d
t
h
e
or
e
msa
n
df
or
mst
or
e
a
l
p
h
y
s
i
c
s
.
Ane
x
a
mp
l
ei
si
nor
d
e
r
.
6.
10 I
n
t
e
gr
a
t
i
o
nByPa
r
t
si
nE
l
e
c
t
r
od
y
n
a
mi
c
s
Th
e
r
ei
son
ee
s
s
e
n
t
i
a
l
t
h
e
or
e
mofv
e
c
t
orc
a
l
c
u
l
u
st
h
a
ti
se
s
s
e
n
t
i
a
l
t
ot
h
ede
v
e
l
op
me
n
t
ofmu
l
t
i
p
ol
e
s–c
omp
u
t
i
n
gt
h
ed
i
p
ol
emome
n
t
.I
nh
i
sb
o
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or
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h
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s
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3
3
J
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R
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6
.
1
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R
Th
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n
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e∇·J=− ∂t=i
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a
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n
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x x
3
3
3
I
R
I
R
i
ω
x=−
i
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dx=−
ρ( )d
(
6
.
2
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d
(
u
v
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=ud
v+vdu
d
(
u
v
)
=
b
b
a
b
ud
v+
vd
u
a
b
a
ud
v
a
b
b
u
v
)
|
a − a vd
=(
u
(
6
.
3
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e
r
ei
ft
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ep
r
od
u
c
t
su
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b
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s
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e
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=∞ a
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h
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r
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t
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u
n
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e
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h
eot
h
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r
”
:
b
a
b
ud
v=−
a
vd
u
(
6
.
4
)
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c
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t
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ne
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t
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e
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s
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r
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l
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2
∇.
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.
dx→nˆ.
.
.
dx
V
(
6
.
5
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V
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nwor
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s
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g
r
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r
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h
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u
n
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h
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x
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r
e
s
s
i
o
ni
sp
r
e
s
e
r
v
e
d
.
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i
l
et
h
edi
v
e
r
ge
n
c
et
h
e
or
e
mi
s
:
3
2
∇·Adx=nˆ·Adx
V
(
6
.
6
)
S/
V
t
h
e
r
ei
sa“
g
r
a
d
i
e
n
tt
h
e
or
e
m”
:
3
V
∇fdx=
2
S/
V
n
ˆ
fdx
(
6
.
7
)
a
n
ds
oon
.
Top
r
ov
eJ
a
c
k
s
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se
x
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e
s
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onwemi
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h
tt
h
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r
e
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yt
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i
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s
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s
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e
r
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n
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r
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g
h
tt
e
n
s
orf
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L
e
tu
sl
ooka
tt
h
ef
ol
l
owi
n
gd
i
v
e
r
g
e
n
c
e
:
∇·(
x
J
)=∇x·J+x
∇·J
∇·J
x+x
=J
(
6
.
8
)
Th
i
sl
ook
sp
r
omi
s
i
n
g
;
i
ti
st
h
ex
c
omp
on
e
n
tofar
e
s
u
l
twemi
g
h
tu
s
e
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v
e
r
,
i
ft
r
yt
o
a
p
p
l
yt
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st
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t
r
i
xd
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d
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cf
or
mi
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a
tl
oo
k
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i
k
ei
tmi
g
h
tb
et
h
er
i
g
h
twa
y
:
∇·
x
(J
)=(
∇x
·)
J+x(
∇·J
)
J+
x(
∇·J
)
=3
(
6
.
9
)
weg
e
tt
h
ewr
o
n
ga
n
s
we
r
.
Toa
s
s
e
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l
et
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er
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gh
ta
n
s
we
r
,
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a
v
et
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u
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e
rt
h
et
h
r
e
es
e
p
a
r
a
t
es
t
a
t
e
me
n
t
s
:
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x
J
)x
ˆ= J
∇·Jx
ˆ
x+x
+∇·(
y
J
)y
ˆ=+J
∇·Jy
ˆ
y+y
+∇·(
z
J
)z
ˆ=+J
∇·Jz
ˆ
z+z
(
6
.
1
0
)
or
x
ˆ
(
x
J
)=J+x(
∇·J
)
i∇·
i
i
(
6
.
1
1
)
wh
i
c
hi
st
h
es
u
m oft
h
r
e
ed
i
v
e
r
ge
n
c
e
s
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i
v
e
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g
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n
c
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t
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l
f
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fwei
n
t
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g
r
a
t
eb
ot
h
s
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d
e
sov
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ra
l
l
s
p
a
c
eweg
e
t
:
∇
3
I
R
x
i
i
x
ˆ
i
3
·(
x
J
)d x =
i
n
J
3
I
R
J
2
(
x
i )d x =
ˆ
i S(
∞) ˆ·
x
0 =
i
i ˆ
3
3
I
R
x ∇ J
3
3
3
( · )
dx (
6
.
1
3
)
J3
x ∇ J 3
3
3
I
R
I
R
dx+
( · )
dx (
6
.
1
4
)
dx+
I
R
J
0 =
x ∇ J 3
( · )
dx (
6
.
1
2
)
3
J
d x+
3
I
R
I
R
x ∇ J 3
R
dx+ I
( · )
dx (
6
.
1
5
)
3
3
wh
e
r
eweh
a
v
eu
s
e
dt
h
ef
a
c
tt
h
a
tJ(
a
n
dρ)h
a
v
ec
omp
a
c
ts
u
p
p
o
r
ta
n
da
r
ez
e
r
o
e
v
e
r
y
wh
e
r
eonas
u
r
f
a
c
ea
ti
n
f
i
n
i
t
y
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Wer
e
a
r
r
a
n
g
et
h
i
sa
n
dg
e
t
:
J
3
I
R
3
dx=−
x ∇ J 3
( · )
d
x
3
I
R
(
6
.
1
6
)
wh
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c
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me
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n
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t
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n
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.
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t
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r
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t
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om t
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t
c
.
)BUTt
h
e
s
ea
s
s
u
mer
e
c
t
i
l
i
n
e
a
rs
u
r
f
a
c
e
sp
a
r
a
l
l
e
l
t
oon
eoft
h
ep
r
i
n
c
i
p
l
e
p
l
a
n
e
s
.
Th
e
r
ea
r
eot
h
e
ra
d’
son
ec
a
nf
or
m.
•Vol
u
meEl
e
me
n
t
:
d
V=d
τ=dxdyd
z
•Gr
a
d
i
e
n
t
:
∂f
∇f=
•Di
v
e
r
g
e
n
c
e
:
∇·
B=
∂f
∂f
ˆ+ ∂y y
ˆ+ ∂z zˆ
∂x x
∂Bx
∂x +
∂By
∂y
∂Bz
+ ∂z
•Cu
r
l
:
∂Bz ∂By
∂Bx
∂Bz
∂By ∂Bx
ˆ+ ∂z − ∂x y
ˆ+ ∂x − ∂y zˆ
∂
y − ∂z x
∇×
B=
•L
a
p
l
a
c
i
a
n
:
2
∂x
∂2f ∂2f ∂
f
2
∇2f
= + +
2
2
∂
y
∂
z
7.
2 Sp
h
e
r
i
c
a
l
Po
l
a
r
•Ve
c
t
or
s
:
r =rr
ˆ
Bu
tNo
t
eWe
l
l
:r
ˆi
sn
owaf
u
n
c
t
i
onof(
θ
,
φ)
!
Si
mi
l
a
r
l
y
:
A=Arr
ˆ
wi
t
hr
ˆaf
u
n
c
t
i
onoft
h
ea
n
g
l
e
s(
θ,
φ)t
h
a
td
e
f
i
n
et
h
edi
r
e
c
t
i
onofA.
Sp
e
c
i
f
i
c
a
l
l
y
:
•Un
i
tv
e
c
t
or
s(
r
e
l
a
t
i
v
et
oCa
r
t
e
s
i
a
nx
ˆ
,
y
ˆ
,
z
ˆ
:
r
ˆ=s
i
nθc
osφx
ˆ+s
i
nθs
i
nφy
ˆ
ˆ
ˆ+cosθsi
nφyˆ−si
nθzˆ
θ =cosθcosφx
ˆ
ˆ
i
nφx
ˆ+cosφy
ˆ
φ =−s
ˆ
Th
ef
a
c
tt
h
a
tr
ˆ
(
θ
,
φ)
,
θ(
θ,
φ)
,φ(
φ)c
omp
l
i
c
a
t
e
sa
l
loft
h
es
p
h
e
r
i
c
a
lc
oor
d
i
n
a
t
e
v
e
c
t
ordi
ffe
r
e
n
t
i
a
l
f
or
ms
,
a
l
t
h
ou
g
hwei
n
d
i
c
a
t
ea
b
ov
ead
i
ffe
r
e
n
t
,
mor
ed
i
r
e
c
twa
y
ofe
v
a
l
u
a
t
i
n
gt
h
e
mt
h
a
na
p
p
l
y
i
n
gd
e
r
i
v
a
t
i
v
e
st
ot
h
eu
n
i
tv
e
c
t
or
st
h
e
ms
e
l
v
e
s
b
e
f
or
ec
omp
l
e
t
i
n
gt
h
et
e
n
s
orc
on
s
t
r
u
c
t
i
onoft
h
ev
e
c
t
ordi
ffe
r
e
n
t
i
a
l
op
e
r
a
t
or
s
.
•Di
r
e
c
tL
e
n
g
t
h
d=d
rˆ+r
d
θˆ+rs
i
nθdφˆ
ℓ
r
θφ
•Di
r
e
c
t
e
dAr
e
a
2
d
A=rs
i
nθd
θd
φ
2
d
An
ˆ=a
d
=rs
i
nθdθd
φr
ˆ=d
Ar
ˆ
An
da
g
a
i
n
,t
h
e
r
ea
r
ema
n
yot
h
e
rp
os
s
i
b
l
ea
d’
s
,f
ore
x
a
mp
l
e
,f
ort
h
eb
ou
n
di
n
g
s
u
r
f
a
c
ef
orh
e
mi
s
p
h
e
r
i
c
a
lv
ol
u
mewh
e
r
eon
ep
i
e
c
eofi
twou
l
db
eac
i
r
c
u
l
a
rs
u
r
f
a
c
e
wi
t
ha
nn
or
ma
ll
i
k
e(
f
ore
x
a
mp
l
e
)z
ˆ
.Th
i
si
sp
r
e
c
i
s
e
l
yt
h
es
u
r
f
a
c
en
e
e
d
e
df
orc
e
r
t
a
i
n
p
r
ob
l
e
msy
ouwi
l
l
t
a
c
k
l
et
h
i
ss
e
me
s
t
e
r
.
•Vol
u
meEl
e
me
n
t
2
d
V=d
τ=rs
i
nθd
θd
φdr=a
d
•Gr
a
d
i
e
n
t
:
1∂
fˆ
∂
f
·dr
r
ˆ
1 ∂fˆ
∂r
nθ∂φφ
r
ˆ+r∂θθ +rsi
∇f=
•Di
v
e
r
g
e
n
c
e
;Th
ed
i
v
e
r
g
e
n
c
ei
sc
on
s
t
r
u
c
t
e
db
yt
h
es
a
mea
r
g
u
me
n
tt
h
a
tp
r
ov
e
st
h
e
d
i
v
e
r
g
e
n
c
et
h
e
or
e
mi
nag
e
n
e
r
a
lc
u
r
v
i
l
i
n
e
a
rc
oor
d
i
n
a
t
es
y
s
t
e
m,
ora
l
t
e
r
n
a
t
i
v
e
l
yp
i
c
k
s
u
pp
i
e
c
e
sf
r
om∇r
ˆ
,
e
t
c
,
h
e
n
c
ei
t
sc
omp
l
e
x
i
t
y
:
2
1∂ rBr
∇·
B=
2
r
s
i
nθBθ)
1 ∂(
+
rsi
nθ
∂r
1 ∂(
Bφ)
+rsi
nθ ∂φ
∂θ
Not
et
h
a
tt
h
i
sf
ol
l
owsf
r
om:
∂
(
g
hBu)
1
∇·
B=
(
fg
h
)
∂u
∂(
fhBv)
+
wi
t
hu=r
,
v=θ
,
w=φ,
a
n
df
c
on
t
r
i
b
u
t
i
onf
r
omr
:
2
∂v
∂w
+
=1
,
g=r
,
h=rs
i
nθ
.Ta
k
et
h
e
2
1∂
(
rBr
)
=
2
2
rsi
nθ
r ∂r
∂r
b
e
c
a
u
s
es
i
nθd
oe
sn
otd
e
p
e
n
donr
,
s
i
mi
l
a
r
l
yf
ort
h
eot
h
e
rt
wop
i
e
c
e
s
.
1
∂(
rs
i
nθBr
)
∂(
fgBw)
•Cu
r
l
Th
ec
u
r
l
i
se
v
a
l
u
a
t
e
di
ne
x
a
c
t
l
yt
h
es
a
mewa
yf
r
omt
h
ee
x
p
r
e
s
s
i
ona
b
ov
e
,
b
u
ti
te
n
d
su
pb
e
i
n
gmu
c
hmor
ec
omp
l
e
x:
s
i
nθBφ)
1 ∂(
∇×
B=
rsi
nθ
∂θ
∂Bθ
1∂
Br
1
∂r
Bφ
ˆ1
i
nθ∂φ − ∂r θ+r
− ∂φ r
ˆ
+r s
∂(
r
Bθ)
∂r − ∂θ φ
•L
a
p
l
a
c
i
a
nTh
eL
a
p
l
a
c
i
a
nf
ol
l
owsb
ya
p
p
l
y
i
n
gt
h
ed
i
v
e
r
g
e
n
c
er
u
l
et
ot
h
eg
r
a
di
e
n
t
r
u
l
ea
n
ds
i
mp
l
i
f
y
i
n
g
:
1∂
2
∂f
2
2
∇f= r∂r r ∂r
1 ∂
∂f
2
i
nθ∂θ si
+rs
nθ ∂θ
7.
3 Cy
l
i
n
d
r
i
c
a
l
Cy
l
i
n
d
r
i
c
a
lc
oor
d
i
n
a
t
e
sa
r
eof
t
e
ng
i
v
e
na
sP=(
s
,φ,z
)s
ot
h
a
tφi
sa
z
i
mu
t
h
a
li
nt
h
es
a
me
s
e
n
s
ea
ss
p
h
e
r
i
c
a
l
p
ol
a
r
,
a
n
ds
ot
h
a
tsi
sd
i
ffe
r
e
n
t
i
a
t
e
df
r
omr
.Howe
v
e
r
,
ma
n
yot
h
e
rs
i
mi
l
a
r
c
on
v
e
n
t
i
on
sa
r
eu
s
e
d.F
ore
x
a
mp
l
e
,P=(
r
,θ
,z
)orP=(
r
,φ,z
)orP=(
ρ,θ,z
)a
r
en
ot
u
n
c
ommon
.Wewi
l
l
u
s
e(
s
,
φ,
z
)i
nt
h
i
sr
e
v
i
e
wt
oa
v
oi
da
smu
c
hc
on
f
u
s
i
ona
sp
os
s
i
b
l
ewi
t
h
s
p
h
e
r
i
c
a
l
p
ol
a
rc
oor
d
i
n
a
t
e
s
.
•Ve
c
t
or
s
:
r =rr
ˆ+z
z
ˆ
Bu
tNo
t
eWe
l
l
:r
ˆi
sn
owaf
u
n
c
t
i
onof(
θ
)
!
Si
mi
l
a
r
l
y
:
A=Arr
ˆ+Azz
ˆ
wi
t
hr
ˆaf
u
n
c
t
i
onoft
h
ea
n
g
l
eAθ =θt
h
a
td
e
f
i
n
e
st
h
ed
i
r
e
c
t
i
onofr
ˆ
.
Sp
e
c
i
f
i
c
a
l
l
y
:
•Un
i
tv
e
c
t
or
s(
r
e
l
a
t
i
v
et
oCa
r
t
e
s
i
a
nx
ˆ
,
y
ˆ
,
z
ˆ
:
s
ˆ=c
osφx
ˆ+s
i
nφy
ˆ
ˆ
=−s
i
nφˆ+c
osφˆ
φ
x
y
z
ˆ=z
ˆ
•Di
r
e
c
tL
e
n
g
t
h
d
ˆ
=dsˆ+s
dφ +dzˆ
∂Br ˆ
ℓ
s
φz
•Di
r
e
c
t
e
dAr
e
a
d
A=s
d
φdz
d
An
ˆ=a
d =sdφdzs
ˆ=dAs
ˆ
An
da
g
a
i
n
,
t
h
e
r
ea
r
ema
n
yot
h
e
rp
os
s
i
b
l
ea
d’
s
,
f
ore
x
a
mp
l
e
:
d
An
ˆ=a
d =sd
φd
sz
ˆ
f
ora
ne
n
dc
a
pofac
y
l
i
n
d
r
i
c
a
l
v
ol
u
me
.
•Vol
u
meEl
e
me
n
t
d
V=d
τ=sd
φdsd
z=a
d
·dz
z
ˆ
f
ort
h
es
e
c
on
doft
h
e
s
ea
r
e
ae
l
e
me
n
t
s
.
•Gr
a
d
i
e
n
t
:
∂f
1∂fˆ
∂f
∇f= ∂ss
ˆ+s∂φφ +∂zzˆ
•Di
v
e
r
g
e
n
c
e
;
Th
ed
i
v
e
r
g
e
n
c
ei
sc
on
s
t
r
u
c
t
e
db
yt
h
es
a
mea
r
g
u
me
n
tt
h
a
tp
r
ov
e
st
h
e
d
i
v
e
r
g
e
n
c
et
h
e
or
e
mi
nag
e
n
e
r
a
l
c
u
r
v
i
l
i
n
e
a
rc
oor
di
n
a
t
es
y
s
t
e
mg
i
v
e
na
b
ov
e
.
1
s
Bs)+1∂(
Bφ)+∂Bz
∇·
B= ∂(
s ∂s
s∂φ∂z
•Cu
r
l
Th
ec
u
r
l
i
se
v
a
l
u
a
t
e
di
ne
x
a
c
t
l
yt
h
es
a
mewa
yf
r
omt
h
ee
x
p
r
e
s
s
i
ona
b
ov
e
,
b
u
ti
te
n
d
su
pb
e
i
n
gmu
c
hmor
ec
omp
l
e
x:
1∂
Bz
∇×
B=
∂Bφ
∂Bs
∂Bz ˆ 1 ∂(
s
Bφ)
s∂φ − ∂z s
s
ˆ+ ∂z − ∂s φ +
∂Bs
∂s − ∂φ zˆ
•L
a
p
l
a
c
i
a
nTh
eL
a
p
l
a
c
i
a
nf
ol
l
owsb
ya
p
p
l
y
i
n
gt
h
ed
i
v
e
r
g
e
n
c
er
u
l
et
ot
h
eg
r
a
di
e
n
t
r
u
l
ea
n
ds
i
mp
l
i
f
y
i
n
g
:
1∂
2
∇f=
∂f
s∂
s s∂s
2
1 ∂f
2
+s
2
∂φ
2
∂f
2
+ ∂z
Ch
a
p
t
e
r8
Th
eDi
r
a
cδF
u
n
c
t
i
o
n
δ(
x
)
f
(
x
)
x
x
Th
eDi
r
a
cδ
f
u
n
c
t
i
oni
su
s
u
a
l
l
yd
e
f
i
n
e
dt
ob
eac
on
v
e
n
i
e
n
t(
s
moot
h
,
i
n
t
e
g
r
a
b
l
e
,
n
a
r
r
ow)
d
i
s
t
r
i
b
u
t
i
one
.
g
.χ
(
x
)t
h
a
ti
ss
y
mme
t
r
i
ca
n
dp
e
a
k
e
di
nt
h
emi
d
d
l
ea
n
dwi
t
hap
a
r
a
me
t
r
i
c
wi
d
t
hx
.
Th
ed
i
s
t
r
i
b
u
t
i
oni
sn
or
ma
l
i
z
e
d(
i
nt
e
r
msofi
t
swi
d
t
h
)s
ot
h
a
ti
t
si
n
t
e
g
r
a
l
i
son
e
:
∞
χ
(
x
)
d
x=1
−
∞
On
et
h
e
nt
a
k
e
st
h
el
i
mi
tx→ 0wh
i
l
ec
on
t
i
n
u
i
n
gt
oe
n
f
or
c
et
h
en
or
ma
l
i
z
a
t
i
on
c
on
d
i
t
i
ont
od
e
f
i
n
et
h
eδf
u
n
c
t
i
on
:
δ(
x
)=l
i
mχ
(
x
)
x
→0
6
3
Th
eδf
u
n
c
t
i
oni
t
s
e
l
fi
st
h
u
sn
ots
t
r
i
c
t
l
ys
p
e
a
k
i
n
ga“
f
u
n
c
t
i
on
”
,
b
u
tr
a
t
h
e
rt
h
el
i
mi
tof
ad
i
s
t
i
b
u
t
i
on
.F
u
r
t
h
e
r
mor
e
,i
ti
sn
e
a
r
l
yu
s
e
l
e
s
si
na
n
dofi
t
s
e
l
f–a
sa“
f
u
n
c
t
i
on
”
s
t
a
n
d
i
n
ga
l
on
ei
tc
a
nb
et
h
ou
g
h
tofa
sa
ni
n
f
i
n
i
t
e
l
yn
a
r
r
ow,
i
n
f
i
n
i
t
e
l
yh
i
g
hp
e
a
ka
r
ou
n
dx
=0wi
t
hac
on
s
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r
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l
s
:
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A−
S1 · ˆ
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t
=
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t V/S
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(
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0
.
7
)
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nt
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se
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n
gt
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t −ǫ
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d
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n
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(
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.
9
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S1 · ˆ
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A=
d
ˆ
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t S2 ǫE·n
nˆ2dA+ d
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E
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J+ǫ dt
(
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0
.
1
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)
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E
·nˆ1d
A=
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A
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t ·n
J+ǫ d
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1
0
.
1
1
)
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r
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eor
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t
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i
n
v=J+ǫ
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d
t
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r
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c
ome
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:
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n
v· ˆ
d
A
C i
d
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= µ
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E
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·n
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A
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0
.
1
3
)
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C
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t
or
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D
d
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C
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A
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0
.
1
4
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nt
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msoft
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eor
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mp
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t
:
(
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0
.
1
5
)
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(
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H)
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l
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m:
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.
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ti
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et
h
a
t
:
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s
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0
.
1
8
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.
(
1
0
.
1
9
)
Th
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(
AL
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(
1
0
.
2
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)
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(
GL
M)
(
1
0
.
2
2
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=0
(
F
L
)
(
1
0
.
2
3
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H−
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E
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e
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r
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e
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s
:
1
∇×
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+
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(
1
0
.
2
4
)
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(
1
0
.
2
5
)
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1
0
.
2
6
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(
1
0
.
2
7
)
I
ti
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ore
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n
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u
c
ht
h
a
t
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B=
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(
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.
2
8
)
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.
2
9
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I
nt
h
a
tc
a
s
e
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on
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b
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t
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r
e
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s
i
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orB:
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∂t
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E+
×
= 0
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e
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e
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s
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t
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e
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A =A0+
(
1
0
.
4
8
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(
1
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.
4
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b
s
t
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t
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t = ∇·
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.
5
0
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2
2
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t=
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t
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ors
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t
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6
9
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(
1
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7
0
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t = 0
(
1
0
.
7
1
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v
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l
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t
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t
f
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.
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a
r
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t
h
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−
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.
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2
)
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qu
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t
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t
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n
t
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t
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ome
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a
ti
s
t
h
emi
s
s
i
n
gh
e
a
t
.
I
ts
e
e
ms
,
t
h
e
n
,
t
h
a
tPoy
n
t
i
n
g
’
st
h
e
or
e
mi
sl
i
k
e
l
yt
ob
ea
p
p
l
i
c
a
b
l
ei
nami
c
r
o
s
c
o
p
i
c
d
e
s
c
r
i
p
t
i
onofp
a
r
t
i
c
l
e
smov
i
n
gi
nav
a
c
u
u
m,wh
e
r
et
h
e
i
ri
n
d
i
v
i
d
u
a
le
n
e
r
g
i
e
sc
a
nb
e
t
r
a
c
k
e
da
n
dt
a
l
l
i
e
d:
∂η
∂t
+
∇·
S =−
J·
E
1
S = µ0E×B
(
1
0
.
9
4
)
(
1
0
.
9
5
)
2
η = 1ǫ0E
2
2
+ 1B
2µ0
(
1
0
.
9
6
)
b
u
tn
otn
e
c
e
s
s
a
r
i
l
ys
ou
s
e
f
u
l
i
nma
c
r
os
c
op
i
cme
d
i
awi
t
hd
y
n
a
mi
c
a
l
di
s
p
e
r
s
i
ont
h
a
twe
d
on
oty
e
tu
n
d
e
r
s
t
a
n
d
.Th
e
r
ewec
a
ni
d
e
n
t
i
f
yt
h
eJ·Et
e
r
ma
st
h
er
a
t
ea
twh
i
c
ht
h
e
me
c
h
a
n
i
c
a
l
e
n
e
r
g
yoft
h
ec
h
a
r
g
e
dp
a
r
t
i
c
l
e
st
h
a
tma
k
eu
pJc
h
a
n
g
e
sa
n
dwr
i
t
e
:
d
E
d
n
ˆd
A
σS·
d
t =
dt(
E
f
i
e
l
d+E
me
c
h
a
n
i
c
a
l
)=−
(
1
0
.
9
7
)
(
wh
e
r
en
ˆi
s
,
r
e
c
a
l
l
,
a
nou
t
wa
r
dd
i
r
e
c
t
e
dn
or
ma
l
)s
ot
h
a
tt
h
i
ss
a
y
st
h
a
tt
h
er
a
t
ea
twh
i
c
h
e
n
e
r
g
yf
l
owsi
n
t
ot
h
ev
ol
u
mec
a
r
r
i
e
db
yt
h
ee
l
e
c
t
r
oma
g
n
e
t
i
cf
i
e
l
de
qu
a
l
st
h
er
a
t
ea
t
wh
i
c
ht
h
et
ot
a
lme
c
h
a
n
i
c
a
lp
l
u
sf
i
e
l
de
n
e
r
g
yi
nt
h
ev
ol
u
mei
n
c
r
e
a
s
e
s
.Th
i
si
sa
ma
r
v
e
l
ou
sr
e
s
u
l
t
!
Mome
n
t
u
mc
a
ns
i
mi
l
a
r
l
yb
ec
on
s
i
de
r
e
d
,
a
g
a
i
ni
nami
c
r
os
c
op
i
cd
e
s
c
r
i
p
t
i
on
.
Th
e
r
ewes
t
a
r
twi
t
hNe
wt
on
’
ss
e
c
on
dl
a
wa
n
dt
h
eL
or
e
n
t
zf
or
c
el
a
w:
p
d
d
t
F=q(
E+
v×B)=
(
1
0
.
9
8
)
s
u
mmi
n
gwi
t
hc
oa
r
s
eg
r
a
i
n
i
n
gi
n
t
oa
ni
n
t
e
g
r
a
l
a
su
s
u
a
l
:
P
d
me
c
h
d
t
3
ρE+J×B)
dx
=V(
(
1
0
.
9
9
)
Asb
e
f
or
e
,
wee
l
i
mi
n
a
t
es
ou
r
c
e
su
s
i
n
gt
h
ei
n
h
omog
e
n
e
ou
sMEs(
t
h
i
st
i
mes
t
a
r
t
i
n
gf
r
om
t
h
eb
e
g
i
n
n
i
n
gwi
t
ht
h
ev
a
c
u
u
mf
or
ms
)
:
P
d
∂E
me
c
h
d
t
∇·E
)
E−ǫ0
=V ǫ0(
1
3
∂t×B+ µ0(
∇ ×B)×Bdx
(
1
0
.
1
0
0
)
or
∂E
2
∂t−cB ×
(
∇×
B).
ρE
+
J×
B=
ǫ0E
(
∇·
E
)
+B×
(
1
0
.
1
0
1
)
Ag
a
i
n
,
wed
i
s
t
r
i
b
u
t
e
:
∂
∂t(
E
×
B)
=
or
∂E
∂E
∂B
∂t ×
B+
E
×
∂t
∂
(
1
0
.
1
0
2
)
∂B
B× ∂t=− ∂t(
E×
B)
+
E
×
∂t
(
1
0
.
1
0
3
)
2
s
u
b
s
t
i
t
u
t
ei
ti
na
b
ov
e
,
a
n
da
d
dcB(
∇·B)=0
:
ρE+J×B
2
= ǫ0 E
(
∇·E
)+cB(
∇·B)
∂
∂B
∂
t
−∂
E×B)+E×
t(
2
−
cB×(
∇×B).
(
1
0
.
1
0
4
)
F
i
n
a
l
l
y
,
s
u
b
s
t
i
t
u
t
i
n
gi
nF
L
:
ρE+J×B
2
= ǫ0 E
(
∇·E
)+cB(
∇·B)
2
−
E×(
∇×E
)−cB×(
∇×B)
∂
−
ǫ0∂t(
E
×
B)
(
1
0
.
1
0
5
)
Re
a
s
s
e
mb
l
i
n
ga
n
dr
e
a
r
r
a
n
g
i
n
g
:
P
d
me
c
h
d
t
d
EB
d
V =ǫ0
+
dtǫ0 V( × )
2
V
E
(
∇
·
E
)
−
E
×
(
∇×
E
)
+
2
cB(
∇·B)−cB×(
∇×B)
d
V
(
1
0
.
1
0
6
)
Th
equ
a
n
t
i
t
yu
n
d
e
rt
h
ei
n
t
e
g
r
a
l
ont
h
el
e
f
th
a
su
n
i
t
sofmome
n
t
u
md
e
n
s
i
t
y
.
Wede
f
i
n
e
:
1
1
g=ǫ0(
E×B)=ǫ0µ
(
E×H)=
c2 S
c2(
0
E
×
H)
=
(
1
0
.
1
0
7
)
t
ob
et
h
ef
i
e
l
d mome
n
t
u
m d
e
n
s
i
t
y
.Pr
ov
i
n
gt
h
a
tt
h
er
i
gh
th
a
n
ds
i
d
e oft
h
i
s
i
n
t
e
r
p
r
e
t
a
t
i
oni
sc
on
s
i
s
t
e
n
twi
t
ht
h
i
si
sa
c
t
u
a
l
l
ya
ma
z
i
n
g
l
yd
i
ffic
u
l
t
.I
ti
ss
i
mp
l
e
rt
oj
u
s
t
d
e
f
i
n
et
h
eMa
x
we
l
l
St
r
e
s
sTe
n
s
or
:
2
Tαβ=ǫ0 EαE
Bβ−
β+cB
α
1
2
(
E·E+cB·B)
δ
2
α
β
(
1
0
.
1
0
8
)
I
nt
e
r
msoft
h
i
s
,
wi
t
hal
i
t
t
l
ewor
kon
ec
a
ns
h
owt
h
a
t
:
P
dP
( fiel
d+ me
c
h
a
n
i
c
a
l
)
α=
ˆ
Tαβnβd
A
dt
(
1
0
.
1
0
9
)
S β
Th
a
ti
s
,
f
ore
a
c
hc
omp
on
e
n
t
,
t
h
et
i
mer
a
t
eofc
h
a
n
g
eoft
h
et
ot
a
l
mome
n
t
u
m(
f
i
e
l
dp
l
u
s
me
c
h
a
n
i
c
a
l
)wi
t
h
i
nt
h
ev
ol
u
mee
qu
a
l
st
h
ef
l
u
xoft
h
ef
i
e
l
dmome
n
t
u
mt
h
r
ou
g
ht
h
e
c
l
os
e
ds
u
r
f
a
c
et
h
a
tc
on
t
a
i
n
st
h
ev
ol
u
me
.
Iwi
s
ht
h
a
tIc
ou
l
dd
ob
e
t
t
e
rwi
t
ht
h
i
s
,b
u
ta
n
a
l
y
z
i
n
gt
h
eMa
x
we
l
lSt
r
e
s
sTe
n
s
or
t
e
r
mwi
s
et
ou
n
d
e
r
s
t
a
n
dh
owi
ti
sr
e
l
a
t
e
dt
of
i
e
l
dmome
n
t
u
mf
l
owi
ss
i
mp
l
yd
i
ffic
u
l
t
.I
t
wi
l
la
c
t
u
a
l
l
yma
k
e mor
es
e
n
s
e
,a
n
db
ee
a
s
i
e
rt
od
e
r
i
v
e
,wh
e
n we f
or
mu
l
a
t
e
e
l
e
c
t
r
od
y
n
a
mi
c
sr
e
l
a
t
i
v
i
s
t
i
c
a
l
l
ys
owewi
l
l
wa
i
tu
n
t
i
l
t
h
e
nt
od
i
s
c
u
s
st
h
i
sf
u
r
t
h
e
r
.
10.
4 Ma
gn
e
t
i
cMon
op
o
l
e
s
L
e
tu
st
h
i
n
kf
oramome
n
ta
b
ou
twh
a
tME
smi
g
h
tb
ec
h
a
n
g
e
di
n
t
oi
fma
g
n
e
t
i
c
mon
op
ol
e
s we
r
ed
i
s
c
ov
e
r
e
d
.We wou
l
dt
h
e
ne
x
p
e
c
ta
l
lf
ou
re
qu
a
t
i
on
st
ob
e
i
n
h
omog
e
n
e
ou
s
:
∇·D
∂D
∇×
H−
∂t
∇·H
∂H
∇×
D+
∂t
GL
E
)
=ρe (
(
1
0
.
1
1
0
)
e
=J
(
1
0
.
1
1
1
)
(
AL
)
=ρm (
GL
M)
(
1
0
.
1
1
2
)
J
m
=−
(
1
0
.
1
1
3
)
(
F
L
)
or
,
i
nav
a
c
u
u
m(
wi
t
hu
n
i
t
sofma
g
n
e
t
i
cc
h
a
r
g
eg
i
v
e
na
sa
mp
e
r
e
me
t
e
r
s
,
a
sop
p
os
e
dt
o
we
b
e
r
s
,
wh
e
r
e1we
b
e
r=µ
mp
e
r
e
me
t
e
r
)
:
0a
1
∇·E
∂E
= ǫ0ρe
(
GL
E
)
(
1
0
.
1
1
4
)
∇×
B−
ǫ0µ0
∂t
e
=µ0J
(
AL
)
(
1
0
.
1
1
5
)
=µ0ρm
(
GL
M)
(
1
0
.
1
1
6
)
∇×
E
+
∇·B
∂B
∂t
µ
J
F
L
)
0
m (
=−
(
1
0
.
1
1
7
)
(
wh
e
r
ewen
ot
et
h
a
ti
fwedi
s
c
ov
e
r
e
da
ne
l
e
me
n
t
a
r
yma
g
n
e
t
i
cmon
op
ol
eofma
g
n
i
t
u
d
e
gs
i
mi
l
a
rt
ot
h
ee
l
e
me
n
t
a
r
ye
l
e
c
t
r
i
cmon
op
ol
a
rc
h
a
r
g
eofewewou
l
da
l
mos
tc
e
r
t
a
i
n
l
y
n
e
e
dt
oi
n
t
r
od
u
c
ea
d
d
i
t
i
on
a
lc
on
s
t
a
n
t
s–ora
r
r
a
n
g
e
me
n
t
soft
h
ee
x
i
s
t
i
n
gon
e
s–t
o
e
s
t
a
b
l
i
s
hi
t
squ
a
n
t
i
z
e
dma
g
n
i
t
u
der
e
l
a
t
i
v
et
ot
h
os
eofe
l
e
c
t
r
i
cc
h
a
r
g
ei
ns
u
i
t
a
b
l
eu
n
i
t
s
a
si
sd
i
s
c
u
s
s
e
ds
h
or
t
l
y
)
.
Th
e
r
ea
r
et
woob
s
e
r
v
a
t
i
on
swen
e
e
dt
oma
k
e
.On
ei
st
h
a
tn
a
t
u
r
ec
ou
l
db
er
i
f
ewi
t
h
ma
g
n
e
t
i
cmon
op
ol
e
sa
l
r
e
a
d
y
.I
nf
a
c
t
,
e
v
e
r
ys
i
n
g
l
ec
h
a
r
g
e
dp
a
r
t
i
c
l
ec
ou
l
dh
a
v
eami
xof
b
ot
he
l
e
c
t
r
i
ca
n
dma
g
n
e
t
i
cc
h
a
r
g
e
.Asl
on
ga
st
h
er
a
t
i
og
/
ei
sac
o
n
s
t
a
n
t
,
wewou
l
db
e
u
n
a
b
l
et
ot
e
l
l
.
Th
i
sc
a
nb
es
h
ownb
yl
ook
i
n
ga
tt
h
ef
ol
l
owi
n
gd
u
a
l
i
t
yt
r
a
n
s
f
or
ma
t
i
o
nwh
i
c
h
“
r
ot
a
t
e
s
”t
h
ema
g
n
e
t
i
cf
i
e
l
di
n
t
ot
h
ee
l
e
c
t
r
i
cf
i
e
l
da
si
tr
ot
a
t
e
st
h
ema
g
n
e
t
i
cc
h
a
r
g
ei
n
t
o
t
h
ee
l
e
c
t
r
i
cc
h
a
r
g
e
:
′
′
os
(
Θ)+Z0Hs
i
n
(
Θ)
E =Ec
(
1
0
.
1
1
8
)
Z0D =Z0Dc
os
(
Θ)+Bs
i
n
(
Θ)
(
1
0
.
1
1
9
)
Z0H =−
Es
i
n
(
Θ)+Z0Hc
os
(
Θ)
(
1
0
.
1
2
0
)
′
′
′
′
′
′
Z0Ds
i
n
(
Θ)+Bc
os
(
Θ)
B =−
wh
e
r
eZ0=
µ0
ǫ0
(
1
0
.
1
2
1
)
i
st
h
ei
mp
e
da
n
c
eoff
r
e
es
p
a
c
e(
a
n
dh
a
su
n
i
t
sofoh
ms
)
,
a
qu
a
n
t
i
t
yt
h
a
t(
a
swes
h
a
l
l
s
e
e
)a
p
p
e
a
r
sf
r
e
qu
e
n
t
l
ywh
e
nma
n
i
p
u
l
a
t
i
n
gME
s
.
Not
et
h
a
twh
e
nt
h
ea
n
g
l
eΘ=0
,
weh
a
v
et
h
eor
di
n
a
r
yME
swea
r
eu
s
e
dt
o.Howe
v
e
r
,
a
l
l
ofou
rme
a
s
u
r
e
me
n
t
soff
or
c
ewou
l
dr
e
ma
i
nu
n
a
l
t
e
r
e
di
fwer
ot
a
t
e
db
yΘ=π/
2a
n
dE
′
=Z0Hi
nt
h
eol
ds
y
s
t
e
m.
Howe
v
e
r
,
i
fwep
e
r
f
or
ms
u
c
har
ot
a
t
i
on
,
wemu
s
ta
l
s
or
ot
a
t
et
h
ec
h
a
r
g
e
d
i
s
t
r
i
b
u
t
i
on
si
ne
x
a
c
t
l
yt
h
es
a
mewa
y
:
′
′
Z0ρe =Z0ρ
os
(
Θ)+ρ
i
n
(
Θ)
ec
ms
′
′
ρ
=
Zρ cos(
Θ)+ρ s
i
n
(
Θ)
− 0e
m
m
(
1
0
.
1
2
2
)
(
1
0
.
1
2
3
)
Z0J
J
os
(
Θ)+J
i
n
(
Θ)
e =−
ec
ms
(
1
0
.
1
2
4
)
′
′
′
′
J
Z0J
i
n
(
Θ)+J
os
(
Θ)
m =−
es
mc
(
1
0
.
1
2
5
)
I
ti
sl
e
f
ta
sa
ne
x
e
r
c
i
s
et
os
h
owt
h
a
tt
h
emon
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ol
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rf
or
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r
el
e
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ti
n
v
a
r
i
a
n
t
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h
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n
g
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omei
nj
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s
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h
er
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g
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omb
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n
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t
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on
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d
e
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t
i
on
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o
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omp
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s
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nan
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t
s
h
e
l
l
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a
tt
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i
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n
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st
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ti
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on
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ot
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et
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t
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od
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n
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mi
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e
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on
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a
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e
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u
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lf
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e
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t
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on
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cs
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r
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t
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h
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sh
a
v
et
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/
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t
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o
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e
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r
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hf
orma
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h
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er
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e
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r
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orp
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r
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tr
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oi
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omt
h
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el
ook
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orp
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r
t
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c
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st
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th
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h
ec
u
r
r
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n
tf
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mer
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l
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t
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et
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od
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rt
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mer
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l
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t
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g
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l
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r
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n
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ore
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mp
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e
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e
or
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s
t
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l
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e
a
l
l
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e
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l
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yl
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k
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or
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h
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et
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ea
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a
s
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ol
ei
nt
h
eu
n
i
v
e
r
s
e
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e
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h
u
n
g
r
yg
r
a
d
u
a
t
es
t
u
de
n
t
swou
l
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n
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t
mi
n
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ft
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n
gt
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ou
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ht
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e
i
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et
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p
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t
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e
r
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of
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l
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t
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er
e
s
u
l
t
st
h
a
th
a
v
ep
r
ov
e
ndu
b
i
ou
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t
a
n
yr
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t
eu
n
r
e
p
e
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t
a
b
l
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e
r
ei
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a
c
kofa
c
t
u
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le
x
p
e
r
i
me
n
t
a
le
v
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de
n
c
ef
ormon
op
ol
e
s
.
L
e
t
’
se
x
a
mi
n
ej
u
s
tab
i
tofwh
yt
h
ei
d
e
aofmon
op
ol
e
si
se
x
c
i
t
i
n
gt
ot
h
e
or
i
s
t
s
.
10.
4.
1 Di
r
a
cMo
n
o
p
o
l
e
s
Con
s
i
d
e
rae
l
e
c
t
r
i
cc
h
a
r
g
eea
tt
h
eo
r
i
g
i
na
n
da
nmon
o
p
ol
a
rc
h
a
r
g
ega
ta
na
r
b
i
t
r
a
r
y
p
oi
n
tont
h
eza
x
i
s
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r
omt
h
eg
e
n
e
r
a
l
i
z
e
df
or
mofME
s
,
wee
x
p
e
c
tt
h
ee
l
e
c
t
r
i
cf
i
e
l
dt
ob
e
g
i
v
e
nb
yt
h
ewe
l
l
k
n
own
:
er
ˆ
2
E= 4πǫ0r
(
1
0
.
1
2
6
)
a
ta
na
r
b
i
t
r
a
r
yp
oi
n
ti
ns
p
a
c
e
.Si
mi
l
a
r
l
y
,
wee
x
p
e
c
tt
h
ema
g
n
e
t
i
cf
i
e
l
doft
h
emon
op
ol
a
r
c
h
a
r
g
egt
ob
e
:
′
gr
ˆ
B=
′
2
4πµ0r
(
1
0
.
1
2
7
)
′
wh
e
r
e
r=
z +r.
Th
emo
me
n
t
u
md
e
n
s
i
t
yoft
h
i
sp
a
i
roff
i
e
l
d
si
sg
i
v
e
na
sn
ot
e
da
b
ov
eb
y
:
1
2
g= c (
E
×
H)
(
1
0
.
1
2
8
)
a
n
di
fon
ed
r
a
wsp
i
c
t
u
r
e
sa
n
du
s
e
son
e
’
sr
i
g
h
th
a
n
dt
od
e
t
e
r
mi
n
ed
i
r
e
c
t
i
on
s
,
i
ti
sc
l
e
a
r
t
h
a
tt
h
ef
i
e
l
dmome
n
t
u
mi
sd
i
r
e
c
t
e
da
r
ou
n
dt
h
ee−ga
x
i
si
nt
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er
i
g
h
th
a
n
d
e
ds
e
n
s
e
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n
f
a
c
tt
h
emome
n
t
u
mf
ol
l
owsc
i
r
c
u
l
a
rt
r
a
c
k
sa
r
ou
n
dt
h
i
sa
x
i
si
ns
u
c
hawa
yt
h
a
tt
h
ef
i
e
l
d
h
a
san
on
z
e
r
os
t
a
t
i
ca
n
gu
l
a
rmome
n
t
u
m.
Th
es
y
s
t
e
mob
v
i
ou
s
l
yh
a
sz
e
r
ot
ot
a
lmome
n
t
u
mf
r
oms
y
mme
t
r
y
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i
sme
a
n
son
e
c
a
nu
s
ea
n
yor
i
g
i
nt
oc
omp
u
t
et
h
ea
n
g
u
l
a
rmome
n
t
u
m.Todos
o,wec
omp
u
t
et
h
e
a
n
g
u
l
a
rmome
n
t
u
md
e
n
s
i
t
ya
s
:
1
r
2
c ×E
×
H
(
1
0
.
1
2
9
)
a
n
di
n
t
e
g
r
a
t
ei
t
:
L
f
i
e
l
d
1
r
2
c
=
=
EH
×( × )
d
V
µ
e1
0
n
ˆ×(
n
ˆ×H)
dV4
πr
µ0e
=− 4π
1
ˆ
(
n
ˆ·H)d
V
r H−n
(
1
0
.
1
3
0
)
ov
e
ra
l
l
s
p
a
c
e
.
Us
i
n
gt
h
ev
e
c
t
ori
d
e
n
t
i
t
y
:
f(
r
)
∂f
r a{−nˆ
(
nˆa·)
}+nˆ
(
nˆa·) ∂r
a
(·∇)
n
ˆ
f(
r
)=
(
1
0
.
1
3
1
)
t
h
i
sc
a
nb
et
r
a
n
s
f
or
me
di
n
t
o:
e
L
f
i
e
l
d=−
B·∇)
n
ˆ
d
V
4π (
(
1
0
.
1
3
2
)
I
n
t
e
g
r
a
t
i
n
gb
yp
a
r
t
s
:
L
e
e
∇ Bn
′
nB n
e
g
ˆ
4 πz
( ·ˆ)
ˆ
dV− 4π Sˆ
4π ( · )
d
A
(
1
0
.
1
3
3
)
Th
es
u
r
f
a
c
et
e
r
mv
a
n
i
s
h
e
sf
r
om s
y
mme
t
r
yb
e
c
a
u
s
enˆi
sr
a
d
i
a
l
l
ya
wa
yf
r
omt
h
e=
or
i
g
i
na
n
da
v
e
r
a
g
e
st
oz
e
r
oonal
a
r
g
es
p
h
e
r
e
.∇·B
ob
t
a
i
n
:
g
δr
(−
z)Th
u
swef
i
n
a
l
l
y
f
i
e
l
d=
L
f
i
e
l
d=
(
1
0
.
1
3
4
)
Th
e
r
ea
r
eav
a
r
i
e
t
yofa
r
g
u
me
n
t
st
h
a
ton
ec
a
ni
n
v
e
n
tt
h
a
tl
e
a
dst
oa
ni
mp
or
t
a
n
t
c
on
c
l
u
s
i
on
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ea
r
g
u
me
n
t
sdi
ffe
ri
nd
e
t
a
i
l
sa
n
di
ns
ma
l
lwa
y
squ
a
n
t
i
t
a
t
i
v
e
l
y
,
a
n
ds
omea
r
e
mor
ee
l
e
g
a
n
tt
h
a
nt
h
i
son
e
.
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tt
h
i
son
ei
sa
d
e
qu
a
t
et
oma
k
et
h
e
p
oi
n
t
.
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fwer
e
qu
i
r
et
h
a
tt
h
i
sf
i
e
l
da
n
g
u
l
a
rmome
n
t
u
mb
eq
u
a
n
t
i
z
e
di
nu
n
i
t
sof
:
e
g
z
4
π ˆ=mz
(
1
0
.
1
3
5
)
wec
a
nc
on
c
l
u
det
h
a
tt
h
ep
r
o
du
c
tofe
gmu
s
tb
equ
a
n
t
i
z
e
d
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i
si
sa
ni
mp
or
t
a
n
t
c
on
c
l
u
s
i
on
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ti
son
eoft
h
ef
e
wa
p
p
r
oa
c
h
e
si
np
h
y
s
i
c
st
h
a
tc
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ng
i
v
eu
si
n
s
i
g
h
ta
st
o
wh
yc
h
a
r
g
ei
squ
a
n
t
i
z
e
d
.
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i
sc
on
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l
u
s
i
onwa
sor
i
g
i
n
a
l
l
ya
r
r
i
v
e
da
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y(
wh
oe
l
s
e
?
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r
a
c
.Howe
v
e
r
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r
a
c
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s
a
r
g
u
me
n
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smor
es
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b
t
l
e
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r
e
a
t
e
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ol
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e
c
tb
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on
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t
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c
t
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t
orp
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n
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a
lt
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tl
e
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o
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ol
a
rf
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l
de
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e
r
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e
r
ei
ns
p
a
c
eb
u
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c
hwa
s
s
i
n
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l
a
ronas
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n
g
l
el
i
n
e
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emode
lf
ort
h
i
sv
e
c
t
orp
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e
n
t
i
a
lwa
st
h
a
tofa
ni
n
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n
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t
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y
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on
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e
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ds
t
r
e
t
c
h
i
n
gi
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omi
n
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i
n
i
t
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l
on
gt
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x
i
s
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i
ss
ol
e
n
oi
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si
nf
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c
ta
s
t
r
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n
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i
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si
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e
n
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et
h
ef
i
r
s
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a
n
t
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ms
t
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i
n
gt
h
e
or
y
.
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edi
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r
e
n
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l
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e
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t
orp
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l
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l
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= 0
(
1
1
.
1
4
)
= 0
(
1
1
.
1
5
)
∂E
∇×
B−
ǫµ
∂t
11.
1.
2 Th
eWa
v
eE
q
u
a
t
i
o
n
Af
t
e
ral
i
t
t
l
ewor
k(
t
a
k
et
h
ec
u
r
l
oft
h
ec
u
r
l
e
qu
a
t
i
on
s
,
u
s
i
n
gt
h
ei
d
e
n
t
i
t
y
:
2
∇×(
∇×a
)=∇(
∇·a
)−∇a
(
1
1
.
1
6
)
a
n
du
s
i
n
gGa
u
s
s
’
ss
ou
r
c
e
f
r
e
eL
a
ws
)wec
a
ne
a
s
i
l
yf
i
n
dt
h
a
tEa
n
dBi
nf
r
e
es
p
a
c
e
s
a
t
i
s
f
yt
h
ewa
v
ee
q
u
a
t
i
o
n
:
∇2u−
1∂2u
=0
(
1
1
.
1
7
)
2 2
v ∂t
(
f
oru=Eoru=B)wh
e
r
e
1
2
(
1
1
.
1
8
)
v=õ
ǫ.
Th
ewa
v
ee
qu
a
t
i
ons
e
p
a
r
a
t
e
sf
orh
a
r
mon
i
cwa
v
e
sa
n
dwec
a
na
c
t
u
a
l
l
ywr
i
t
et
h
e
f
ol
l
owi
n
gh
omog
e
n
e
ou
sPDEf
orj
u
s
tt
h
es
p
a
t
i
a
l
p
a
r
tofEorB:
2
ω
∇2
2
+v
2
2
2
2
E=∇ +k E=0
ω2
2
∇
2
+v
B=∇ +k B=0
−
i
ωt
wh
e
r
et
h
et
i
med
e
p
e
n
d
e
n
c
ei
si
mp
l
i
c
i
t
l
ye a
n
dwh
e
r
ev=ω/
k
.
Th
i
si
sc
a
l
l
e
dt
h
eh
omoge
n
e
ou
sHe
l
mh
o
l
t
ze
qu
a
t
i
on(
HHE
)a
n
dwe
’
l
l
s
p
e
n
dal
otoft
i
me
s
t
u
d
y
i
n
gi
ta
n
di
t
si
n
h
omog
e
n
e
ou
sc
ou
s
i
n
.Not
et
h
a
ti
tr
e
d
u
c
e
si
nt
h
ek→ 0l
i
mi
tt
ot
h
e
f
a
mi
l
i
a
rh
omog
e
n
e
ou
sL
a
p
l
a
c
ee
qu
a
t
i
on
,
wh
i
c
hi
sb
a
s
i
c
a
l
l
ya
s
p
e
c
i
a
l
c
a
s
eoft
h
i
sPDE.
3
Ob
s
e
r
v
i
n
gt
h
a
t:
i
k
n
ˆ
·
x=
i
k
n
ˆ
·
x
∇e
i
k
n
ˆ
e
(
1
1
.
1
9
)
2
i
ωt
I
nc
a
s
ey
ou
’
v
ef
or
g
ot
t
e
n
:
Tr
yas
ol
u
t
i
ons
u
c
ha
su
(
x
,
t
)=X(
x
)
Y(
y
)
Z(
z
)
T(
t
)
,
or(
wi
t
hab
i
tofi
n
s
p
i
r
a
t
i
on
)E
(
x
)
e
−
i
nt
h
edi
ffe
r
e
n
t
i
a
le
qu
a
t
i
on
.Di
v
i
d
eb
yu
.Youe
n
du
pwi
t
hab
u
n
c
hoft
e
r
mst
h
a
tc
a
ne
a
c
hb
ei
d
e
n
t
i
f
i
e
da
sb
e
i
n
g
c
on
s
t
a
n
ta
st
h
e
yde
p
e
n
donx
,
y
,
z
,
ts
e
p
a
r
a
t
e
l
y
.F
oras
u
i
t
a
b
l
ec
h
oi
c
eofc
on
s
t
a
n
t
son
eob
t
a
i
n
st
h
ef
ol
l
owi
n
gPDEf
or
s
p
a
t
i
a
l
p
a
r
tofh
a
r
mon
i
cwa
v
e
s
.
3
Ye
s
,
y
ous
h
ou
l
dwor
kt
h
i
sou
tt
e
r
mwi
s
ei
fy
ou
’
v
en
e
v
e
rd
on
es
ob
e
f
or
e
.
Don
’
tj
u
s
tt
a
k
emywor
df
ora
n
y
t
h
i
n
g.
wh
e
r
en
ˆi
sau
n
i
tv
e
c
t
or
,
wec
a
ne
a
s
i
l
ys
e
et
h
a
tt
h
ewa
v
ee
qu
a
t
i
onh
a
s(
a
mon
gma
n
y
,
3
ma
n
yot
h
e
r
s
)as
ol
u
t
i
ononI
Rt
h
a
tl
ook
sl
i
k
e
:
i
(
k
n
ˆ
·
x
−ωt
)
0
u
(
x
,
t
)=u
e
(
1
1
.
2
0
)
wh
e
r
et
h
ewa
v
en
u
mb
e
rk=k
n
ˆh
a
st
h
ema
g
n
i
t
u
d
e
ω
k= v=√ µǫω
(
1
1
.
2
1
)
a
n
dp
oi
n
t
si
nt
h
edi
r
e
c
t
i
onofp
r
op
a
g
a
t
i
onoft
h
i
sp
l
a
n
ewa
v
e
.
11.
1.
3 Pl
a
n
eWa
v
e
s
Pl
a
n
ewa
v
e
sc
a
np
r
op
a
g
a
t
ei
na
n
yd
i
r
e
c
t
i
on
.An
ys
u
p
e
r
p
os
i
t
i
onoft
h
e
s
ewa
v
e
s
,
f
ora
l
l
p
os
s
i
b
l
eω,
k
,
i
sa
l
s
oas
ol
u
t
i
ont
ot
h
ewa
v
ee
qu
a
t
i
on
.Howe
v
e
r
,
r
e
c
a
l
l
t
h
a
tEa
n
dBa
r
e
n
oti
n
de
p
e
n
d
e
n
t
,
wh
i
c
hr
e
s
t
r
i
c
t
st
h
es
ol
u
t
i
oni
ne
l
e
c
t
r
od
y
n
a
mi
c
ss
ome
wh
a
t
.
Tog
e
taf
e
e
l
f
ort
h
ei
n
t
e
r
de
p
e
n
d
e
n
c
eofEa
n
dB,
l
e
t
’
sp
i
c
kk=±
k
x
ˆs
ot
h
a
te
.
g
.
:
E
(
x
,
t
)
i
(
k
x
−ωt
)+
i
(
−k
x
−ωt
)
+
−e
e
E
=E
(
1
1
.
2
2
)
B(
x
,
t
)
i
(
k
x
−ωt
)+
i
(
−k
x
−ωt
)
+
e
B−e
=B
(
1
1
.
2
3
)
wh
i
c
ha
r
ep
l
a
n
ewa
v
e
st
r
a
v
e
l
l
i
n
gt
ot
h
er
i
g
h
torl
e
f
ta
l
on
gt
h
ex
a
x
i
sf
ora
n
yc
omp
l
e
xE
,
+
E
B+,
B−.
I
non
ed
i
me
n
s
i
on
,
a
tl
e
a
s
t
,
i
ft
h
e
r
ei
sn
od
i
s
p
e
r
s
i
onwec
a
nc
on
s
t
r
u
c
ta
−,
f
ou
r
i
e
rs
e
r
i
e
soft
h
e
s
es
ol
u
t
i
on
sf
orv
a
r
i
ou
skt
h
a
tc
on
v
e
r
g
e
st
oa
n
ywe
l
l
–b
e
h
a
v
e
d
f
u
n
c
t
i
onofas
i
n
g
l
ev
a
r
i
a
b
l
e
.
[
Not
ei
np
a
s
s
i
n
gt
h
a
t
:
u
(
x
,
t
)=f(
x−v
t
)+g
(
x+v
t
)
(
1
1
.
2
4
)
f
ora
r
b
i
t
r
a
r
ys
moot
hf(
z
)a
n
dg
(
z
)i
st
h
emo
s
tg
e
n
e
r
a
l
s
ol
u
t
i
onoft
h
e1
d
i
me
n
s
i
on
a
l
wa
v
e
e
qu
a
t
i
on
.
An
ywa
v
e
f
or
mt
h
a
tp
r
e
s
e
r
v
e
si
t
ss
h
a
p
ea
n
dt
r
a
v
e
l
sa
l
on
gt
h
ex
a
x
i
sa
ts
p
e
e
dv
i
sas
ol
u
t
i
ont
ot
h
eon
ed
i
me
n
s
i
on
a
l
wa
v
ee
qu
a
t
i
on(
a
sc
a
nb
ev
e
r
i
f
i
e
ddi
r
e
c
t
l
y
,
ofc
ou
r
s
e
)
.
Howb
or
i
n
g
!
Th
e
s
ep
a
r
t
i
c
u
l
a
rh
a
r
mon
i
cs
ol
u
t
i
on
sh
a
v
et
h
i
sf
or
m(
v
e
r
i
f
yt
h
i
s
)
.
]
I
ft
h
e
r
ei
sd
i
s
p
e
r
s
i
on(
wh
e
r
et
h
ev
e
l
oc
i
t
yoft
h
ewa
v
e
si
saf
u
n
c
t
i
onoft
h
ef
r
e
qu
e
n
c
y
)
t
h
e
nt
h
ef
ou
r
i
e
rs
u
p
e
r
p
o
s
i
t
i
oni
sn
ol
on
g
e
rs
t
a
b
l
ea
n
dt
h
el
a
s
te
qu
a
t
i
onn
ol
on
ge
rh
o
l
d
s
.
E
a
c
hf
ou
r
i
e
rc
omp
on
e
n
ti
ss
t
i
l
la
ne
x
p
on
e
n
t
i
a
l
,b
u
ta
l
lt
h
ev
e
l
oc
i
t
i
e
soft
h
ef
ou
r
i
e
r
c
omp
on
e
n
t
sa
r
ed
i
ffe
r
e
n
t
.Asac
on
s
e
qu
e
n
c
e
,a
n
yi
n
i
t
i
a
l
l
yp
r
e
p
a
r
e
dwa
v
ep
a
c
k
e
ts
p
r
e
a
d
s
ou
ta
si
tp
r
op
a
g
a
t
e
s
.We
’
l
l
l
ooka
tt
h
i
ss
h
or
t
l
y(
i
nt
h
eh
ome
wor
k
)i
ns
omed
e
t
a
i
lt
os
e
eh
ow
t
h
i
swor
k
sf
orav
e
r
ys
i
mp
l
e(
g
a
u
s
s
i
a
n
)wa
v
ep
a
c
k
e
tb
u
tf
orn
owwe
’
l
l
mov
eon
.
Not
et
h
a
tEa
n
dBa
r
ec
on
n
e
c
t
e
db
yh
a
v
i
n
gt
os
a
t
i
s
f
yMa
x
we
l
l
’
se
qu
a
t
i
on
se
v
e
ni
f
t
h
ewa
v
ei
st
r
a
v
e
l
l
i
n
gi
nj
u
s
ton
ed
i
r
e
c
t
i
on(
s
a
y
,
i
nt
h
ed
i
r
e
c
t
i
onofau
n
i
tv
e
c
t
orn
ˆ
)
;
we
c
a
n
n
otc
h
oos
et
h
ewa
v
ea
mp
l
i
t
u
d
e
ss
e
p
a
r
a
t
e
l
y
.
Su
p
p
os
e
i
(
k
n
ˆ
·
x
−ωt
)
E
x
(
)= E
e
,
t
i
(
k
n
ˆ
·
x
−ωt
)
Bx
(,
t
) = Be
wh
e
r
eE
,B,a
n
dn
ˆa
r
ec
on
s
t
a
n
tv
e
c
t
or
s(
wh
i
c
hma
yb
ec
omp
l
e
x
,a
tl
e
a
s
tf
ort
h
e
mome
n
t
)
.
2
2
Not
et
h
a
ta
p
p
l
y
i
n
g(
∇ +k)t
ot
h
e
s
es
ol
u
t
i
on
si
nt
h
eHHEl
e
a
d
su
st
o:
2
ω
2
2
2
knˆ·nˆ=µǫω = v
(
1
1
.
2
5
)
a
st
h
ec
on
d
i
t
i
onf
oras
ol
u
t
i
on
.Th
e
nar
e
a
ln
ˆ·n
ˆ=1l
e
a
dst
ot
h
ep
l
a
n
ewa
v
es
ol
u
t
i
on
ω
i
n
d
i
c
a
t
e
da
b
ov
e
,wi
t
hk= v,wh
i
c
hi
st
h
emos
tf
a
mi
l
i
a
rf
or
moft
h
es
ol
u
t
i
on(
b
u
tn
ot
t
h
eon
l
yon
e
)
!
Th
i
sh
a
smos
t
l
yb
e
e
n“
ma
t
h
e
ma
t
i
c
s
”
,f
ol
l
owi
n
gmor
eorl
e
s
sd
i
r
e
c
t
l
yf
r
om t
h
ewa
v
e
e
qu
a
t
i
on
.Th
es
a
mer
e
a
s
on
i
n
gmi
g
h
th
a
v
eb
e
e
na
p
p
l
i
e
dt
os
ou
n
dwa
v
e
s
,wa
t
e
rwa
v
e
s
,
wa
v
e
sonas
t
r
i
n
g
,or“
wa
v
e
s
”u
(
x
,t
)ofn
ot
h
i
n
gi
np
a
r
t
i
c
u
l
a
r
.Nowl
e
t
’
su
s
es
omep
h
y
s
i
c
s
a
n
ds
e
ewh
a
ti
tt
e
l
l
su
sa
b
ou
tt
h
ep
a
r
t
i
c
u
l
a
re
l
e
c
t
r
oma
g
n
e
t
i
cwa
v
e
st
h
a
tf
ol
l
ow f
r
om
Ma
x
we
l
l
’
se
qu
a
t
i
on
st
u
r
n
e
di
n
t
ot
h
ewa
v
ee
qu
a
t
i
on
.Th
e
s
ewa
v
e
sa
l
ls
a
t
i
s
f
ye
a
c
hof
Ma
x
we
l
l
’
se
qu
a
t
i
on
ss
e
p
a
r
a
t
e
l
y
.
F
ore
x
a
mp
l
e
,
f
r
omGa
u
s
s
’
L
a
wswes
e
ee
.
g
.
t
h
a
t
:
∇
·
E
·
Eei(knˆ·x−ωt)
∇·E
=0
=0
·
x
−ωt
)
i
(
k
n
ˆ
∇e
=0
i
k
Ene
=0
·
x
−ωt
)
i
(
knˆ
(
1
1
.
2
6
)
·
or(
d
i
v
i
d
i
n
gou
tn
on
z
e
r
ot
e
r
msa
n
dt
h
e
nr
e
p
e
a
t
i
n
gt
h
er
e
a
s
on
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n
gf
orB)
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ˆ·E=0a
n
dn
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.
(
1
1
.
2
7
)
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t
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onp
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e
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e
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t
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n
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ss
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h
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n
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s
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gon
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et
h
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r
le
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a
r
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d
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g
e
t
s
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√
B= µǫn
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(
1
1
.
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8
)
×
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(
t
h
ei
c
a
n
c
e
l
s
,
k
/
ω=1
/
v= ǫµ
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e
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h
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e
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ds
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r
e
n
g
t
h
s
:
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E1=ǫˆ
ˆ
1E
0,B
1=ǫ
2 õ
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1
1
.
2
9
)
a
n
d
′
′
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E2=ǫˆ
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0,B
2=−
1 õ
(
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1
.
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0
)
4
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oop
s
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omp
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on
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st
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t
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os
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ond
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e(
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me
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omp
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ed
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t
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on
,
n
ˆ
:
1
∗
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1
(
1
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.
3
2
)
∗
= 2µ E×B
=
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2
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2
µ |E0|nˆ
= 2
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ewe
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l
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h
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omb
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n
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on
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)
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1
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.
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4
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5
)
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m.
5
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k
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n
k
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h
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or
e
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h
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tk=k
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omp
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x
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l
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ne
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g
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h
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omp
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orofy
ou
rc
h
oi
c
e
s
u
c
ht
h
a
t
n
ˆ·n
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.
(
1
1
.
3
6
)
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dy
oug
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tt
h
a
t
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t
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n
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yt
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eg
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emos
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s
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ook
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ory
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c
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oug
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e
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t
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n
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ory
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eh
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n
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t
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n
dt
h
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tn
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sc
omp
l
e
x
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e
ni
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t
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e
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s
:
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nˆ
R+i
I
2
5
He
h
,
h
e
h
.
2
(
1
1
.
3
7
)
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(
1
1
.
3
8
)
nˆ
nˆ
.
R·
I=0
(
1
1
.
3
9
)
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n
ˆ
s
tb
eor
t
h
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on
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l
t
on
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dt
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ed
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ffe
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n
c
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i
rs
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a
r
e
smu
s
tb
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e
.
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ore
x
a
mp
l
e
:
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Ia
√
ˆ
(
1
1
.
4
0
)
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ˆ
ˆ
n
j
R=2in
I=1
wor
k
s
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sd
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n
f
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t
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e
n
e
r
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c
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l
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n
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c
s
f
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n
c
t
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on
s
)
:
nˆ=eˆ1coshθ+i
eˆ2si
nhθ
(
1
1
.
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1
)
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e
r
et
h
eu
n
i
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e
c
t
or
sa
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eor
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h
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l
s
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st
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r
a
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u
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h
a
tn·E=0i
s
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i
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A+eˆ3B
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n
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r
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omp
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n
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et
h
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ti
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omp
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e
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on
e
n
t
i
a
l
p
a
r
toft
h
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i
e
l
d
sb
e
c
ome
s
:
n
ˆx
i
(
k
e
n
ˆx n
ˆx
−k ·e
i
(
k
=e
·−ωt
)
I
.
R·−
ωt
)
(
1
1
.
4
3
)
Th
i
si
n
h
o
moge
n
e
ou
sp
l
a
v
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v
ee
x
p
on
e
n
t
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a
l
l
yg
r
owsord
e
c
a
y
si
ns
omedi
r
e
c
t
i
onwh
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l
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r
e
ma
i
n
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n
ga“
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l
a
n
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v
e
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nt
h
eot
h
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r(
p
e
r
p
e
n
di
c
u
l
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r
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r
e
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t
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on
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or
t
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n
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t
e
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y
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t
u
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ep
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t
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c
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n
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at
h
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h
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or(
i
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g
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yn
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u
s
ti
ma
g
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n
e
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ne
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e
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t
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od
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c
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or
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h
emome
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t
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u
tr
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me
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h
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ti
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ef
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e
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ni
n
t
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ti
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e
l
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y
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ma
t
h
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et
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d
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t
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r
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h
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e
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r
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l
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.
11.
1.
4 Pol
a
r
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a
t
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nofPl
a
n
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r
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·
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i
(
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4
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e
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i
(
k
·
x
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E
(
x
,
t
)=(
E+ǫ
ˆ
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++E
−
−
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.
5
2
)
I
ti
sl
e
f
ta
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ov
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s
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h
a
ta
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y
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r
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l
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mp
l
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t
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d
e
s
!
6
I
’
ml
e
a
v
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e
sp
a
r
a
me
t
e
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t
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11.
2 Re
f
l
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c
t
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o
na
n
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a
c
t
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na
taPl
a
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f
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Su
p
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p
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c
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ti
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i
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ome
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yf
orr
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ma
n
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c
e
√
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e
t
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n
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ǫ=ω/
v
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f
r
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c
t
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n
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st
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′
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v.
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et
h
a
tt
h
ef
r
e
qu
e
n
c
yoft
h
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v
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si
st
h
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a
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t
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h
a
t
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sy
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e
l
d
st
h
ef
ol
l
o
wi
n
gf
or
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ort
h
ev
a
r
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ou
swa
v
e
s
:
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n
c
i
d
e
n
tWa
v
e
i
(
k
·
x
−ωt
)
0
E=E
e
B = √µǫ k×E
k
(
1
1
.
5
3
)
(
1
1
.
5
4
)
Re
f
r
a
c
t
e
dWa
v
e
i
(
k·
x
−
ωt
)
′ =E
′
0
E
e′
B′ =
′×E
′
′′ k
′
µǫ
i
(
k·
x
−
ωt
)
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′
0
=E
e ′′
B′
′
= √µǫ
(
1
1
.
5
6
)
k
Re
f
l
e
c
t
e
dWa
v
e
E
′
′
(
1
1
.
5
5
)
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′
′
′
k×E
(
1
1
.
5
7
)
(
1
1
.
5
8
)
k
Ou
rg
oa
li
st
oc
omp
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t
e
l
yu
n
d
e
r
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t
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omp
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r
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v
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h
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n
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i
d
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n
twa
v
e
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i
si
sd
on
eb
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t
c
h
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n
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h
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c
r
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st
h
eb
ou
n
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r
f
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e
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et
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s
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t
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t
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h
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h
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t
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l
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c ma
t
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h
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n
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d wi
t
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h
a
n
g
i
n
g
)
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a
r
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z
a
t
i
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nt
h
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d
i
u
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e
s
et
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n
d
sofma
t
c
h
i
n
gl
e
a
dt
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i
s
t
i
n
c
ta
n
dwe
l
l
k
n
ownr
e
s
u
l
t
s
.
11.
2.
1 Ki
n
e
ma
t
i
c
sa
n
dSn
e
l
l
’
sL
a
w
Th
ep
h
a
s
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c
t
or
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l
l
t
h
r
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v
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smu
s
tb
ee
qu
a
l
ont
h
ea
c
t
u
a
l
b
ou
n
d
a
r
yi
t
s
e
l
f
,
h
e
n
c
e
:
′
′
′
(
kx
·)
kx
·)
kx
·)
z
=
0=(
z
=
0=(
z
=
0
(
1
1
.
5
9
)
a
sak
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n
e
ma
t
i
cc
o
n
s
t
r
a
i
n
tf
ort
h
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v
et
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ec
on
s
i
s
t
e
n
t
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a
ti
s
,
t
h
i
sh
a
sn
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h
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n
gt
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o
wi
t
h“
p
h
y
s
i
c
s
”p
e
rs
e
,
i
ti
sj
u
s
tama
t
h
e
ma
t
i
c
a
l
r
e
qu
i
r
e
me
n
tf
ort
h
ewa
v
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e
s
c
r
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p
t
i
ont
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k
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s
e
qu
e
n
t
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n
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u
s
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r
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ome
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r
yofB0×n
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e
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t
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n
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′
′
E
0+E
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0
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1
1
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6
7
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si
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on
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n
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os
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n(
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9
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n
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r
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te
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t
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on
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′
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E
µ 0− 0
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e
c
ta
l
lt
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ms
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µ
i
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′
′
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θi)
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′
2
n
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′
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4
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′
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n
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i
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′
E
0 =E
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6
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i
n
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n
nc
os
(
θ
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i
′
E
0 =E
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θ
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i
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′
E
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θ
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i
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c
os
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θ
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i
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2
′
2
2
2
n −n s
i
n(
θ
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i
n −n s
i
n(
θi
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(
11.
80)
(
11.
81)
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i
n(
θ
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i
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el
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e
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n
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E
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3
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on
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edy
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mi
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ome
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t
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or
t
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n
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t
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v
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na
c
t
u
a
l
p
h
y
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c
a
l
me
di
a
.
11.
3.
1 St
a
t
i
cCa
s
e
Re
c
a
l
l
,
(
f
r
oms
e
c
t
i
on
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n
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nJ
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c
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h
a
twh
e
nt
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t
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i
cf
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l
dp
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e
t
r
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t
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me
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i
u
mma
d
eofb
ou
n
dc
h
a
r
g
e
s
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tp
o
l
a
r
i
z
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st
h
os
ec
h
a
r
g
e
s
.Th
ec
h
a
r
g
e
st
h
e
ms
e
l
v
e
s
t
h
e
np
r
od
u
c
eaf
i
e
l
dt
h
a
top
p
os
e
s
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n
dh
e
n
c
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ys
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p
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r
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e
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h
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p
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ek
e
ya
s
s
u
mp
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nt
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e
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on
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st
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z
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t
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m
wa
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rf
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n
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t
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l
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di
nt
h
ev
i
c
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n
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t
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h
ea
t
oms
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i
n
e
a
r
i
t
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e
s
p
on
s
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se
a
s
i
l
ymo
de
l
l
e
db
ya
s
s
u
mi
n
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a
r
mon
i
c(
l
i
n
e
a
r
)
r
e
s
t
or
i
n
gf
or
c
e
:
2
F=−mωx
0
(
1
1
.
1
0
1
)
a
c
t
i
n
gt
op
u
l
lac
h
a
r
g
eei
n
t
oan
e
wn
e
u
t
r
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le
qu
i
l
i
b
r
i
u
mi
nt
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ep
r
e
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n
c
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ne
l
e
c
t
r
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c
f
i
e
l
dv
Ea
c
t
i
n
gonap
r
e
s
u
me
dc
h
a
r
g
ee
.
Th
ef
i
e
l
de
x
e
r
t
saf
or
c
e
Fe=e
E
,
s
o:
2
eE−mωx0 =0
(
1
1
.
1
0
2
)
i
st
h
ec
on
d
i
t
i
onf
ore
q
u
i
l
i
b
r
i
u
m.Th
edi
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ol
emome
n
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h
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s(
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r
e
s
u
me
d
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e
c
u
l
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r
s
y
s
t
e
mi
s
2
pmol=x
e=
2
e
1e
E=
2
2
ǫ0E=γ
ǫ0E
mol
mω0
ǫ0mω0
wh
e
r
eγ
i
st
h
e“
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e
c
u
l
a
rp
ol
a
r
i
z
a
b
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l
i
t
y
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ns
u
i
t
a
b
l
eu
n
i
t
s
.
mo
l
(
1
1
.
1
0
3
)
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a
l
mol
e
c
u
l
e
s
,
ofc
ou
r
s
e
,
h
a
v
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n
yb
ou
n
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h
a
r
g
e
s
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e
a
c
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c
ha
te
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l
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b
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u
m
h
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sa
na
p
p
r
ox
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ma
t
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yl
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n
e
a
rr
e
s
t
or
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n
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or
c
ewi
t
hi
t
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a
t
u
r
a
lf
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e
qu
e
n
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y
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e
n
e
r
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l
mode
l
ofmol
e
c
u
l
a
rp
ol
a
r
i
z
a
b
i
l
i
t
yi
s
:
γ =
mol
2
e
1
i
ǫ0
2
mi
ωi .
i
(
1
1
.
1
0
4
)
Th
i
si
sf
oras
i
n
gl
emol
e
c
u
l
e
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c
t
u
a
lme
di
u
mc
on
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i
s
t
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e
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l
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e
ru
n
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t
v
ol
u
me
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r
om t
h
el
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n
e
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ra
p
p
r
ox
i
ma
t
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ony
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t
a
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ne
qu
a
t
i
onf
ort
h
et
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l
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ol
a
r
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z
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t
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on(
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i
p
ol
emome
n
tp
e
ru
n
i
tv
ol
u
me
)oft
h
ema
t
e
r
i
a
l
:
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3P
P=Nγ
E+
molǫ
0
(
1
1
.
1
0
5
)
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e
qu
a
t
i
on4
.
6
8
)wh
e
r
et
h
ef
a
c
t
orof1
/
3c
ome
sf
r
oma
v
e
r
a
g
i
n
gt
h
el
i
n
e
a
rr
e
s
p
on
s
eov
e
r
a“
s
p
h
e
r
i
c
a
l
”mol
e
c
u
l
e
.
Th
i
sc
a
nb
ep
u
ti
nma
n
yf
or
ms
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ore
x
a
mp
l
e
,
u
s
i
n
gt
h
ed
e
f
i
n
i
t
i
onoft
h
e
(
d
i
me
n
s
i
on
l
e
s
s
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l
e
c
t
r
i
cs
u
s
c
e
p
t
i
b
i
l
i
t
y
:
P=ǫ0χ
E
e
wef
i
n
dt
h
a
t
:
χ
e=
(
1
1
.
1
0
6
)
Nγ
mol
Nγ
.
(
1
1
.
1
0
7
)
MOL
1− 3
Th
es
u
s
c
e
p
t
i
b
i
l
i
t
yi
son
eoft
h
emos
tof
t
e
nme
a
s
u
r
e
dord
i
s
c
u
s
s
e
dqu
a
n
t
i
t
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e
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p
h
y
s
i
c
a
l
me
d
i
ai
nma
n
yc
on
t
e
x
t
sofp
h
y
s
i
c
s
.
Howe
v
e
r
,a
swe
’
v
ej
u
s
ts
e
e
n
,i
nt
h
ec
on
t
e
x
tofwa
v
e
swewi
l
lmos
tof
t
e
nh
a
v
e
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c
a
s
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ont
ou
s
ep
ol
a
r
i
z
a
b
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l
i
t
yi
nt
e
r
msoft
h
ep
e
r
mi
t
t
i
v
i
t
yoft
h
eme
d
i
u
m,
ǫ.
Re
c
a
l
l
t
h
a
t
:
D=ǫE=ǫ0E+P=ǫ0(
1+χ
)
E
(
1
1
.
1
0
8
)
e
F
r
omt
h
i
swec
a
ne
a
s
i
l
yf
i
n
dǫi
nt
e
r
mofχ
:
e
ǫ=ǫ0(
1+χe)
(
1
1
.
1
0
9
)
F
r
om ak
n
owl
e
d
g
eofǫ(
i
nt
h
er
e
g
i
meofop
t
i
c
a
lf
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e
qu
e
n
c
i
e
swh
e
r
eµ≈µ
orma
n
y
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ma
t
e
r
i
a
l
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n
t
e
r
e
s
t
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a
ne
a
s
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l
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t
a
i
n
,
e
.
g
.
t
h
ei
n
d
e
xofr
e
f
r
a
c
t
i
on
:
c
or
√µǫ
ǫ
n= v =√µ0ǫ0 ≈
n=
ǫ0 ≈ 1+χ
e
γ
MOL
1+2N3
Nγ
MOL
1−
(
1
1
.
1
1
0
)
(
1
1
.
1
1
1
)
3
i
fNa
n
dγ
r
ek
n
ownora
tl
e
a
s
ta
p
p
r
ox
i
ma
t
e
l
yc
omp
u
t
a
b
l
eu
s
i
n
gt
h
e(
s
u
r
p
r
i
s
i
n
g
l
y
mo
la
a
c
c
u
r
a
t
e
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x
p
r
e
s
s
i
ona
b
ov
e
.
Somu
c
hf
ors
t
a
t
i
cp
ol
a
r
i
z
a
b
i
l
i
t
yofi
n
s
u
l
a
t
or
s–i
ti
sr
e
a
di
l
yu
n
de
r
s
t
a
n
da
b
l
ei
n
t
e
r
msofr
e
a
lp
h
y
s
i
c
sofp
u
s
h
e
sa
n
dp
u
l
l
s
,
a
n
dt
h
es
e
mi
qu
a
n
t
i
t
a
t
i
v
emod
e
l
son
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s
e
s
t
ou
n
d
e
r
s
t
a
n
di
twor
kqu
i
t
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l
l
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v
e
r
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e
a
lf
i
e
l
d
sa
r
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n
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ts
t
a
t
i
c
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n
dr
e
a
lma
t
e
r
i
a
l
s
a
r
e
n
’
ta
l
l
i
n
s
u
l
a
t
or
s
.
Soweg
ot
t
a
a
)Modi
f
yt
h
emod
e
l
t
oma
k
ei
tdy
n
a
mi
c
.
b
)E
v
a
l
u
a
t
et
h
emod
e
l
(
mor
eorl
e
s
sa
sa
b
ov
e
,
b
u
twe
’
l
l
h
a
v
et
owor
kh
a
r
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e
r
)
.
c
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d
e
r
s
t
a
n
dwh
a
t
’
sg
oi
n
gon
.
L
e
t
’
sg
e
ts
t
a
r
t
e
d
.
11.
3.
2 Dy
n
a
mi
cCa
s
e
Th
eob
v
i
ou
sg
e
n
e
r
a
l
i
z
a
t
i
onoft
h
es
t
a
t
i
cmod
e
lf
ort
h
ep
ol
a
r
i
z
a
t
i
oni
st
oa
s
s
u
mead
a
mpe
d
l
i
n
e
a
rr
e
s
p
on
s
et
oah
a
r
mo
n
i
c(
p
l
a
n
ewa
v
e
)d
r
i
v
i
n
ge
l
e
c
t
r
i
cf
i
e
l
d
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a
ti
s
,e
v
e
r
ymol
e
c
u
l
e
wi
l
lb
ev
i
e
we
da
sac
ol
l
e
c
t
i
onofd
a
mp
e
d
,dr
i
v
e
n(
c
h
a
r
g
e
d
)h
a
r
mon
i
cos
c
i
l
l
a
t
or
s
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g
n
e
t
i
c
a
n
dn
on
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i
n
e
a
re
ffe
c
t
swi
l
l
b
en
e
g
l
e
c
t
e
d
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i
si
sv
a
l
i
df
orav
a
r
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e
t
yofma
t
e
r
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a
l
ss
u
b
j
e
c
t
e
dt
o
1
1
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we
a
k
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a
r
mon
i
cE
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i
e
l
d
s wh
i
c
hi
np
r
a
c
t
i
c
e(
wi
t
hop
t
i
c
a
lf
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e
qu
e
n
c
i
e
s
)me
a
n
sn
e
a
r
l
y
e
v
e
r
y
t
h
i
n
gb
u
tl
a
s
e
rl
i
g
h
t
.
1
2
Th
ee
qu
a
t
i
onofmot
i
on f
oras
i
n
g
l
ed
a
mp
e
d
,
d
r
i
v
e
nh
a
r
mon
i
c
a
l
l
yb
ou
n
dc
h
a
r
g
e
d
e
l
e
c
t
r
oni
s
:
2
¨ ˙
mx +γx+ωx0 =−
e
Ex
(,
t
)
˙
(
1
1
.
1
1
2
)
wh
e
r
eγi
st
h
eda
mp
i
n
gc
on
s
t
a
n
t(
s
o−
mγ
xi
st
h
ev
e
l
oc
i
t
yd
e
p
e
n
d
e
n
td
a
mp
i
n
gf
or
c
e
)
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f
wea
s
s
u
met
h
a
tt
h
ee
l
e
c
t
r
i
cf
i
e
l
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n
d
xa
r
eh
a
r
mon
i
ci
nt
i
mea
tf
r
e
qu
e
n
c
yω(
orf
ou
r
i
e
r
t
r
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n
s
f
or
mt
h
ee
qu
a
t
i
ona
n
df
i
n
di
t
ss
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u
t
i
onf
oras
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n
g
l
ef
ou
r
i
e
rc
omp
on
e
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ome
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b
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c
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e
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e
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n
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et
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e
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me
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e
rNe
wt
on
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a
w,
don
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ty
ou
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r
eh
o
p
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o.
.
.
1
3
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c
e
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t
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l
yh
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ouc
a
nde
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et
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t
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tl
e
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e
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e
n
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nqu
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f
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s
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et
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h
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n
gk
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c
s
,
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n
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v
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r
.
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e
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on
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t
h
i
ns
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a
b
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n
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,
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n
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equ
a
n
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mos
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t
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,
t
h
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o
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n
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t
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r
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l
l
y
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equ
a
n
t
u
mme
c
h
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n
i
c
a
l
.
11.
3.
3 Th
i
n
gst
oNo
t
e
Be
f
or
eweg
oon
,
wes
h
ou
l
du
n
d
e
r
s
t
a
n
daf
e
wt
h
i
n
g
s
:
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)ǫi
sn
owc
o
mp
l
e
x
!
Th
ei
ma
g
i
n
a
r
yp
a
r
ti
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x
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l
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c
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t
l
yc
on
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e
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t
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h
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a
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on
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t
a
n
t
.
b
)Con
s
e
qu
e
n
t
l
ywec
a
nn
ows
e
eh
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h
ei
n
de
xofr
e
f
r
a
c
t
i
on
√µǫ
c
n= v =
√µ0ǫ0 ,
(
1
1
.
1
1
6
)
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nb
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l
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mp
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x
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omp
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r
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bs
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t
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or
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t
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n
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om t
h
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OM (
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r
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a
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r
s
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c
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rh
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e
)
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s
ma
k
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se
n
e
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g
yc
on
s
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e
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n
e
r
g
ya
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s
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yt
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on
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at
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e“
f
r
i
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t
i
on
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l
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mp
i
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or
c
ei
sr
e
mov
e
df
r
om t
h
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Mf
i
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l
da
si
t
p
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op
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g
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t
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st
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g
ht
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eme
d
i
u
m.Th
i
s(
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omp
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s
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n
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ort
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l
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s
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r
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l
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on
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s
t
s
.
c
)Th
et
e
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m
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2 2
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a
saf
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mt
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ty
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ts
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ou
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eme
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t
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p
l
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td
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e
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on
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e
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ti
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s
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f
u
lt
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e
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h
i
si
n
t
oaf
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c
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a
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ma
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h
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c
a
s
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ont
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t
et
h
e
mi
nr
e
a
lp
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b
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e
b
r
ag
i
v
e
su
s
:
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2
2
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1
9
2
=
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d
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fNi
s“
s
ma
l
l
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∼1
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e
c
u
l
e
s
/
c
cf
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a
s
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ma
l
l
(
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u
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a
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n
dt
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d
i
u
mi
sn
e
a
r
l
yt
r
a
n
s
p
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r
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n
ta
tmos
tf
r
e
qu
e
n
c
i
e
s
.
2
3
e
)i
fNi
s“
l
a
r
g
e
”(
∼1
0 mol
e
c
u
l
e
s
/
c
cf
oral
i
qu
i
dors
ol
i
d
)χ
a
nb
equ
i
t
el
a
r
g
ei
n
ec
p
r
i
n
c
i
p
l
e
,
a
n
dn
e
a
rar
e
s
on
a
n
c
ec
a
nb
equ
i
t
el
a
r
g
ea
n
dc
omp
l
e
x
!
Th
e
s
ep
oi
n
t
sa
n
dmor
er
e
qu
i
r
ean
e
wl
a
n
g
u
a
g
ef
ort
h
e
i
rc
on
v
e
n
i
e
n
td
e
s
c
r
i
p
t
i
on
.
Wewi
l
l
n
owp
a
u
s
eamome
n
tt
od
e
v
e
l
opon
e
.
11.
3.
4 An
oma
l
o
u
sDi
s
p
e
r
s
i
on
,
a
n
dRe
s
o
n
a
n
tAb
s
or
p
t
i
on
F
i
g
u
r
e1
1
.
4
:Ty
p
i
c
a
lc
u
r
v
e
si
n
d
i
c
a
t
i
n
gt
h
er
e
a
la
n
di
ma
g
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n
a
r
yp
a
r
t
sofǫ/
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ora
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t
om
wi
t
ht
h
r
e
ev
i
s
i
b
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er
e
s
on
a
n
c
e
s
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et
h
er
e
g
i
on
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n
o
ma
l
ou
s(
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e
s
c
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n
d
i
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g
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e
a
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s
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s
i
oni
nt
h
ei
mme
d
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on
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.
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a
tme
a
n
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h
a
ta
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t
f
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n
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Su
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t
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e
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h
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i
st
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r
mc
a
nd
omi
n
a
t
et
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er
e
s
toft
h
es
u
m.
Wed
e
f
i
n
e
:
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ma
ld
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s
p
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c
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e
a
s
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n
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hi
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i
n
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l
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a
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i
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eb
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e
a
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o
ma
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r
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r
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ome
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s
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e
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me
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r
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l
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s
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r
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b
et
h
i
sn
e
x
t
.
11.
3.
5 At
t
e
n
u
a
t
i
onb
yac
omp
l
e
xǫ
Su
p
p
os
ewewr
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t
e(
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orag
i
v
e
nf
r
e
qu
e
n
c
y
)
k=β+i
Th
e
n
i
k
x
α
.
2
i
βx−
E
()=e =e e
ωx
(
1
1
.
1
1
7
)
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−α
x
2
x
(
1
1
.
1
1
8
)
a
n
dt
h
ei
n
t
e
n
s
i
t
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h
e(
p
l
a
n
e
)wa
v
ef
a
l
l
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k
ee .αme
a
s
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r
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st
h
eda
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i
n
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h
e
p
l
a
n
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v
ei
nt
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eme
d
i
u
m.
L
e
t
’
st
h
i
n
kab
i
ta
b
o
u
tk
:
k= ω = ω n
v c
wh
e
r
e
:
(
1
1
.
1
1
9
)
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n=c
/
v=
(
1
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0
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nmos
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r
a
n
s
p
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r
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n
t
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t
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a
l
s
,
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Th
u
s
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n
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i
ss
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mp
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la
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c
t
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ti
se
a
s
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os
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et
h
a
t
:
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2
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Rek =β −
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n
d
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4 =c
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m
c
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on
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g
a
i
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u
emos
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nt
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a
l
s
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nt
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u
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i
t
e
:
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(
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1
2
4
)
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Reǫ(
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n
d
ǫ
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c
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i
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h
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h
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omt
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h
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v
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orwh
i
c
hi
s“
u
n
i
v
e
r
s
a
l
”
.
11.
3.
6 L
o
wF
r
e
q
u
e
n
c
yBe
h
a
v
i
or
Ne
a
rω=0t
h
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a
l
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t
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t
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v
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e
h
a
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p
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d
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onwh
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h
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rorn
ott
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r
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r
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on
a
n
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e
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t
h
e
r
e
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ft
h
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r
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s
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h
e
nǫ(
ω≈0
)c
a
nb
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g
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t
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omp
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omp
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h
a
ta
t
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ft
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h
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a
r
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dt
h
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p
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r
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e
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ou
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l
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t
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ou
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t
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db
yf
)
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Th
e
ni
fwes
t
a
r
tou
twi
t
h
:
b
2
Ne
ǫ(
ω) = ǫ0 1+ m
f
i
2 2
ωi−ω −i
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)
i(
i
2
Ne
f
b
2 2
ωb −ω −i
ωγ
)
b
b (
= ǫ0 1+m
2
+ Nme
2
f
f
−
ω −i
ωγ
f)
f(
2
Nef
f
γ
ω)
=ǫb+i
ǫ0 mω(
0−i
(
1
1
.
1
2
6
)
wh
e
r
eǫbi
sn
owon
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yt
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ec
on
t
r
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u
t
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ou
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p
ol
e
s
.
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a
nu
n
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e
r
s
t
a
n
dt
h
i
sf
r
om
∇×
H=
J
+
d
D
d
t
(
1
1
.
1
2
7
)
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x
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l
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/
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sL
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w)
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t
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r
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l
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r
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u
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t
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c
or
d
i
n
gt
oOh
m’
sL
a
w:
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=
σE
.
(
1
1
.
1
2
8
)
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fwea
s
s
u
meah
a
r
mon
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ct
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med
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ea
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or
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l
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t
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cc
on
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t
a
n
tǫb,
weg
e
t
:
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ωǫb)E
∇×
H=(
σ
i
ω ǫb+i ω E
=−
.
(
1
1
.
1
2
9
)
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h
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h
e
rh
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n
d
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ni
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ra
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r
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t
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et
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l
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on
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h
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l
dE
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nt
h
i
s
c
a
s
e
:
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H=−
i
ωǫE
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Ne
i
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=−
f
f
γ
ω) E
0−i
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(
1
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.
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0
)
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qu
a
t
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n
gt
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et
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rt
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nt
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f
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n
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e
l
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t
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ort
h
ec
on
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c
t
i
v
i
t
y
:
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ǫ0
2
nfe
1
m
(
γ
ω)
0−i
.
(
1
1
.
1
3
1
)
Th
i
si
st
h
eDr
u
d
eMo
d
e
l
wi
t
hn
h
en
u
mb
e
rof“
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r
e
e
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l
e
c
t
r
on
sp
e
ru
n
i
tv
ol
u
me
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t
f=f
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i
sp
r
i
ma
r
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l
yu
s
e
f
u
lf
ort
h
ei
n
s
i
g
h
tt
h
a
ti
tg
i
v
e
su
sc
on
c
e
r
n
i
n
gt
h
e“
c
on
du
c
t
i
v
i
t
y
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e
i
n
g
c
l
os
e
l
yr
e
l
a
t
e
dt
ot
h
ez
e
r
of
r
e
qu
e
n
c
yc
omp
l
e
xp
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r
toft
h
ep
e
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mi
t
t
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v
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t
y
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et
h
a
ta
tω=
0i
ti
sp
u
r
e
l
yr
e
a
l
,
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si
ts
h
ou
l
db
e
,
r
e
c
ov
e
r
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n
gt
h
eu
s
u
a
l
Oh
m’
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a
w.
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on
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l
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d
et
h
a
tt
h
ed
i
s
t
i
n
c
t
i
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e
t
we
e
nd
i
e
l
e
c
t
r
i
c
sa
n
dc
on
d
u
c
t
or
si
sama
t
t
e
rof
p
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r
s
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t
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omt
h
ep
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r
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t
a
t
i
cc
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s
e
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yf
r
omt
h
es
t
a
t
i
cc
a
s
e
,
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c
on
d
u
c
t
i
v
i
t
y
”
i
ss
i
mp
l
yaf
e
a
t
u
r
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e
s
on
a
n
ta
mp
l
i
t
u
d
e
s
.
I
ti
sama
t
t
e
roft
a
s
t
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e
t
h
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rade
s
c
r
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p
t
i
on
i
sb
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t
t
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d
ei
nt
e
r
mso
fd
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l
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c
t
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cc
on
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t
a
n
t
sa
n
dc
on
d
u
c
t
i
v
i
t
yorc
omp
l
e
xd
i
e
l
e
c
t
r
i
c
.
11.
3.
7Hi
ghF
r
e
qu
e
n
c
yL
i
mi
t
;
Pl
a
s
maF
r
e
qu
e
n
c
y
Wa
ya
b
o
v
et
h
eh
i
g
h
e
s
tr
e
s
on
a
n
tf
r
e
qu
e
n
c
yt
h
ed
i
e
l
e
c
t
r
i
cc
on
s
t
a
n
tt
a
k
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mp
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f
or
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c
t
or
i
n
gou
tω>
>ωia
n
dd
oi
n
gt
h
es
u
mt
ot
h
el
owe
s
ts
u
r
v
i
v
i
n
g
or
de
ri
nωp/
ω.
Asb
e
f
or
e
,
wes
t
a
r
tou
twi
t
h
:
ǫ(
ω)= ǫ0
2
Ne
f
i
2 2
ωi−ω −i
ωγ
)
i(
i
1+ m
=ǫ0
≈ǫ0
2
Ne
f
i
1−
ω2m
γ
i
2
ωi
2
1+i ω − ω2)
i(
NZe
2
1− ω m
≈ǫ0
wh
e
r
e
2
ω
(
1
1
.
1
3
2
)
p
2
1− ω
2
2
n
e
p
m
(
1
1
.
1
3
3
)
ω= .
Th
i
si
sc
a
l
l
e
dt
h
ep
l
a
s
maf
r
e
qu
e
n
c
y
,
a
n
di
td
e
p
e
n
d
son
l
yonn=NZ,
t
h
et
ot
a
l
n
u
mb
e
rof
e
l
e
c
t
r
on
sp
e
ru
n
i
tv
ol
u
me
.
Th
ewa
v
en
u
mb
e
ri
nt
h
i
sl
i
mi
ti
sg
i
v
e
nb
y
:
2
2
2
c
k= ω −ωp
(
1
1
.
1
3
4
)
2 22
(
orω =ωp +
ck)
.Th
i
si
sc
a
l
l
e
dad
i
s
p
e
r
s
i
o
nr
e
l
a
t
i
onω(
k
)
.Al
a
r
g
ep
or
t
i
onof
c
on
t
e
mp
or
a
r
ya
n
df
a
mou
sp
h
y
s
i
c
si
n
v
ol
v
e
sc
a
l
c
u
l
a
t
i
n
gd
i
s
p
e
r
s
i
on r
e
l
a
t
i
on
s(
or
e
qu
i
v
a
l
e
n
t
l
ys
u
s
c
e
p
t
i
b
i
l
i
t
i
e
s
,
r
i
g
h
t
?
)f
r
omf
i
r
s
tp
r
i
n
c
i
p
l
e
s
.
I
nc
e
r
t
a
i
np
h
y
s
i
c
a
l
s
i
t
u
a
t
i
on
s(
s
u
c
ha
sap
l
a
s
maort
h
ei
on
os
p
h
e
r
e
)a
l
l
t
h
ee
l
e
c
t
r
on
sa
r
e
e
s
s
e
n
t
i
a
l
l
y“
f
r
e
e
”(
i
nad
e
g
e
n
e
r
a
t
e“
g
a
s
”s
u
r
r
ou
n
d
i
n
gt
h
ep
os
i
t
i
v
ec
h
a
r
g
e
s
)a
n
dr
e
s
on
a
n
t
d
a
mp
i
n
gi
sn
e
g
l
i
b
l
e
.I
nt
h
a
tc
a
s
et
h
i
sr
e
l
a
t
i
onc
a
nh
ol
df
orf
r
e
qu
e
n
c
i
e
swe
l
lb
e
l
owωp(
b
u
t
we
l
la
b
ov
et
h
es
t
a
t
i
cl
i
mi
t
,
s
i
n
c
ep
l
a
s
ma
sa
r
el
owf
r
e
qu
e
n
c
y“
c
on
d
u
c
t
or
s
”
)
.Wa
v
e
si
n
c
i
d
e
n
t
onap
l
a
s
maa
r
er
e
f
l
e
c
t
e
da
n
dt
h
ef
i
e
l
d
si
n
s
i
d
ef
a
l
l
offe
x
p
on
e
n
t
i
a
l
l
ya
wa
yf
r
omt
h
es
u
r
f
a
c
e
.
Not
et
h
a
t
α
p≈
2ωp
c
(
1
1
.
1
3
5
)
s
h
owsh
owe
l
e
c
t
r
i
cf
l
u
xi
se
x
p
e
l
l
e
db
yt
h
e“
s
c
r
e
e
n
i
n
g
”e
l
e
c
t
r
on
s
.
Th
er
e
f
l
e
c
t
i
v
i
t
yofme
t
a
l
si
sc
a
u
s
e
db
ye
s
s
e
n
t
i
a
l
l
yt
h
es
a
meme
c
h
a
n
i
s
m.Ath
i
g
h
f
r
e
qu
e
n
c
i
e
s
,
t
h
edi
e
l
e
c
t
r
i
cc
on
s
t
a
n
tofame
t
a
l
h
a
st
h
ef
or
m
2
ω
p
2
ǫ(
ω)≈ǫ0(
ω)− ω
(
1
1
.
1
3
6
)
2
2 ∗
∗
wh
e
r
eωp =n
e/
m i
st
h
e“
p
l
a
s
maf
r
e
qu
e
n
c
y
”oft
h
ec
on
d
u
c
t
i
one
l
e
c
t
r
on
s
.m i
st
h
e
“
e
ffe
c
t
i
v
e ma
s
s
”oft
h
ee
l
e
c
t
r
on
s
,i
n
t
r
odu
c
e
dt
od
e
s
c
r
i
b
et
h
ee
ffe
c
t
s ofb
i
n
d
i
n
g
p
h
e
n
ome
n
ol
og
i
c
a
l
l
y
.
Me
t
a
l
sr
e
f
l
e
c
ta
c
c
or
d
i
n
gt
ot
h
i
sr
u
l
e(
wi
t
hav
e
r
ys
ma
l
lf
i
e
l
dp
e
n
e
t
r
a
t
i
onl
e
n
g
t
hof
“
s
k
i
nde
p
t
h
”
)a
sl
on
ga
st
h
ed
i
e
l
e
c
t
r
i
cc
on
s
t
a
n
ti
sn
e
g
a
t
i
v
e
;
i
nt
h
eu
l
t
r
a
v
i
ol
e
ti
tb
e
c
ome
s
p
os
i
t
i
v
ea
n
dme
t
a
l
sc
a
nb
e
c
omet
r
a
n
s
p
a
r
e
n
t
.J
u
s
ton
eofma
n
yp
r
ob
l
e
msi
n
v
ol
v
e
di
n
ma
k
i
n
gh
i
g
hu
l
t
r
a
v
i
ol
e
t
,
x
–r
a
ya
n
dg
a
mmar
a
yl
a
s
e
r
s—i
ti
ss
oh
a
r
dt
oma
k
eami
r
r
or
!
10
8
6
4
2
0
0
2
4
6
8
1
0
F
i
g
u
r
e1
1
.
5
:
Th
ed
i
s
p
e
r
s
i
onr
e
l
a
t
i
onf
orap
l
a
s
ma
.
F
e
a
t
u
r
e
st
on
ot
e
:
Ga
pa
tk=0
,
a
s
y
mp
t
ot
i
c
a
l
l
yl
i
n
e
a
rb
e
h
a
v
i
or
.
11.
4 Pe
n
e
t
r
a
t
i
o
no
fWa
v
e
sI
n
t
oaCo
n
d
u
c
t
o
r–Sk
i
n
De
p
t
h
11.
4.
1 Wa
v
eAt
t
e
n
u
a
t
i
o
ni
nTwoL
i
mi
t
s
Re
c
a
l
l
f
r
oma
b
ov
et
h
a
t
:
∇×H=−
i
ωǫE=−
i
ω
σ
ǫb+iω E
.
(
1
1
.
1
3
7
)
1+i σ
(
1
1
.
1
3
8
)
Th
e
n
:
2
2
2
2
k =ω =µǫω =µǫbω
Al
s
ok=β+i
α
2
v2
ωǫb
s
ot
h
a
t
2
α
2
k=
2
β− 4
σ
2
ωǫb
+i
α
β=µ
ǫbω 1+i
(
1
1
.
1
3
9
)
Oop
s
.
Tode
t
e
r
mi
n
eαa
n
dβ,
weh
a
v
et
ot
a
k
et
h
es
qu
a
r
er
ootofac
omp
l
e
xn
u
mb
e
r
.
How
d
oe
st
h
a
twor
ka
g
a
i
n
?Se
et
h
ea
p
p
e
n
d
i
xonCo
mp
l
e
xNu
mb
e
r
s
.
.
.
I
nma
n
yc
a
s
e
swec
a
np
i
c
kt
h
er
i
g
h
tb
r
a
n
c
hb
ys
e
l
e
c
t
i
n
gt
h
eon
ewi
t
ht
h
er
i
g
h
t
(
d
e
s
i
r
e
d
)b
e
h
a
v
i
oronp
h
y
s
i
c
a
l
g
r
ou
n
d
s
.
I
fwer
e
s
t
r
i
c
tou
r
s
e
l
v
e
st
ot
h
et
wos
i
mp
l
ec
a
s
e
s
wh
e
r
eωi
sl
a
r
g
eorσi
sl
a
r
g
e
,i
ti
st
h
eon
ei
nt
h
ep
r
i
n
c
i
p
l
eb
r
a
n
c
h(
u
p
p
e
rh
a
l
fp
l
a
n
e
,
a
b
ov
eab
r
a
n
c
hc
u
ta
l
on
gt
h
er
e
a
l
a
x
i
s
.F
r
omt
h
el
a
s
te
qu
a
t
i
ona
b
ov
e
,
i
fweh
a
v
eap
oor
c
on
du
c
t
or(
ori
ft
h
ef
r
e
qu
e
n
c
yi
smu
c
hh
i
g
h
e
rt
h
a
nt
h
ep
l
a
s
maf
r
e
qu
e
n
c
y
)a
n
dα≪ β,
t
h
e
n
:
β ≈ √ µǫbω
µ
(
1
1
.
1
4
0
)
ǫbσ
(
1
1
.
1
4
1
)
α ≈
−
i
βn
ˆ
·
E
a
n
dt
h
ea
t
t
e
n
u
a
t
i
on(
r
e
c
a
l
lt
h
a
tE=E
e α2 e
)i
si
n
d
e
p
e
n
d
e
n
toff
r
e
qu
e
n
c
y
.Th
eot
h
e
r
0
l
i
mi
tt
h
a
ti
sr
e
l
a
t
i
v
e
l
ye
a
s
yi
sago
o
dc
on
du
c
t
or
,
σ≫ωǫb.
I
nt
h
a
t
c
a
s
et
h
ei
ma
gi
n
a
r
yt
e
r
md
omi
n
a
t
e
sa
n
dwes
e
et
h
a
t
α
β≈ 2
(
1
1
.
1
4
2
)
or
µσω
β ≈
2
(
1
1
.
1
4
3
)
α ≈
2µσω
(
1
1
.
1
4
4
)
Th
u
s
µσω
k=(
1+i
)
(
1
1
.
1
4
5
)
2
i
k
(
n
ˆ
·
x
−i
ωt
Re
c
a
l
l
t
h
a
ti
fwea
p
p
l
yt
h
e∇op
e
r
a
t
ort
oE
e
weg
e
t
:
s
oE
n
dH0a
r
en
ot
0a
i
np
h
a
s
e(
u
s
i
n
gt
h
e
∇·E = 0
i
k
E
n
ˆ = 0
0·
i
π/
2
f
a
c
tt
h
a
ti
=e )
.
E
nˆ = 0
0·
a
n
d
∂B
− ∂t
= ∇×E
µσω
i
ωµH0
(
n
ˆ×E
)
(
1+i
)
0
=i
1 σω
H0 = ω
1
=ω
1
2
n
ˆ×E
) √2(
0
µ(
1+i
)
σω
i
π/
4
n
ˆ×E
)
e
0
µ(
(
1
1
.
1
4
6
)
(
1
1
.
1
4
7
)
I
nt
h
ec
a
s
eofs
u
p
e
r
c
on
d
u
c
t
or
s
,σ→ ∞ a
n
dt
h
ep
h
a
s
ea
n
g
l
eb
e
t
we
e
nt
h
e
mi
sπ/
4
.I
n
t
h
i
sc
a
s
eH0≫E(
s
h
owt
h
i
s
!
)a
n
dt
h
ee
n
e
r
g
yi
smos
t
l
yma
gn
e
t
i
c
.
α−
1
F
i
n
a
l
l
y
,
n
ot
ewe
l
lt
h
a
tt
h
equ
a
n
t
i
t
y 2 =δi
sa
ne
x
po
n
e
n
t
i
a
ld
a
mp
i
n
gl
e
n
gt
ht
h
a
t
d
e
s
c
r
i
b
e
sh
owr
a
p
i
d
l
yt
h
ewa
v
ea
t
t
e
n
u
a
t
e
sa
si
tmov
e
si
n
t
ot
h
ec
on
du
c
t
i
n
gme
d
i
u
m.δ
i
sc
a
l
l
e
dt
h
es
k
i
nde
p
t
ha
n
dwes
e
et
h
a
t
:
2 1
2
= =
α β
µσω
δ=
(
1
1
.
1
4
8
)
Wewi
l
l
e
x
a
mi
n
et
h
i
squ
a
n
t
i
t
yi
ns
omed
e
t
a
i
l
i
nt
h
es
e
c
t
i
on
sonwa
v
e
g
u
i
de
sa
n
dop
t
i
c
a
l
c
a
v
i
t
i
e
s
,
wh
e
r
ei
tp
l
a
y
sa
ni
mp
or
t
a
n
tr
ol
e
.
11.
5 Kr
a
me
r
s
Kr
on
i
gRe
l
a
t
i
on
s
Wef
i
n
dKKr
e
l
a
t
i
on
sb
yp
l
a
y
i
n
gl
oop
e
dg
a
me
swi
t
hF
ou
r
i
e
rTr
a
n
s
f
or
ms
.Web
e
g
i
nwi
t
h
t
h
er
e
l
a
t
i
onb
e
t
we
e
nt
h
ee
l
e
c
t
r
i
cf
i
e
l
da
n
dd
i
s
p
l
a
c
e
me
n
ta
ts
omep
a
r
t
i
c
u
l
a
rf
r
e
qu
e
n
c
y
ω:
Dx
(,
ω)=ǫ(
ω)
E
x
(,
ω)
(
1
1
.
1
4
9
)
wh
e
r
ewen
ot
et
h
et
wo(
f
or
wa
r
da
n
db
a
c
k
wa
r
d
)f
ou
r
i
e
rt
r
a
n
s
f
or
mr
e
l
a
t
i
on
s
:
1
Dx( ,
t
)=
2
π
Dx
(,
ω)=
1
√
2π
a
n
dofc
ou
r
s
e
:
E
x
(,
ω)=
√
2
π
1
1
Dx
(,
t
)=
√
=
√
=ǫ0
−∞
2π
−∞
(
1
1
.
1
5
0
)
∞
′
Dx
(,
′
i
ωtd
′
t
)
e
t
(
1
1
.
1
5
1
)
−∞
−i
ωt
∞
,
ω)
e dω
E
x
(
(
1
1
.
1
5
2
)
−∞
∞
′
E
x
(,
′
i
ωtd
′
t
)
e
t
(
1
1
.
1
5
3
)
−∞
−i
ωt
∞
2π
1
d
ω
−∞
√
2π
Th
e
r
e
f
or
e
:
,
ω)
e
Dx
(
1
Ex( ,
t
)=
−i
ωt
∞
√
ǫ(
ω)
Ex
(,
ω)
e dω
∞
∞
−i
ωt
ǫ(
ω)
e
d
ω 1
√
E
x
(,
t
)+
2π
∞
E
x
(
′
′
i
ωtd
′
,
t
)
e
t
−∞
t−τ)
d
τ
G(
τ)
Ex
(,
−∞
wh
e
r
eweh
a
v
ei
n
t
r
od
u
c
e
dt
h
es
u
s
c
e
p
t
i
b
i
l
i
t
yk
e
r
n
e
l
:
(
1
1
.
1
5
4
)
1
∞
ǫ(
ω)
ǫ0 −1
G(
τ
)= 2π
−∞
1
∞
e d
ω= 2π
−
i
ωτ
χ
(
ω)
e
e
−i
ωτ
−∞
d
ω (
1
1
.
1
5
5
)
(
n
ot
i
n
gt
h
a
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ω)=ǫ0(
1+χ
(
ω)
)
)
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i
se
qu
a
t
i
oni
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on
l
oc
a
li
nt
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meu
n
l
e
s
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sa
e
d
e
l
t
af
u
n
c
t
i
on
,
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i
c
hi
nt
u
r
ni
st
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u
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l
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h
ed
i
s
p
e
r
s
i
oni
sc
on
s
t
a
n
t
.
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n
d
e
r
s
t
a
n
dt
h
i
s
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on
s
i
d
e
rt
h
es
u
s
c
e
p
t
i
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l
i
t
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e
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n
e
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oras
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e
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n
c
e
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l
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e
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on
a
n
c
e
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r
ej
u
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ts
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e
r
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os
i
t
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on
)
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nt
h
i
sc
a
s
e
,
r
e
c
a
l
l
t
h
a
t
:
ǫ
χ
e= ǫ0
s
o
2
G(
τ
)=
ωp
2
ωp
2 2
γ0ω
−1=ω0 −ω −i
1
∞
2
2
γ
ω
0
2π −∞ ω0 −ω −i
−i
ωτd
e
ω
(
1
1
.
1
5
6
)
(
1
1
.
1
5
7
)
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i
si
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ni
n
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e
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r
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n
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ri
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e
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t
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e
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a
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t
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cf
or
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n
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er
oot
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e
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t
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h
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et
h
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a
c
t
or
e
d
de
n
omi
n
a
t
ori
nt
e
r
msoft
h
er
oot
s
:
i
γ±
ω1,2= −
2
2
−γ +4ω0
2
(
1
1
.
1
5
8
)
or
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−
i
γ ν
ω
1 γ
=
(
1
1
.
1
5
9
)
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2 ± 0 − 4ω0
2 ± 0
1,
2
wh
e
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eν
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on
ga
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/
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u
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l
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yt
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e
,
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e
me
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e
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n
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t
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t
h
e
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e
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r
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nt
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a
n
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e
c
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u
s
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nt
h
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g
i
n
a
l
h
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r
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l
l
a
t
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twa
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i
s
s
i
p
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t
i
v
e
.
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i
si
si
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or
t
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n
t
.
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e
n
2
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i
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=−
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e
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nt
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r
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ra
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v
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n
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n
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nt
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e
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l
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a
l
l
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u
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li
nb
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a
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e
s
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e
c
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u
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et
h
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r
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e
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n
s
f
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n
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e
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mp
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l
l
t
i
me
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n
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l
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et
h
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n
t
e
g
r
a
n
di
nt
h
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HP,
τ>0a
n
di
fwed
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h
er
e
s
toft
h
e(
f
a
i
r
l
y
s
t
r
a
i
g
h
t
f
or
wa
r
d
)a
l
g
e
b
r
aweg
e
t
:
G(
τ)=ωe
2−γτ s
i
n
(
ν
0
)
p
2
ν
0
Θ(
τ)
(
1
1.
1
61
)
wh
e
r
et
h
el
a
t
t
e
ri
saHe
a
v
i
s
i
def
u
n
c
t
i
ont
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n
f
or
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et
h
eτ>0c
on
s
t
r
a
i
n
t
.
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rl
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i
t
t
l
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e
r
c
i
s
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s
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omp
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n
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u
c
h
y
’
st
h
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e
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g
a
i
n
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s
t
a
r
tb
yn
ot
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n
gt
h
a
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n
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n
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r
ea
l
lr
e
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l
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e
nwec
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ni
n
t
e
g
r
a
t
eb
yp
a
r
t
sa
n
d
f
i
n
dt
h
i
n
g
sl
i
k
e
:
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ω)
′
G(
0
)
G(
0
)
2
ǫ0 −1=i ω − ω
+.
.
.
(
1
1
.
1
6
2
)
∗ ∗
f
r
om wh
i
c
hwec
a
nc
on
c
l
u
d
et
h
a
tǫ(
−
ω)=ǫ(
ω )a
n
dt
h
el
i
k
e
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et
h
ee
v
e
n
/
od
d
i
ma
g
i
n
a
r
y
/
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e
a
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i
l
l
a
t
i
oni
nt
h
es
e
r
i
e
s
.ǫ(
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st
h
e
r
e
f
or
ea
n
a
l
y
t
i
ci
nt
h
eUHPa
n
dwe
c
a
nwr
i
t
e
:
′
ǫ(
z
)
ǫ(
ω)
ǫ0 −1
1
1=
ǫ0 −
2
πiC
′
ω −z
′
d
ω
(
1
1
.
1
6
3
)
Wel
e
tz=ω+i
δwh
e
r
eδ→ 0
ord
e
f
or
mt
h
ei
n
t
e
g
r
a
lab
i
tb
e
l
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h
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i
n
g
u
l
a
r
+(
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oi
n
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h
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(
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x
i
s
)
.
F
r
omt
h
ePl
e
ml
j
Re
l
a
t
i
on
:
1
1
′
πδ(
ω −ω)
(
1
1
.
1
6
4
)
ω i
δ =P ω
ω +i
′
′
− −
−
(
s
e
ee
.
g
.Wy
l
d
,
Ar
f
k
i
n
)
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fwes
u
b
s
t
i
t
u
t
et
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n
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ot
h
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n
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e
g
r
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la
b
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on
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er
e
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la
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i
s
on
l
y
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l
t
a
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n
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ta
n
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b
t
r
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c
ti
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t
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a
n
c
e
laf
a
c
t
orof1
/
2t
h
a
tt
h
u
s
a
p
p
e
a
r
s
,
weg
e
t
:
ω
′
ǫ(
ω) =1+ 1P
ω)
∞ ǫ(
ǫ0
1
− dω′
(
1
1
.
1
6
5
)
′
ω
ǫ0
−∞ ω−
i
π
Al
t
h
ou
g
ht
h
i
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ook
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i
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n
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e
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l
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e
c
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h
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nt
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e
n
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n
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t
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e
a
l
l
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er
e
a
lp
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r
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h
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n
t
e
g
r
a
n
db
e
c
ome
st
h
ei
ma
g
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n
a
r
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r
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h
er
e
s
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l
ta
n
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v
i
c
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e
r
s
a
.
Th
a
ti
s
:
′
Re
ǫ(
ω)
ǫ0
ǫ(
ω)
1
= 1+ P
π
1
ǫ(
ω)
∞
I
m
−∞
∞
Re
ǫ0
′
′
ω−
ω
′
ǫ(
ω)
ǫ0
′
d
ω
(
1
1
.
1
6
6
)
−1
′
ω−
ω
d
ω
I
m ǫ0
=− π P −∞
(
1
1
.
1
6
7
)
Th
e
s
ea
r
et
h
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a
me
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s
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g Re
l
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t
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on
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11.
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a
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k
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on
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p
r
ob
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ms
:
7
.
4
,
7
.
6
,
7
.
1
9
,
7
.
2
1
Al
s
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r
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t =i
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(
1
2
.
8
)
(
1
2
.
9
)
become
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(
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i µcω∇×E
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(
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.
1
2
)
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he
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c ≈ −σ n
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(
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2
.
1
3
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1
4
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∂
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µcω ∂ξ − σ (
n
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1
2
∂Hc)
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n
ˆ× ∂ξ
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i
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ˆ×Hc
)=−
n
ˆ×Hc +i
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ˆ×Hc =0
)
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∂ξ
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i
2
2
∂
n
ˆ×Hc)+ δ (
n
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2
(
1
2
.
1
5
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n
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ˆ = 0
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| = 0
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(
1
2
.
1
6
)
(
1
2
.
1
7
)
(
1
2
.
1
8
)
(
1
2
.
1
9
)
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e
r
e
:
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n
ˆ×Hc)×n
ˆ=H|
|
a
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da
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u
t
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ot
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i
sf
or
mi
st
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e
n
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(
1
2
.
2
0
)
√
±−κ2ξ
Hc(
ξ
)=H0e
(
1
2
.
2
1
)
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et
h
es
u
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e
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e
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nt
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s
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r
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ort
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i
sp
a
r
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i
c
u
l
a
rp
r
ob
l
e
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2
−
κ=
2
i
2
1
− δ =±δ(
−
1+i
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(
1
2
.
2
2
)
(
d
r
a
wt
h
i
sou
ti
np
i
c
t
u
r
e
s
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n
tt
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ont
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r
f
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on
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t
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c
e
n
di
n
gf
r
omt
h
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l
e
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t
r
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cme
d
i
u
m,
wh
i
c
hi
st
h
ep
os
i
t
i
v
eb
r
a
n
c
h
:
√
Hc=H0e
2
−κ
ξ
=H0e
1
−1
+i
)
ξ
δ(
ξ
ξ
−
= H0e
i
δ
eδ
(
1
2
.
2
3
)
i
ξ
/
δ
−
ωt
(
c
on
s
i
d
e
re
)
.
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e
e
dt
of
i
n
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ne
x
p
r
e
s
s
i
onf
orE
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i
c
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yb
a
c
k
s
u
b
s
t
i
t
u
t
i
n
gi
n
t
o
c
Amp
e
r
e
’
sL
a
w:
∂Hc
1
E c
=− σ n
ˆ× ∂ξ
1
E c
1
−
1+i
)n
ˆ×H0e
=− δσ(
−1
+i
)
ξ
δ(
µcω
ξ
−
1−i
)
(
n
ˆ×H0)
e
2
σ(
=
ξ
i
δ
eδ
(
1
2
.
2
4
)
Not
ewe
l
l
t
h
edi
r
e
c
t
i
on
!
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v
i
ou
s
l
yn
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nt
h
i
sa
p
p
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i
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t
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o
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s
tl
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nt
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l
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n
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h
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ors
u
r
f
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c
e
,
j
u
s
tl
i
k
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|
b
e
f
or
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s
c
u
s
s
e
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i
e
l
d
si
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oodc
on
d
u
c
t
or
)
:
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,
Hcn
oti
np
h
a
s
e
,
b
u
tou
tofp
h
a
s
eb
yπ/
4
.
c
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p
i
dd
e
c
a
ya
swa
v
ep
e
n
e
t
r
a
t
e
ss
u
r
f
a
c
e
.
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σ“
l
a
r
g
e
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,
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s
ma
l
l
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oe
n
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r
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yp
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d
c⊥ Hc⊥ n
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t
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n
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h
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r
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f
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e
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a
t
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n
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p
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y
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st
h
e
yg
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c
a
l
l
:
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E−E
)=0
c
(
1
2
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2
5
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a
tt
h
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r
f
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e
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t
h
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sy
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ds
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ξ
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c≈
1−i
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n
ˆ×H0)
e
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i
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δ
(
1
2
.
2
6
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u
s
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t
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et
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r
f
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n
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ox
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lt
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h
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r
f
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c
e
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n
dt
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e
r
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ma
l
lB⊥
t
ot
h
es
u
r
f
a
c
eoft
h
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meg
e
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e
r
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l
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rofma
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t
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d
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sE
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n
c
eb
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n
dH|
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h
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r
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c
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s
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l
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n
t
o
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|=0a
|=0a
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on
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c
t
or
!
dPi
n
1
∗
µcωδ
E
c×Hc)=
d
A =− 2Ren ·(
H0|
4|
2
(
1
2
.
2
7
)
wh
e
r
eweHOPEt
h
a
ti
tt
u
r
n
si
n
t
oh
e
a
t
.
L
e
t
’
ss
e
e
:
µcωσ
J
=
σE
=
−ξ
(
1
−
i
)
/
δ
1−i
)
(
n
ˆ×H0)
e
2(
(
1
2
.
2
8
)
s
ot
h
a
tt
h
et
i
mea
v
e
r
a
g
e
dp
owe
rl
os
si
s(
f
r
omOh
m’
sL
a
w)
:
dP
1d
P
dV = Adξ
P
1
∗
1
∗
= 2J·E = 2σJ·J
1 ∞
∗
(
1
2
.
2
9
)
ξ
J·J
= A 2σ 0 d
µcω
∞
H0|
2
= A 2|
µcω
0
−2
ξ
/
δ
d
ξ
e
2
H0|
= A 4|
(
1
2
.
3
0
)
wh
i
c
hj
u
s
th
a
p
p
e
n
st
oc
or
r
e
s
p
on
dt
ot
h
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l
u
xoft
h
ep
oi
n
t
i
n
gv
e
c
t
ort
h
r
ou
g
has
u
r
f
a
c
eA!
F
i
n
a
l
l
y
,
wen
e
e
dt
od
e
f
i
n
et
h
e“
s
u
r
f
a
c
ec
u
r
r
e
n
t
”
:
∞
Keff=
0
J
d
ξ=(
n
ˆ×H)
(
1
2
.
3
1
)
wh
e
r
eHi
sd
e
t
e
r
mi
n
e
dj
u
s
tou
t
s
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de
(
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n
s
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d
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e
c
t
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on
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u
c
t
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mi
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et
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r
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s
ta
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n
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pt
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nt
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r
f
a
c
el
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e
r
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n
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h
a
ti
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l
l
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k
sou
t
.
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e
f
u
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l
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se
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omp
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n
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or
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te
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ta
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b
l
e
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r
om l
e
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t
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t
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h
a
ta
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t
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h
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l
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t
r
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i
g
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t
f
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r
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,
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i
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a
l
!
12.
2 Mu
t
i
l
a
t
e
dMa
x
we
l
l
’
sE
q
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o
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s
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r
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r
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ooka
tt
h
ep
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i
onofwa
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u
me
sofs
p
a
c
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n
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omewa
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d
u
c
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s
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lg
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r
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s
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met
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u
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d
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l
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ea
b
l
et
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h
i
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nt
h
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r
s
l
e
e
p
:
2
2
(
∇ +µǫω ) EorB
(
1
2
.
3
2
)
=0
Wel
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t
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ma
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k
e
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i
k
z
−i
ωt
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x
(,
t
) = E(
ρ,
φ)
e
(
1
2
.
3
3
)
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(,
t
) = B(
ρ,
φ)
e
(
1
2
.
3
4
)
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k
z
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ωt
s
ot
h
a
tt
h
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ee
qu
a
t
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onb
e
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ome
s
:
2
2
2
∇⊥+(
µǫω −k)
2
2
EorB
(
1
2
.
3
5
)
=0
∂
(
Not
et
h
a
t∇⊥=∇ −∂z22)
.
Re
s
ol
v
ef
i
e
l
d
si
n
t
oc
omp
on
e
n
t
s⊥a
n
d|
|
t
oz
:
E =E
ˆ+(
z
ˆ×E
)×z
ˆ=
zz
(
1
2
.
3
6
)
E
z+E
⊥
B =Bzz
ˆ+(
z
ˆ×B)×z
ˆ=
Bz+B⊥
(
1
2
.
3
7
)
(
1
2
.
3
8
)
(
d
e
f
i
n
i
n
gE
n
dE
t
c
.i
nf
a
i
r
l
yob
v
i
ou
swa
y
s
)
.Now wet
r
yt
owr
i
t
eMa
x
we
l
l
’
s
za
⊥e
e
qu
a
t
i
on
si
nt
e
r
msoft
h
e
s
ef
i
e
l
dc
omp
on
e
n
t
s
,a
s
s
u
mi
n
gt
h
a
tt
h
eo
n
l
yz
d
e
p
e
n
d
e
n
c
e
±
i
k
z
p
e
r
mi
t
t
e
di
se .
Th
i
si
s
n
’
tt
r
i
v
i
a
l
t
od
o–l
e
t
’
ss
t
a
r
twi
t
hF
a
r
a
da
y
’
sl
a
w,
f
ore
x
a
mp
l
e
:
∂B
ωB
∂
t =i
∇×E=−
I
fwep
r
oj
e
c
tou
tt
h
ezc
omp
on
e
n
tofb
ot
hs
i
de
sweg
e
t
:
z
ˆ·
ωBz
z
ˆ·(
∇×E
) =i
∂E
∂E
∂E
∂E
z
y
x
z
∂
y − ∂z x
ˆ+ ∂z − ∂x
y
ˆ+
∂E
∂E
y
x
∂x − ∂y z
ˆ
ωBz
=i
∂E
∂E
y
x
∂x − ∂y
ωBz
=i
z
ˆ·(
∇⊥×E
⊥)
ωBz
=i
(
1
2
.
3
9
)
a
son
l
yt
h
e⊥c
omp
on
e
n
t
soft
h
ec
u
r
l
c
on
t
r
i
b
u
t
et
ot
h
ezdi
r
e
c
t
i
on
.
Si
mi
l
a
r
l
y
:
∂E
z
z
ˆ×
z
ˆ×(
∇×E
)
∂E
y
∂E
x
=i
ω(
z
ˆ×B)
∂E
z
∂y − ∂z x
ˆ+ ∂z − ∂x y
ˆ
∂E
∂E
y
x
∂y
∂
z
z
ˆ =i
ω(
z
ˆ×B⊥)
∂x − ∂y
∂E
z−
∂E
y
y
ˆ−
+
∂E
∂E
x
z
∂z − ∂x
x
ˆ=i
ω(
z
ˆ×B⊥)
∂E
⊥
ω(
z
ˆ×B⊥)=
∂z +i
∇
⊥E
z
(
1
2
.
4
0
)
(
wh
e
r
ez
ˆ×B=z
ˆ×B⊥,
ofc
ou
r
s
e
)
.
Ou
c
h
!L
ook
sl
i
k
ewor
k
i
n
gt
h
r
ou
g
ht
h
ec
u
r
lt
e
r
mwi
s
ei
sac
e
r
t
a
i
na
mou
n
tofp
a
i
n
!
Howe
v
e
r
,n
owt
h
a
twe
’
v
ed
on
ei
ton
c
e(
a
n
ds
e
eh
owi
tg
oe
s
)Amp
e
r
e
’
sl
a
ws
h
ou
l
db
e
s
t
r
a
i
g
h
t
f
or
wa
r
d:
i
ωDz
z
ˆ·(
∇×H) = −
i
ωµǫE
z
ˆ·(
∇⊥×B⊥) = −
z
a
n
d
z
ˆ×(
∇×H)=−
i
ω(
z
ˆ×D)
∂B⊥
∂z
−i
ωµǫ(
z
ˆ×E
⊥) =
∇
⊥B
z
F
i
n
a
l
l
y
,
weh
a
v
eGa
u
s
s
’
sL
a
w(
s
)
:
∇·E = 0
∂E
z
∇⊥·
E
⊥+ ∂
z =0
∇⊥·
E
⊥
∂E
z
= − ∂z
a
n
di
d
e
n
t
i
c
a
l
l
y
,
∂
Bz
∇⊥·B⊥=−
∂z
L
e
t
’
sc
ol
l
e
c
ta
l
l
oft
h
e
s
ei
nj
u
s
ton
ep
l
a
c
en
ow:
∇⊥·
E
⊥
= −∂Ez
(
1
2
.
4
1
)
∇⊥·
B⊥
= −∂Bz
(
1
2
.
4
2
)
∂z
z
ˆ·(
∇⊥×B⊥)
z
ˆ·(
∇⊥×E
⊥)
∂ B⊥
ωµǫ(
z
ˆ×E
⊥)
∂
z −i
∂
E
⊥
∂z
= −
i
ωµǫE
z
= i
ωBz
= ∇⊥Bz
+i
ω(
z
ˆ×B⊥)
∂z
= ∇⊥Ez
(
1
2
.
4
3
)
(
1
2
.
4
4
)
(
1
2
.
4
5
)
(
1
2
.
4
6
)
Ge
e
,
on
l
yaf
e
wp
a
ge
sofa
l
g
e
b
r
at
oob
t
a
i
ni
nas
h
o
r
t
e
n
e
dwa
ywh
a
tJ
a
c
k
s
onj
u
s
tp
u
t
s
downi
nt
h
r
e
es
h
or
tl
i
n
e
s
.Hop
e
f
u
l
l
yt
h
ep
oi
n
ti
sc
l
e
a
r–t
o“
g
e
t
”al
otoft
h
i
sy
ouh
a
v
et
o
s
oon
e
rorl
a
t
e
rwor
ki
ta
l
l
ou
t
,
h
owe
v
e
rl
on
gi
tma
yt
a
k
ey
ou
,
ory
ou
’
l
l
e
n
du
pme
mor
i
z
i
n
g(
or
t
r
y
i
n
gt
o)a
l
l
ofJ
a
c
k
s
on
’
sr
e
s
u
l
t
s
.Some
t
h
i
n
gt
h
a
tmos
tn
or
ma
l
h
u
ma
n
sc
ou
l
dn
e
v
e
rdoi
na
l
i
f
e
t
i
meoft
r
y
i
n
g
.
.
.
Ba
c
kt
owor
k
,
a
st
h
e
r
ei
ss
t
i
l
l
p
l
e
n
t
yt
odo.
12.
3 TE
MWa
v
e
s
Nowwec
a
ns
t
a
r
tl
ook
i
n
ga
twa
v
e
f
or
msi
nv
a
r
i
ou
sc
a
v
i
t
i
e
s
.Su
p
p
os
ewel
e
tEz=Bz=0
.
Th
e
nt
h
ewa
v
ei
nt
h
ec
a
v
i
t
yi
sap
u
r
et
r
a
n
s
v
e
r
s
ee
l
e
c
t
r
oma
gn
e
t
i
c(
TE
M)wa
v
ej
u
s
tl
i
k
e
ap
l
a
n
ewa
v
e
,e
x
c
e
p
tt
h
a
ti
th
a
st
os
a
t
i
s
f
yt
h
eb
ou
n
d
a
r
yc
on
d
i
t
i
on
sofap
e
r
f
e
c
t
c
on
du
c
t
ora
tt
h
ec
a
v
i
t
yb
ou
n
da
r
y
!
Not
ef
r
omt
h
ee
qu
a
t
i
on
sa
b
ov
et
h
a
t
:
∇⊥·
E
⊥
= 0
∇⊥×
E⊥
f
r
omwh
i
c
hwec
a
ni
mme
d
i
a
t
e
l
ys
e
et
h
a
t
:
= 0
2
∇⊥ E
0
⊥=
(
1
2
.
4
7
)
a
n
dt
h
a
t
E
∇φ
⊥=−
(
1
2
.
4
8
)
2
f
ors
omes
u
i
t
a
b
l
ep
ot
e
n
t
i
a
lt
h
a
ts
a
t
i
s
f
i
e
s∇⊥φ =0
.Th
es
ol
u
t
i
onl
ook
sl
i
k
ea
p
r
op
a
g
a
t
i
n
ge
l
e
c
t
r
o
s
t
a
t
i
cwa
v
e
.
F
r
omt
h
ewa
v
ee
qu
a
t
i
onwes
e
et
h
a
t
:
2
2
µǫω =k
or
√
k=±
ω µǫ
wh
i
c
hi
sj
u
s
tl
i
k
eap
l
a
n
ewa
v
e(
wh
i
c
hc
a
np
r
op
a
g
a
t
ei
ne
i
t
h
e
rd
i
r
e
c
t
i
on
,
r
e
c
a
l
l
)
.
(
1
2
.
4
9
)
(
1
2
.
5
0
)
Ag
a
i
nr
e
f
e
r
r
i
n
gt
oou
rl
i
s
tofmu
t
i
l
a
t
e
dMa
x
we
l
l
e
qu
a
t
i
on
sa
b
ov
e
,
wes
e
et
h
a
t
:
i
k
E
i
ω(
z
ˆ×B⊥)
⊥ =−
ωµǫ
D⊥=
−
z
ˆ×H⊥)
k(
√
D⊥=
± µǫ(
z
ˆ×H⊥)
(
1
2
.
5
1
)
orwor
k
i
n
gt
h
eot
h
e
rwa
y
,
t
h
a
t
:
√
B⊥=
±
z
ˆ×E
⊥)
µǫ(
(
1
2
.
5
2
)
s
owec
a
ne
a
s
i
l
yf
i
n
don
ef
r
omt
h
eot
h
e
r
.
TEM wa
v
e
sc
a
n
n
otb
es
u
s
t
a
i
n
e
di
nac
y
l
i
n
d
e
rb
e
c
a
u
s
et
h
es
u
r
r
ou
n
d
i
n
g(
p
e
r
f
e
c
t
,
r
e
c
a
l
l
)c
on
d
u
c
t
ori
se
qu
i
p
ot
e
n
t
i
a
l
.Th
e
r
e
f
or
eE
sz
e
r
oa
si
sB⊥.Howe
v
e
r
,
t
h
e
ya
r
et
h
e
⊥i
d
o
mi
n
a
n
twa
ye
n
e
r
g
yi
st
r
a
n
s
mi
t
t
e
dd
ownac
oa
x
i
a
l
c
a
b
l
e
,
wh
e
r
eap
ot
e
n
t
i
a
l
di
ffe
r
e
n
c
e
i
sma
i
n
t
a
i
n
e
db
e
t
we
e
nt
h
ec
e
n
t
r
a
lc
o
n
du
c
t
ora
n
dt
h
ec
oa
x
i
a
ls
h
e
a
t
h
e
.I
nt
h
i
sc
a
s
et
h
e
f
i
e
l
dsa
r
ev
e
r
ys
i
mp
l
e
,
a
st
h
eEi
sp
u
r
e
l
yr
a
d
i
a
l
a
n
dt
h
eBf
i
e
l
dc
i
r
c
l
e
st
h
ec
on
d
u
c
t
or(
s
o
t
h
ee
n
e
r
g
yg
oe
swh
i
c
hwa
y
?
)wi
t
hn
ozc
omp
on
e
n
t
s
.
F
i
n
a
l
l
y
,
n
ot
et
h
a
ta
l
l
f
r
e
qu
e
n
c
i
e
sa
r
ep
e
r
mi
t
t
e
df
oraTE
Mwa
v
e
.I
ti
sn
ot“
qu
a
n
t
i
z
e
d”
b
yt
h
ea
p
p
e
a
r
a
n
c
eofe
i
g
e
n
v
a
l
u
e
sdu
et
oac
on
s
t
r
a
i
n
i
n
gb
ou
n
d
a
r
yv
a
l
u
ep
r
ob
l
e
m.
12.
4 TEa
n
dTMWa
v
e
s
Not
ewe
l
lt
h
a
tweh
a
v
ewr
i
t
t
e
nt
h
emu
t
i
l
a
t
e
dMa
x
we
l
lEqu
a
t
i
on
ss
ot
h
a
tt
h
ezc
omp
on
e
n
t
s
a
r
ea
l
lo
nt
h
er
i
gh
th
a
n
ds
i
d
e
.I
ft
h
e
ya
r
ek
n
ownf
u
n
c
t
i
on
s
,
a
n
di
ft
h
eon
l
yzde
p
e
n
de
n
c
ei
s
t
h
ec
omp
l
e
xe
x
p
on
e
n
t
i
a
l(
s
owec
a
nd
oa
l
lt
h
ez
de
r
i
v
a
t
i
v
e
sa
n
dj
u
s
tb
r
i
n
gd
owna±
i
k
)t
h
e
n
t
h
et
r
a
n
s
v
e
r
s
ec
omp
o
n
e
n
t
sE
n
d
⊥a
B⊥a
r
ed
e
t
e
r
mi
n
e
d!
+
i
k
z
−i
ωt
I
nf
a
c
t(
f
orp
r
op
a
g
a
t
i
oni
nt
h
e+zd
i
r
e
c
t
i
on
,
e
)
:
i
k
E
ω(
z
ˆ×B⊥)=∇⊥Ez
⊥+i
i
k
(
z
ˆ×E
+i
ωz
ˆ×(
z
ˆ×B⊥)=z
ˆ×∇⊥E
⊥)
z
i
k
(
z
ˆ×E
ωB⊥+z
ˆ×∇⊥E
⊥) =i
z
i
ωµǫE
z
ˆ·(
∇⊥×B⊥) =−
z
(
1
2
.
5
3
)
(
1
2
.
5
4
)
a
n
d
i
k
B⊥−i
ωµ
ǫ(
z
ˆ×E
⊥)=∇⊥B
z
i
k
B⊥−∇⊥Bz = i
ωµǫ(
z
ˆ×E
⊥)
2
k
k
k
(
z
ˆ×E
⊥)
iωµǫB⊥− ωµǫ ∇⊥Bz = i
2
k
k
ωB⊥+z
ˆ×∇⊥E
z
iωµǫB⊥− ωµǫ ∇⊥Bz = i
(
1
2
.
5
5
)
or
i
B⊥
2 2
= µǫω −k
i
k
∇⊥Bz+µǫω(
z
ˆ×∇⊥Ez)
(
1
2
.
5
6
)
2 2
E
∇⊥E
z
ˆ×∇⊥Bz)
⊥
z−ω(
= µǫω −k k
(
1
2
.
5
7
)
(
wh
e
r
ewes
t
a
r
t
e
dwi
t
ht
h
es
e
c
on
de
qu
a
t
i
ona
n
de
l
i
mi
n
a
t
e
dz
ˆ×B⊥ t
og
e
tt
h
es
e
c
on
d
e
qu
a
t
i
onj
u
s
tl
i
k
et
h
ef
i
r
s
t
)
.
Nowc
ome
st
h
er
e
l
a
t
i
v
e
l
yt
r
i
c
k
yp
a
r
t
.Re
c
a
l
lt
h
eb
ou
n
d
a
r
yc
on
d
i
t
i
on
sf
orap
e
r
f
e
c
t
c
on
d
u
c
t
or
:
n
ˆ×(
E−E
)=n
ˆ×E
c
n
ˆ·(
B−Bc)=n
ˆ·B
= 0
= 0
n
ˆ×H
= K
n
ˆ·D
= Σ
Th
e
yt
e
l
l
u
sb
a
s
i
c
a
l
l
yt
h
a
tE(
D)i
ss
t
r
i
c
t
l
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e
r
p
e
n
di
c
u
l
a
rt
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h
es
u
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f
a
c
ea
n
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h
a
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s
s
t
r
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c
t
l
yp
a
r
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e
l
t
ot
h
es
u
r
f
a
c
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h
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d
u
c
t
ora
tt
h
es
u
r
f
a
c
eoft
h
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on
d
u
c
t
or
.
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i
sme
a
n
st
h
a
ti
ti
sn
o
tn
e
c
e
s
s
a
r
yf
orE
rBzb
o
t
ht
ov
a
n
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s
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v
e
r
y
wh
e
r
ei
n
s
i
d
et
h
e
zo
d
i
e
l
e
c
t
r
i
c(
a
l
t
h
ou
g
hb
ot
hc
a
n
,
ofc
ou
r
s
e
,
a
n
dr
e
s
u
l
ti
naTEMwa
v
eorn
owa
v
ea
ta
l
l
)
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l
t
h
a
t
i
ss
t
r
i
c
t
l
yr
e
qu
i
r
e
db
yt
h
eb
ou
n
d
a
r
yc
on
d
i
t
i
on
si
sf
or
E
z|
S=0
(
1
2
.
5
8
)
ont
h
ec
on
du
c
t
i
n
gs
u
r
f
a
c
eS(
i
tc
a
non
l
yh
a
v
ean
or
ma
lc
omp
on
e
n
ts
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h
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omp
on
e
n
tmu
s
tv
a
n
i
s
h
)
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ec
on
d
i
t
i
ononBzi
se
v
e
nwe
a
k
e
r
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tmu
s
tl
i
ep
a
r
a
l
l
e
lt
ot
h
e
s
u
r
f
a
c
ea
n
db
ec
on
t
i
n
u
ou
sa
c
r
os
st
h
es
u
r
f
a
c
e(
wh
e
r
eHc
a
nd
i
s
c
on
t
i
n
u
ou
s
l
yc
h
a
n
g
e
b
e
c
a
u
s
eofK)
.
Th
a
ti
s
:
∂Bz
0
∂n|
S=
(
1
2
.
5
9
)
Wet
h
e
r
e
f
or
eh
a
v
et
wop
os
s
i
b
i
l
i
t
i
e
sf
orn
o
n
z
e
r
oEzorBzt
h
a
tc
a
na
c
ta
ss
ou
r
c
e
t
e
r
mi
nt
h
emu
t
i
l
a
t
e
dMa
x
we
l
l
Equ
a
t
i
on
s
.
12.
4.
1 TMWa
v
e
s
Bz = 0
(
1
2
.
6
0
)
Ez|
S = 0
(
1
2
.
6
1
)
Th
ema
g
n
e
t
i
cf
i
e
l
di
ss
t
r
i
c
t
l
yt
r
a
n
s
v
e
r
s
e
,
b
u
tt
h
ee
l
e
c
t
r
i
cf
i
e
l
di
nt
h
ezd
i
r
e
c
t
i
onon
l
yh
a
s
t
ov
a
n
i
s
ha
tt
h
eb
ou
n
d
a
r
y–e
l
s
e
wh
e
r
ei
tc
a
nh
a
v
eazc
omp
on
e
n
t
.
Th
u
s
:
i
2 2
E
ǫω −k
⊥ = µ
1
2
2
2
2
k
∇⊥E
z
ˆ×∇⊥Bz)
z−ω(
(
µǫω −k)
E
k
∇⊥E
⊥ = i
z
µǫω −k)
E
⊥ = ∇⊥E
z
i
k (
(
1
2
.
6
2
)
wh
i
c
hl
ook
sj
u
s
tp
e
r
f
e
c
tt
os
u
b
s
t
i
t
u
t
ei
n
t
o:
i
2
2
B⊥ = µǫω −k k
∇⊥Bz+µǫω(
z
ˆ×∇⊥Ez)
2 2
(
µ
ǫω −k)
B⊥ = i
µ
ǫω(
z
ˆ×∇⊥E
z)
µǫω
2
2
2
2
(
µǫω −k)
B⊥ = k (
µǫω −k)
(
z
ˆ×E
⊥)
g
i
v
i
n
gu
s
:
(
1
2
.
6
3
)
µǫω
B⊥=
±
or(
a
st
h
eb
ookwou
l
dh
a
v
ei
t
)
:
z
ˆ×E
⊥)
k(
(
1
2
.
6
4
)
ǫω
z
ˆ×E
⊥)
H ⊥=± k(
(
1
2
.
6
5
)
(
wh
e
r
ea
su
s
u
a
l
t
h
et
wos
i
g
n
si
n
d
i
c
a
t
et
h
edi
r
e
c
t
i
onofwa
v
ep
r
op
a
g
a
t
i
on
)
.
Ofc
ou
r
s
e
,wes
t
i
l
lh
a
v
et
of
i
n
da
tl
e
a
s
ton
eoft
h
et
wof
i
e
l
d
sf
ort
h
i
st
odou
sa
n
y
g
ood.
Ordowe
?L
ook
i
n
ga
b
ov
ewes
e
e
:
2
2
(
µǫω −k)
E
⊥
E
k
∇⊥ψ
=i
±
i
k
=
2
2
(
µǫω −k)
⊥
∇ ψ
⊥
(
1
2
.
6
6
)
i
k
z
Wh
e
r
eψ(
x
,
y
)
e =E
Th
i
smu
s
ts
a
t
i
s
f
yt
h
et
r
a
n
s
v
e
r
s
ewa
v
ef
u
n
c
t
i
on
:
z.
2
2
2
∇⊥ +(
µǫω −k)ψ=0
a
n
dt
h
eb
ou
n
d
a
r
yc
on
d
i
t
i
on
sf
oraTMwa
v
e
:
(
1
2
.
6
7
)
ψ|
0
S=
(
1
2
.
6
8
)
TEWa
v
e
s
E
z = 0
(
1
2
.
6
9
)
∂Bz
S = 0
∂n |
(
1
2
.
7
0
)
Th
ee
l
e
c
t
r
i
cf
i
e
l
di
ss
t
r
i
c
t
l
yt
r
a
n
s
v
e
r
s
e
,
b
u
tt
h
ema
g
n
e
t
i
cf
i
e
l
di
nt
h
ez
d
i
r
e
c
t
i
onc
a
nb
e
n
on
z
e
r
o.Doi
n
ge
x
a
c
t
l
yt
h
es
a
mea
l
g
e
b
r
aont
h
es
a
met
woe
qu
a
t
i
on
sa
sweu
s
e
di
nt
h
e
TMc
a
s
e
,
weg
e
ti
n
s
t
e
a
d:
k
z
ˆ×E
⊥)
H⊥=± µω (
(
1
2
.
7
1
)
a
l
on
gwi
t
h
i
k
z
B =
±
i
k
⊥
(
µǫω −k)
2
2
∇ ψ
(
1
2
.
7
2
)
⊥
wh
e
r
eψ(
x
,
y
)
e =Bza
n
d
2
2
2
µǫω −k) ψ=0
∇⊥ +(
(
1
2
.
7
3
)
a
n
dt
h
eb
ou
n
d
a
r
yc
on
d
i
t
i
on
sf
oraTEwa
v
e
:
∂ψ
0
S=
∂n|
(
1
2
.
7
4
)
12.
4.
2 Su
mma
r
yofTE
/
TMwa
v
e
s
Th
et
r
a
n
s
v
e
r
s
ewa
v
ee
qu
a
t
i
ona
n
db
ou
n
d
a
r
yc
on
d
i
t
i
on(
d
i
r
i
c
h
l
e
torn
e
u
ma
n
n
)
a
r
ea
ne
i
ge
n
v
a
l
u
ep
r
o
b
l
e
m.
Wec
a
ns
e
et
wot
h
i
n
g
sr
i
g
h
ta
wa
y
.
F
i
r
s
tofa
l
l
:
2
2
µǫω ≥k
(
1
2
.
7
5
)
orwen
ol
on
g
e
rh
a
v
eawa
v
e
,
weh
a
v
ea
ne
x
p
on
e
n
t
i
a
lf
u
n
c
t
i
ont
h
a
tc
a
n
n
otb
ema
d
et
o
s
a
t
i
s
f
yt
h
eb
ou
n
d
a
r
yc
on
d
i
t
i
on
sont
h
ee
n
t
i
r
es
u
r
f
a
c
e
.
Al
t
e
r
n
a
t
i
v
e
l
y
,
ω2
1
2 2
2
v
p =k ≥ µ
ǫ=v
(
1
2
.
7
6
)
wh
i
c
hh
a
st
h
el
ov
e
l
yp
r
op
e
r
t
y(
a
sap
h
a
s
ev
e
l
oc
i
t
y
)ofb
e
i
n
gf
a
s
t
e
rt
h
a
nt
h
es
p
e
e
dof
l
i
g
h
ti
nt
h
eme
di
u
m!
Top
r
oc
e
e
df
u
r
t
h
e
ri
nou
ru
n
d
e
r
s
t
a
n
d
i
n
g
,
wen
e
e
dt
ol
ooka
ta
na
c
t
u
a
l
e
x
a
mp
l
e–we
’
l
lf
i
n
dt
h
a
ton
l
yc
e
r
t
a
i
nk
nf
orn=1
,2
,3
.
.
.
n
l
lp
e
r
mi
tt
h
e
n=k
0
c
u
t
offwi
b
ou
n
d
a
r
yc
on
d
i
t
i
on
st
ob
es
ol
v
e
d,
a
n
dwe
’
l
l
l
e
a
r
ns
omei
mp
or
t
a
n
tt
h
i
n
g
s
a
b
ou
tt
h
ep
r
op
a
g
a
t
i
n
gs
ol
u
t
i
on
sa
tt
h
es
a
met
i
me
.
12.
5 Re
c
t
a
n
gu
l
a
rWa
v
e
gu
i
d
e
s
Re
c
t
a
n
g
u
l
a
rwa
v
e
g
u
i
de
sa
r
ei
mp
or
t
a
n
tf
ort
wor
e
a
s
on
s
.F
i
r
s
tofa
l
l
,t
h
eL
a
p
l
a
c
i
a
n
op
e
r
a
t
ors
e
p
a
r
a
t
e
sn
i
c
e
l
yi
nCa
r
t
e
s
i
a
nc
oor
d
i
n
a
t
e
s
,
s
ot
h
a
tt
h
eb
ou
n
d
a
r
yv
a
l
u
ep
r
ob
l
e
m
t
h
a
tmu
s
tb
es
ol
v
e
di
sb
ot
hf
a
mi
l
i
a
ra
n
ds
t
r
a
i
g
h
t
f
or
wa
r
d
.Se
c
on
d
,t
h
e
ya
r
ee
x
t
r
e
me
l
y
c
ommoni
na
c
t
u
a
la
p
p
l
i
c
a
t
i
oni
np
h
y
s
i
c
sl
a
b
or
a
t
or
i
e
sf
orp
i
p
i
n
ge
.
g
.mi
c
r
owa
v
e
s
a
r
ou
n
da
se
x
p
e
r
i
me
n
t
a
l
p
r
ob
e
s
.
I
nCa
r
t
e
s
i
a
nc
oor
d
i
n
a
t
e
s
,
t
h
ewa
v
ee
qu
a
t
i
onb
e
c
ome
s
:
2
∂2
∂
2
2
2
2
µǫω −k)ψ=0
∂x +∂y +(
(
1
2
.
7
7
)
Th
i
swa
v
ee
qu
a
t
i
ons
e
p
a
r
a
t
e
sa
n
ds
ol
u
t
i
on
sa
r
ep
r
od
u
c
t
sofs
i
n
,c
osore
x
p
on
e
n
t
i
a
l
f
u
n
c
t
i
on
si
ne
a
c
hv
a
r
i
a
b
l
es
e
p
a
r
a
t
e
l
y
.Tod
e
t
e
r
mi
n
ewh
i
c
hc
omb
i
n
a
t
i
ont
ou
s
ei
ts
u
ffic
e
st
o
l
ooka
tt
h
eBC’
sb
e
i
n
gs
a
t
i
s
f
i
e
d.F
orTMwa
v
e
s
,on
es
ol
v
e
sf
orψ=E
u
b
j
e
c
tt
oE
,
zs
z|
S=0
wh
i
c
hi
sa
u
t
oma
t
i
c
a
l
l
yt
r
u
ei
f
:
mπx
nπy
s
i
n
a
b
E
x
,
y
)=ψmn(
x
,
y
)=E
i
n
z(
0s
(
1
2
.
7
8
)
wh
e
r
eaa
n
dba
r
et
h
edi
me
n
s
i
on
soft
h
exa
n
dys
i
d
e
soft
h
eb
ou
n
da
r
yr
e
c
t
a
n
g
l
ea
n
d
wh
e
r
ei
np
r
i
n
c
i
p
l
em,
n=0
,
1
,
2
.
.
.
.
Howe
v
e
r
,
t
h
ewa
v
e
n
u
mb
e
rofa
n
yg
i
v
e
nmod
e(
g
i
v
e
nt
h
ef
r
e
qu
e
n
c
y
)i
sd
e
t
e
r
mi
n
e
df
r
om:
m2
2
2
2
k =µǫω −π
2
n
2
a
2
+b
+
(
1
2
.
7
9
)
2
wh
e
r
ek >0f
ora“
wa
v
e
”t
oe
x
i
s
tt
op
r
op
a
g
a
t
ea
ta
l
l
.
I
fe
i
t
h
e
ri
n
de
xmorni
sz
e
r
o,
t
h
e
r
e
i
sn
owa
v
e
,
s
ot
h
ef
i
r
s
tmod
et
h
a
tc
a
np
r
op
a
g
a
t
eh
a
sad
i
s
p
e
r
s
i
onr
e
l
a
t
i
onof
:
1
2
2
2 1
2
s
ot
h
a
t
:
a
k
ǫω −π (
11=µ
2
b )
+
(
1
2
.
8
0
)
1
1
)
ω≥ √µǫ a + b =ωc,
TM(
(
1
2
.
8
1
)
π
1
1
2
2
E
a
c
hc
omb
i
n
a
t
i
onofp
e
r
mi
t
t
e
dma
n
dni
sa
s
s
oc
i
a
t
e
dwi
t
hac
u
t
offoft
h
i
ss
or
t–wa
v
e
s
wi
t
hf
r
e
qu
e
n
c
i
e
sg
r
e
a
t
e
rt
h
a
nore
qu
a
lt
ot
h
ec
u
t
offc
a
ns
u
p
p
or
tp
r
op
og
a
t
i
oni
na
l
lt
h
e
mode
swi
t
hl
owe
rc
u
t
offf
r
e
qu
e
n
c
i
e
s
.
I
fwer
e
p
e
a
tt
h
ea
r
g
u
me
n
ta
b
ov
ef
orTEwa
v
e
s(
a
si
sdon
ei
nJ
a
c
k
s
on
,
wh
i
c
hi
swh
y
I
d
i
dTMh
e
r
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e
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l
l
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el
e
db
yn
e
a
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t
r
i
b
u
t
i
on
;
h
owe
v
e
r
,
t
h
i
si
sav
e
r
yi
n
v
ol
v
e
di
s
s
u
et
h
a
twewi
l
l
e
x
a
mi
n
ei
nd
e
t
a
i
l
l
a
t
e
ri
nt
h
es
e
me
s
t
e
r
.
Th
ef
or
moft
h
ec
h
a
r
g
ed
i
s
t
r
i
b
u
t
i
onwewi
l
l
s
t
u
dyf
ort
h
en
e
x
tf
e
wwe
e
k
si
s
:
ρx(,
t
)
Jx(,
t
)
−
i
ωt
()
e
=ρx
−
i
ωt
x
()
e
=J
.
(
1
3
.
1
)
(
1
3
.
2
)
Th
es
p
a
t
i
a
ld
i
s
t
r
i
b
u
t
i
oni
se
s
s
e
n
t
i
a
l
l
y“
a
r
b
i
t
r
a
r
y
”
.Ac
t
u
a
l
l
y
,
wewa
n
ti
tt
oh
a
v
ec
o
mp
a
c
t
s
u
p
p
o
r
twh
i
c
hj
u
s
tme
a
n
st
h
a
ti
tdoe
s
n
’
te
x
t
e
n
dt
oi
n
f
i
n
i
t
yi
na
n
yd
i
r
e
c
t
i
on
.
L
a
t
e
rwewi
l
l
a
l
s
owa
n
ti
tt
ob
es
ma
l
l
wi
t
hr
e
s
p
e
c
tt
oawa
v
e
l
e
n
g
t
h
.
13.
1.
1 Qu
i
c
k
i
eRe
v
i
e
wofCh
a
p
t
e
r6
Re
c
a
l
lt
h
ef
ol
l
owi
n
gmor
p
h
sofMa
x
we
l
l
’
se
qu
a
t
i
on
s
,t
h
i
st
i
mewi
t
ht
h
es
ou
r
c
e
sa
n
d
e
x
p
r
e
s
s
e
di
nt
e
r
msofp
ot
e
n
t
i
a
l
sb
yme
a
n
soft
h
eh
omog
e
n
e
ou
se
qu
a
t
i
on
s
.
1
4
5
Ga
u
s
s
’
sL
a
wf
orma
g
n
e
t
i
s
mi
s
:
(
1
3
.
3
)
∇·
B=
0
Th
i
si
sa
ni
de
n
t
i
t
yi
fwed
e
f
i
n
eB=∇×A:
∇·
(
∇×
A)
=
0
(
1
3
.
4
)
Si
mi
l
a
r
l
y
,
F
a
r
a
d
a
y
’
sL
a
wi
s
∂B
∇×
E
+
∇
∂t
= 0
(
1
3
.
5
)
= 0
(
1
3
.
6
)
∂t) = 0
(
1
3
.
7
)
A
E+ ∂∇×
×
∂t
∂A
∇×
(
E
+
a
n
di
ss
a
t
i
s
f
i
e
da
sa
ni
d
e
n
t
i
t
yb
yas
c
a
l
a
rp
ot
e
n
t
i
a
l
s
u
c
ht
h
a
t
:
∂A
E+ ∂t = −
∇φ
(
1
3
.
8
)
∂A
E = −
∇φ−
∂t
(
1
3
.
9
)
Nowwel
ooka
tt
h
ei
n
h
omog
e
n
e
ou
se
qu
a
t
i
on
si
nt
e
r
msoft
h
ep
ot
e
n
t
i
a
l
s
.
Amp
e
r
e
’
sL
a
w:
∂E
(
1
3
.
1
0
)
= µ(
J+ǫ ∂
t)
∇×B
∂E
∇×
(
∇×
A)
(
1
3
.
1
1
)
= µ(
J+ǫ ∂
t)
2
∂E
∂
t
∇(
∇·A)−∇A
= µJ+µ
ǫ
2
∂A
∂
φ −µ
ǫ
= µJ−µǫ∇ ∂
t
2
∂
t
2
∇(
∇·A)−∇A
∂
A
2
∂
φ
∇2A−µ
ǫ
µ
J+∇(
∇·A+µ
ǫ ∂
)
t
2 = −
∂t
Si
mi
l
a
r
l
yGa
u
s
s
’
sL
a
wf
ort
h
ee
l
e
c
t
r
i
cf
i
e
l
db
e
c
ome
s
:
ρ
∇·E
∇·−
∇φ−
2φ+
∇
=ǫ
∂A
ρ
∂t
=ǫ
−
= ρ
ǫ
∂∇·
A
(
1
3
.
1
2
)
(
1
3
.
1
3
)
(
1
3
.
1
4
)
(
1
3
.
1
5
)
(
1
3
.
1
6
)
(
1
3
.
1
7
)
I
nt
h
et
h
eL
or
e
n
t
zg
a
u
g
e
,
∂Φ
∂t =0
∇·
A+
µǫ
(
1
3
.
1
8
)
t
h
ep
ot
e
n
t
i
a
l
ss
a
t
i
s
f
yt
h
ef
ol
l
owi
n
gi
n
h
omoge
n
e
ou
swa
v
ee
qu
a
t
i
on
s
:
2
∂Φ
2
∇Φ−µ
ǫ ∂t
2
∂A
ρ
=−ǫ
(
1
3
.
1
9
)
∇A−µǫ ∂t
=−
µ
J
(
1
3
.
2
0
)
2
2
2
wh
e
r
eρa
n
dJa
r
et
h
ec
h
a
r
g
ed
e
n
s
i
t
ya
n
dc
u
r
r
e
n
td
e
n
s
i
t
ydi
s
t
r
i
b
u
t
i
on
s
,r
e
s
p
e
c
t
i
v
e
l
y
.
F
ort
h
et
i
meb
e
i
n
gwewi
l
l
s
t
i
c
kwi
t
ht
h
eL
or
e
n
t
zg
a
u
g
e
,
a
l
t
h
ou
g
ht
h
eCou
l
ombg
a
u
g
e
:
(
1
3
.
2
1
)
∇·
A=
0
i
smor
ec
on
v
e
n
i
e
n
tf
orc
e
r
t
a
i
np
r
ob
l
e
ms
.I
ti
sp
r
ob
a
b
l
ywor
t
hr
e
mi
n
d
i
n
gy
’
a
l
lt
h
a
tt
h
e
L
or
e
n
t
zg
a
u
g
ec
on
d
i
t
i
oni
t
s
e
l
fi
sr
e
a
l
l
yj
u
s
ton
eou
tofawh
ol
ef
a
mi
l
yofc
h
oi
c
e
s
.
Re
c
a
l
l
t
h
a
t(
ormor
ep
r
op
e
r
l
y
,
ob
s
e
r
v
et
h
a
ti
ni
t
sr
ol
ei
nt
h
e
s
ewa
v
ee
qu
a
t
i
on
s
)
1
2
µǫ= v
(
1
3
.
2
2
)
wh
e
r
evi
st
h
es
p
e
e
dofl
i
g
h
ti
nt
h
eme
d
i
u
m.F
ort
h
et
i
meb
e
i
n
g
,
l
e
t
’
sj
u
s
ts
i
mp
l
i
f
yl
i
f
ea
b
i
ta
n
da
g
r
e
et
owor
ki
nav
a
c
u
u
m:
1
µ0ǫ0= c
2
s
ot
h
a
t
:
2
1∂Φ
ρ
∂t
= −ǫ0
∇2Φ− c2
2
(
1
3
.
2
3
)
2
2
(
1
3
.
2
4
)
1∂A
2
2
µ
J
0
∇A− c ∂t
= −
(
1
3
.
2
5
)
I
f
/
wh
e
nwel
ooka
twa
v
es
ou
r
c
e
se
mb
e
d
d
e
di
nad
i
e
l
e
c
t
r
i
cme
d
i
u
m,
wec
a
na
l
wa
y
s
c
h
a
n
g
eb
a
c
ka
st
h
eg
e
n
e
r
a
l
f
or
ma
l
i
s
mwi
l
l
n
otb
ea
n
yd
i
ffe
r
e
n
t
.
13.
2 Gr
e
e
n
’
sF
u
n
c
t
i
o
n
sf
o
rt
h
eWa
v
eE
qu
a
t
i
on
Asb
yn
owy
ous
h
ou
l
df
u
l
l
yu
n
d
e
r
s
t
a
n
df
r
omwor
k
i
n
gwi
t
ht
h
ePoi
s
s
one
qu
a
t
i
on
,on
e
v
e
r
yg
e
n
e
r
a
lwa
yt
os
ol
v
ei
n
h
omog
e
n
e
ou
sp
a
r
t
i
a
ldi
ffe
r
e
n
t
i
a
le
qu
a
t
i
on
s(
PDE
s
)i
st
o
1
b
u
i
l
daGr
e
e
n
’
sf
u
n
c
t
i
on a
n
dwr
i
t
et
h
es
ol
u
t
i
ona
sa
ni
n
t
e
g
r
a
l
e
qu
a
t
i
on
.
1
Not
et
h
a
tt
h
i
se
x
p
r
e
s
s
i
ons
t
a
n
dsf
or
:“
Th
eg
e
n
e
r
a
l
i
z
e
dp
oi
n
ts
ou
r
c
ep
ot
e
n
t
i
a
l
/
f
i
e
l
dde
v
e
l
op
e
db
yGr
e
e
n
.
”A
n
u
mb
e
rofp
e
op
l
ec
r
i
t
i
c
i
z
et
h
ev
a
r
i
ou
swa
y
sofr
e
f
e
r
r
i
n
gt
oi
t–Gr
e
e
nf
u
n
c
t
i
on(
wh
a
tc
ol
orwa
st
h
a
ta
g
a
i
n
?wh
a
t
s
h
a
deofGr
e
e
n
?
)
,Gr
e
e
n
sf
u
n
c
t
i
on(
af
u
n
c
t
i
onma
deofl
e
t
t
u
c
ea
n
ds
p
i
n
a
c
ha
n
dk
a
l
e
?
)
,“
a
”Gr
e
e
n
’
sf
u
n
c
t
i
on(
a
s
i
n
g
u
l
a
rr
e
p
r
e
s
e
n
t
a
t
i
v
eofap
l
u
r
a
lc
l
a
s
sr
e
f
e
r
e
n
c
e
da
sas
i
n
g
u
l
a
rob
j
e
c
t
)
.Al
lh
a
v
ep
r
ob
l
e
ms
.It
e
n
dt
og
owi
t
ht
h
e
l
a
t
t
e
roft
h
e
s
ea
si
ts
e
e
msl
e
a
s
tod
dt
ome
.
L
e
t
’
sv
e
r
yqu
i
c
k
l
yr
e
v
i
e
wt
h
eg
e
n
e
r
a
lc
o
n
c
e
p
t(
f
oraf
u
r
t
h
e
rdi
s
c
u
s
s
i
ondon
’
tf
or
g
e
t
WI
YF,
MWI
YF
)
.
Su
p
p
os
eDi
sag
e
n
e
r
a
l
(
s
e
c
on
dor
d
e
r
)l
i
n
e
a
rp
a
r
t
i
a
l
d
i
ffe
r
e
n
t
i
a
l
op
e
r
a
t
or
3
one
.
g
.
I
Ra
n
don
ewi
s
h
e
st
os
ol
v
et
h
ei
n
h
omog
e
n
e
ou
se
qu
a
t
i
on
:
Df
x
()=ρx
()
(
1
3
.
2
6
)
f
orf.
I
fon
ec
a
nf
i
n
das
ol
u
t
i
onGx
(−
x0)t
ot
h
ea
s
s
oc
i
a
t
e
dd
i
ffe
r
e
n
t
i
a
le
qu
a
t
i
onf
ora
2
po
i
n
ts
ou
r
c
ef
u
n
c
t
i
on :
DGx
(x
,
)=δx
(−
x0)
0
(
1
3
.
2
7
)
t
h
e
n(
s
u
b
j
e
c
tt
ov
a
r
i
ou
sc
on
d
i
t
i
on
s
,s
u
c
ha
st
h
ea
b
i
l
i
t
yt
oi
n
t
e
r
c
h
a
n
g
et
h
edi
f
f
e
r
e
n
t
i
a
l
op
e
r
a
t
ora
n
dt
h
ei
n
t
e
g
r
a
t
i
on
)t
os
ol
u
t
i
ont
ot
h
i
sp
r
ob
l
e
mi
saF
r
e
d
h
ol
mI
n
t
e
gr
a
lE
qu
a
t
i
on(
a
c
on
v
ol
u
t
i
onoft
h
eGr
e
e
n
’
sf
u
n
c
t
i
onwi
t
ht
h
es
ou
r
c
et
e
r
ms
)
:
3
f
x
()=χ
x
()+
3
Gx
(x
,
)
ρ
x
(0)
dx
0
0
(
1
3
.
2
8
)
I
R
wh
e
r
eχ
x
()i
sa
na
r
b
i
t
r
a
r
ys
ol
u
t
i
ont
ot
h
ea
s
s
oc
i
a
t
e
dh
omog
e
n
e
ou
sPDE
:
D[
χ
x
()
]
=0
Th
i
ss
ol
u
t
i
onc
a
ne
a
s
i
l
yb
ev
e
r
i
f
i
e
d
:
f
x
()=χx
()+
(
1
3
.
2
9
)
3
3
Gx
(x
,
ρx
( 0)
dx
0)
0
(
1
3
.
3
0
)
I
R
χ
x
()
]
+DI
R3
Df
x
()=D[
3
ρx(0)
dx
0
3
ρx
(0)
dx
0
Gx(x
,0)
(
1
3
.
3
1
)
(
1
3
.
3
2
)
3
3
DGx
(x
,
ρx
( 0)
dx
0)
0
3
δx
(−x0)
ρx
( 0)
dx
0
Df
x
()=0+
I
R
Df
x
()=0+
I
R
3
(
1
3
.
3
3
)
(
1
3
.
3
4
)
Df
x
()=ρx
()
(
1
3
.
3
5
)
I
ts
e
e
ms
,t
h
e
r
e
f
or
e
,t
h
a
twes
h
ou
l
dt
h
or
ou
gh
l
yu
n
d
e
r
s
t
a
n
dt
h
ewa
y
sofb
u
i
l
d
i
n
g
Gr
e
e
n
’
sf
u
n
c
t
i
on
si
ng
e
n
e
r
a
l
f
orv
a
r
i
ou
si
mp
or
t
a
n
tPDEs
.I
’
mu
n
c
e
r
t
a
i
nofh
owmu
c
hof
t
h
i
st
od
owi
t
h
i
nt
h
e
s
en
ot
e
s
,h
owe
v
e
r
.Th
i
si
s
n
’
tr
e
a
l
l
y“
E
l
e
c
t
r
ody
n
a
mi
c
s
”
,i
ti
s
ma
t
h
e
ma
t
i
c
a
lp
h
y
s
i
c
s
,on
e of t
h
ef
u
n
d
a
me
n
t
a
lt
ool
s
e
t
sy
ou n
e
e
dt
od
o
El
e
c
t
r
od
y
n
a
mi
c
s
,qu
a
n
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13.
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1 Poi
s
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qu
a
t
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qu
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t
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on
:
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2
∇ φ=− ǫ0
(
1
3
.
3
6
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∇Gx
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, 0)=δx
(
1
3
.
3
7
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i
st
h
es
ol
u
t
i
ont
o:
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u
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(
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3
.
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8
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l
oc
a
t
e
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t
x0.
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n
c
e
:
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()=χ0x
(
ρx
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1
dx
0
)+
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(
1
3
.
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9
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e
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r
e
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(
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6
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x
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t
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nt
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yoft
h
e
s
e
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dx
0
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i
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x
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x
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9
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∇ +k Gx
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1
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5
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1
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5
7
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0
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1
3
.
5
6
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(
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5
8
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t
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t
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t
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e
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l
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h
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e
m)
.
F
r
omAx
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a
ne
a
s
i
l
yf
i
n
dBo
rH:
B=
µ
B=
∇×
A
0
(
1
3
.
6
9
)
(
b
yd
e
f
i
n
i
t
i
on
)
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t
s
i
d
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r
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g
h(
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e
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et
h
ec
u
r
r
e
n
t
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r
ea
l
l
z
e
r
o)
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e
r
e
’
sl
a
wt
e
l
l
su
st
h
a
t
:
(
1
3
.
7
0
)
∇×H=−
i
ωD
or
or
∇×B
=−
i
ωµ
ǫ0E
0
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=−
i c E=i cE
c
ω
2
k
(
1
3
.
7
1
)
(
1
3
.
7
2
)
E=i k∇×B
(
1
3
.
7
3
)
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n
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ei
n
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e
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la
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ov
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r
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e
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h
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ormos
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e
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on
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et
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t
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h
ti
n
t
ot
h
er
a
di
a
t
i
v
ep
r
oc
e
s
s
e
s
.
13.
3.
1 Th
eZo
n
e
s
Su
p
p
os
et
h
es
ou
r
c
el
i
v
e
si
n
s
i
d
ear
e
g
i
onofma
x
i
mu
ms
i
z
ed≪ λwh
e
r
eλ=2
πc
/
ω.By
t
h
a
tIme
a
nt
h
a
tas
p
h
e
r
eofr
a
d
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u
sd(
a
b
ou
tt
h
eor
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g
i
n
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omp
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t
e
l
yc
on
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a
i
n
sa
l
l
c
h
a
r
g
e
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u
r
r
e
n
td
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t
r
i
b
u
t
i
on
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.
Th
e
nwec
a
nd
e
f
i
n
et
h
r
e
ez
on
e
sofa
p
p
r
ox
i
ma
t
i
on
:
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en
e
a
r(
s
t
a
t
i
c
)z
on
e
d<
<r<
<λ
b
)Th
ei
n
t
e
r
me
d
i
a
t
e(
i
n
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u
c
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i
on
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on
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c
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ef
a
r(
r
a
d
i
a
t
i
on
)z
on
e
d<
<r∼λ
d<
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Th
ef
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ffe
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e
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l
l
b
r
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c
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h
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omi
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r
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r
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ft
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orr
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c
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emos
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h
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ti
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ef
a
rz
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ei
swh
e
r
ewea
l
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om t
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om t
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d
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g
e
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e
r
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l
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r
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e
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ma
l
l
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ma
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n
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h
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rz
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h
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r
a
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c
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l
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n
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l
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t
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re
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ookl
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i
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st
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c
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a
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t
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omc
omp
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l
l
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e
n
dmos
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on
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i
de
r
i
n
gs
ol
u
t
i
on
si
nt
h
ef
a
rz
on
e
.
13.
3.
2 Th
eNe
a
rZon
e
Su
p
p
os
et
h
a
twea
r
ei
nt
h
en
e
a
rz
o
n
e
.
Th
e
nb
yde
f
i
n
i
t
i
on
′
k
x
|−
x|
<
<1
a
n
d
i
k
x
|−
x′|≈1
e
Th
i
sma
k
e
st
h
ei
n
t
e
g
r
a
le
qu
a
t
i
oni
n
t
ot
h
e“
s
t
a
t
i
c
”f
or
ma
l
r
e
a
d
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on
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r
e
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n
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h
a
p
t
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r5(
c
f
.e
qu
a
t
i
on(
5
.
3
2
)
)
.
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e
et
h
a
t−
1
/
4
πx
|
−
x|
i
sj
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s
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h
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n
’
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n
c
t
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h
eg
oodol
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s
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dc
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u
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tl
i
k
ei
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eg
oodol
dd
a
y
s
:
− 1
′
G0x
(x
,)=
′
ℓ
r
ℓ
+
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L
2
ℓ+1r
ˆ
′∗
YL(
ˆ
r
)
YL(
r).
(
1
3
.
7
4
)
Not
eWe
l
l
:I
wi
l
l
u
s
eL≡(
ℓ
,
m)f
r
e
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l
ya
n
dwi
t
h
ou
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r
n
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nt
h
i
sc
ou
r
s
e
.
Th
es
u
mi
s
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e
ra
l
l
ℓ
,
m.
Hop
e
f
u
l
l
y
,
b
yn
owy
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a
tt
h
e
yr
u
nov
e
r
.
I
f
4
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l
l
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a
r
nt
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r
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tc
e
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t
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i
ne
x
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e
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t
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e
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eme
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n
ot
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e
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h
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p
t
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h
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c
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l
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i
s
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o
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t
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i
sme
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n
st
h
a
t(
i
fy
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i
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l
i
mAx
()=
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ℓ
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Y
Y (
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r
) Jx
(′
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+
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r
L(
k
r
→0
L
∗ 3′
(
r̂)dr
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L
(
1
3
.
7
5
)
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l
l
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e
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r
e
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e
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l
ows
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ort
h
a
tr
e
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h
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ty
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u
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t
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r
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e
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l
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e
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t
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h
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t
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nt
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omet
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t
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a
t
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l
t
h
en
e
a
rz
on
et
h
e“
s
t
a
t
i
cz
on
e
”
.
13.
3.
3 Th
eF
a
rZo
n
e
E
x
a
c
t
l
yt
h
eop
p
os
i
t
ei
st
r
u
ei
nt
h
ef
a
rz
on
e
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r
ek
r>
>1a
n
dt
h
ee
x
p
on
e
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i
a
l
os
c
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l
l
a
t
e
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r
a
p
i
d
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y
.
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a
na
p
p
r
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ma
t
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r
g
u
me
n
toft
h
ee
x
p
on
e
n
t
i
a
l
a
sf
ol
l
ows
:
xx
|−
′
2 ′
2
r
+r
=
|
2r
nx′
2
−
·
′
2
r
2
rnx·′+ r
1
′
·+O
= r−nx
r
=r1−
1
/
2
(
1
3
.
7
6
)
′
wh
e
r
eweh
a
v
ea
s
s
u
me
dt
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l
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t
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on(
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)
:
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(
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(
1
3
.
7
9
)
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(
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,
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.
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(
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t:
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n
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eor
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r
e
:
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(
r
)
ℓ
(
k
r
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n
d
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=j
(
1
3
.
8
4
)
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=n
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3
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8
7
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8
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9
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t
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x) = xc
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π
l
i
mn
ℓ
(
x
) = si
x
n
(
x−(
ℓ+1
)
)
.
x
2
(
1
3
.
9
3
)
(
1
3
.
9
4
)
→∞
Not
et
h
a
tb
ot
hs
ol
u
t
i
on
sa
r
er
e
g
u
l
a
r(
g
ot
oz
e
r
os
moot
h
l
y
)a
ti
n
f
i
n
i
t
ya
n
da
r
et
h
es
a
me
(
t
r
i
g
)f
u
n
c
t
i
ons
h
i
f
t
e
db
yπ/
2ov
e
rxt
h
e
r
e
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et
h
a
tt
h
e
ya
r
en
o
ts
qu
a
r
ei
n
t
e
g
r
a
b
l
eon
3
I
R(
f
ory
ou
rqu
a
n
t
u
mc
ou
r
s
e
)b
u
ta
r
es
t
i
l
lb
e
t
t
e
rt
h
a
np
l
a
n
ewa
v
e
si
nt
h
a
tr
e
g
a
r
d
.
Some
t
h
i
n
gt
ot
h
i
n
ka
b
o
u
t.
.
.
Ha
n
k
e
l
F
u
n
c
t
i
on
s
E
x
a
mi
n
i
n
gt
h
ea
s
y
mp
t
ot
i
cf
or
ms
,we s
e
et
h
a
tt
wo p
a
r
t
i
c
u
l
a
rc
omp
l
e
xl
i
n
e
a
r
c
omb
i
n
a
t
i
on
soft
h
es
t
a
t
i
on
a
r
ys
ol
u
t
i
onh
a
v
et
h
eb
e
h
a
v
i
or
,
a
ti
n
f
i
n
i
t
y
,
ofa
nou
t
g
oi
n
gor
i
n
c
omi
n
gs
p
h
e
r
i
c
a
l
wa
v
ewh
e
nt
h
et
i
mede
p
e
n
d
e
n
c
ei
sr
e
s
t
or
e
d
:
+
h
x
)
ℓ(
−
h
x
)
ℓ(
1
(
x
)+i
n
(
x
)
ℓ
ℓ
=j
(
=h
x
)
)
ℓ(
(
1
3
.
9
5
)
(
x
)−i
n
(
x
)
ℓ
ℓ
=j
2
(
=h
x
)
)
ℓ(
(
1
3
.
9
6
)
t
h
es
p
h
e
r
i
c
a
lh
a
n
k
e
lf
u
n
c
t
i
o
n
soft
h
ef
i
r
s
t(
+
)(
ou
t
g
oi
n
g
)a
n
ds
e
c
o
n
d(
−
)(
i
n
c
omi
n
g
)
ℓ
+
1
k
i
n
ds
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t
hoft
h
e
s
es
ol
u
t
i
on
sa
r
es
i
n
g
u
l
a
ra
tt
h
eor
i
g
i
nl
i
k
e1
/
x (
wh
y
?
)a
n
db
e
h
a
v
e
l
i
k
e
+
l
i
mh(
x
)
x
→∞
ℓ
−
i
)
=(
−
l
i
mh(
x
)
x
→∞
ℓ
+
1
ℓ
eix
x
−i
x
ℓ
+1e
=(
i
)
x
(
1
3
.
9
7
)
(
1
3
.
9
8
)
a
ti
n
f
i
n
i
t
y
.
Twop
a
r
t
i
c
u
l
a
r
l
yu
s
e
f
u
l
s
p
h
e
r
i
c
a
l
h
a
n
k
e
l
f
u
n
c
t
i
on
st
ok
n
owa
r
et
h
ez
e
r
ot
h
or
d
e
ron
e
s
:
+
h
x
)
0(
−
h
x
)
0(
=
i
x
e
i
x
−i
x
e
(
1
3
.
9
9
)
=
(
1
3
.
1
0
0
)
−
i
x
Pl
a
n
eWa
v
eE
x
p
a
n
s
i
on
Pl
a
n
ewa
v
e
sa
n
df
r
e
es
p
h
e
r
i
c
a
lwa
v
e
sb
ot
hf
or
ma
n(
on
–s
h
e
l
l
)c
omp
l
e
t
eor
t
h
n
or
ma
l
3
s
e
to
nI
R(
wi
t
horwi
t
h
o
u
tt
h
eor
i
g
i
n
)
.Th
a
tme
a
n
st
h
a
ton
emu
s
tb
ea
b
l
et
oe
x
p
a
n
don
e
i
nt
e
r
msoft
h
eot
h
e
r
.
Pl
a
n
ewa
v
e
sc
a
nb
ee
x
p
a
n
d
e
di
nt
e
r
msoff
r
e
es
p
h
e
r
i
c
a
l
wa
v
e
sb
y
:
i
k
·
r=e
i
k
rc
os
(
Θ)
e
=
ℓ
ˆ
∗
4
πiYL(
k)
j
(
k
r
)
YL(
ˆ
r
).
ℓ
L
(
1
3
.
1
0
1
)
Th
i
si
sd
u
et
oL
or
dRa
y
l
e
i
g
ha
n
di
ss
ome
t
i
me
sc
a
l
l
e
dt
h
eRa
y
l
e
i
g
he
x
p
a
n
s
i
on
.
Re
c
a
l
l
t
h
a
tΘi
st
h
ea
n
g
l
eb
e
t
wi
x
tt
h
e
r
a
n
dt
h
eka
n
dt
h
a
tc
os
(
Θ)=c
os
(
−
Θ)
.
Th
e
r
ei
ss
i
mi
l
a
r
l
ya
n(
i
n
t
e
g
r
a
l
)e
x
p
r
e
s
s
i
onf
orj
(
k
r
)i
nt
e
r
msofa
ni
n
t
e
g
r
a
lov
e
rt
h
e
ℓ
i
k
·
r
e b
u
twewi
l
ln
otu
s
ei
th
e
r
e
.I
tf
ol
l
owsf
r
omt
h
ec
omp
l
e
t
e
n
e
s
sr
e
l
a
t
i
ononp
a
g
e2
1
4
i
nWy
l
d
,
t
h
eRa
y
l
e
i
g
he
x
p
a
n
s
i
on
,a
n
dt
h
ec
omp
l
e
t
e
n
e
s
sr
e
l
a
t
i
ononp
a
g
e2
1
2
.De
r
i
v
ei
t
f
orh
ome
wor
k(
orf
i
n
di
ts
ome
wh
e
r
ea
n
dc
op
yi
t
,
b
u
ty
ous
h
ou
l
d
n
’
th
a
v
et
o)
.Ch
e
c
ky
ou
r
r
e
s
u
l
tb
yf
i
n
d
i
n
gi
ts
ome
wh
e
r
e
.It
h
i
n
ki
tmi
g
h
tb
es
ome
wh
e
r
ei
nWy
l
d
,
b
u
tIk
n
owi
ti
s
e
l
s
e
wh
e
r
e
.
Th
i
swi
l
l
b
eh
a
n
d
e
di
n
.
±
13.
4.
2 J
(
r
)
,
NL(
r
)
,
a
n
dHL(
r
)
L
F
orc
on
v
e
n
i
e
n
c
e
,
wed
e
f
i
n
et
h
ef
ol
l
owi
n
g
:
J
(
r
)=j
(
k
r
)
YL(
ˆ
r
)
L
ℓ
(
1
3
.
1
0
2
)
NL(
r
) =n
(
k
r
)
YL(
ˆ
r
)
ℓ
(
1
3
.
1
0
3
)
±
±
HL(
r
) =h
k
r
)
YL(
ˆ
r
)
ℓ(
(
1
3
.
1
0
4
)
2
Th
e
s
ea
r
et
h
eb
a
s
i
cs
ol
u
t
i
on
st
ot
h
eHHEt
h
a
ta
r
ea
l
s
oe
i
g
e
n
f
u
n
c
t
i
on
sofL a
n
dL
z.
2
Cl
e
a
r
l
yt
h
e
r
ei
sa
ni
mp
l
i
c
i
tl
a
b
e
lofk(
ork)f
ort
h
e
s
es
ol
u
t
i
on
s
.Ag
e
n
e
r
a
ls
ol
u
t
i
on(
on
as
u
i
t
a
b
l
ed
oma
i
n
)c
a
nb
ec
on
s
t
r
u
c
t
e
dou
tofal
i
n
e
a
rc
omb
i
n
a
t
i
onofa
n
yt
wooft
h
e
m.
13.
4.
3 Ge
n
e
r
a
l
Sol
u
t
i
o
n
st
ot
h
eHHE
On“
s
p
h
e
r
i
c
a
l
”doma
i
n
s(
t
h
ei
n
t
e
r
i
ora
n
de
x
t
e
r
i
orofas
p
h
e
r
e
,
ori
nas
p
h
e
r
i
c
a
l
s
h
e
l
l
)t
h
e
c
omp
l
e
t
e
l
yg
e
n
e
r
a
l
s
ol
u
t
i
ont
ot
h
eHHEc
a
nt
h
e
r
e
f
or
eb
ewr
i
t
t
e
ni
ns
t
a
t
i
o
n
a
r
yf
or
ma
s
:
ALJ
(
r
)+BLNL(
r
)
L
(
1
3
.
1
0
5
)
L
or(
f
ors
c
a
t
t
e
r
i
n
gt
h
e
or
y
,
mos
t
l
y
)i
nt
h
eo
u
t
go
i
n
gwa
v
ef
or
m
+
CLJ
(
r
)+SLHL(
r
)
.
L
L
(
1
3
.
1
0
6
)
I
n
s
i
d
eas
p
h
e
r
e
,
BLa
n
dSLmu
s
tb
ez
e
r
o.Ou
t
s
i
d
eas
p
h
e
r
e
,
ori
nas
p
h
e
r
i
c
a
l
a
n
n
u
l
u
s
,
a
l
l
t
h
ec
oe
ffic
i
e
n
t
sc
a
nb
en
on
–z
e
r
ou
n
l
i
k
et
h
es
i
t
u
a
t
i
onf
ort
h
eL
a
p
l
a
c
ee
qu
a
t
i
on(
wh
y
?
)
.
[
Th
i
ss
h
ou
l
dp
r
ov
ok
ed
e
e
pt
h
ou
g
h
t
sa
b
ou
tt
h
ef
u
n
da
me
n
t
a
ls
i
g
n
i
f
i
c
a
n
c
eoft
h
e
L
a
p
l
a
c
ee
qu
a
t
i
on
.Ar
et
h
e
r
ea
n
y“
r
e
a
l
l
y
”s
t
a
t
i
on
a
r
ys
ou
r
c
e
si
nt
h
ed
y
n
a
mi
c
a
l
,
c
ov
a
r
i
a
n
t
,
u
n
i
v
e
r
s
e
?Dowee
x
p
e
c
tt
oh
a
v
eac
on
t
r
i
b
u
t
i
ont
ot
h
ez
e
r
of
r
e
qu
e
n
c
yc
h
a
r
g
e
/
c
u
r
r
e
n
t
d
e
n
s
i
t
yd
i
s
t
r
i
b
u
t
i
oni
na
n
yr
e
g
i
onofs
p
a
c
e
?Wh
a
twou
l
dt
h
i
sc
or
r
e
s
p
on
dt
o?
]
13.
4.
4 Gr
e
e
n
’
sF
u
n
c
t
i
on
sa
n
dF
r
e
eSp
h
e
r
i
c
a
l
Wa
v
e
s
7
Wee
x
p
e
c
t
,f
orp
h
y
s
i
c
a
lr
e
a
s
on
st
h
a
tt
h
ewa
v
ee
mi
t
t
e
db
yat
i
med
e
p
e
n
d
e
n
ts
ou
r
c
e
s
h
ou
l
db
e
h
a
v
el
i
k
ea
nou
t
g
oi
n
gwa
v
ef
a
rf
r
om t
h
es
ou
r
c
e
.Not
et
h
a
ti
n
s
i
d
et
h
e
b
ou
n
d
i
n
gs
p
h
e
r
eoft
h
es
ou
r
c
et
h
a
tn
e
e
dn
otb
et
r
u
e
.E
a
r
l
i
e
ri
nt
h
i
sc
h
a
p
t
e
r
,
weu
s
e
da
n
“
ou
t
g
oi
n
gwa
v
eGr
e
e
n
’
sf
u
n
c
t
i
on
”t
oc
on
s
t
r
u
c
tt
h
es
ol
u
t
i
ont
ot
h
eI
HEwi
t
ht
h
i
s
a
s
y
mp
t
ot
i
cb
e
h
a
v
i
or
.
We
l
l
,
l
oa
n
db
e
h
ol
d:
′
xx
i
k
±
x x′
h
π0 (
k
|−
)=∓ 4
G±(,
F
ors
t
a
t
i
on
a
r
ywa
v
e
s(
u
s
e
f
u
l
i
nqu
a
n
t
u
mt
h
e
or
y
)
|
)
(
1
3
.
1
0
7
)
k
′
,)=
G0x( x
4πn0 (
k
x
|−
x
′
|
)
.
(
1
3
.
1
0
8
)
Th
i
se
x
t
r
e
me
l
yi
mp
or
t
a
n
tr
e
l
a
t
i
onf
or
mst
h
ec
on
n
e
c
t
i
onb
e
t
we
e
nf
r
e
es
p
h
e
r
i
c
a
l
wa
v
e
s(
r
e
v
i
e
we
da
b
ov
e
)a
n
dt
h
ei
n
t
e
g
r
a
le
qu
a
t
i
ons
ol
u
t
i
on
swea
r
ei
n
t
e
r
e
s
t
e
di
n
c
on
s
t
r
u
c
t
i
n
g
.
Th
i
sc
on
n
e
c
t
i
onf
ol
l
owsf
r
omt
h
ea
d
d
i
t
i
ont
h
e
or
e
msormu
l
t
i
p
ol
a
re
x
p
a
n
s
i
on
sof
t
h
ef
r
e
es
p
h
e
r
i
c
a
l
wa
v
e
sde
f
i
n
e
da
b
ov
e
.
F
ort
h
es
p
e
c
i
a
l
c
a
s
eofL=(
0
,
0
)t
h
e
s
ea
r
e
:
1
N0(
r−r
)=n
(
k
|
r−r
) √4π
0
′
′
√
=4
π
a
n
d
±
H(
r
0
′
−
±
′
r
)=h(
kr r
) 1
0
√
|−
4π
∗
L
√
NL(
r
)
J
(
r
)
>
L
<
±
= 4π
(
1
3
.
1
0
9
)
∗
H(
r )
J (
r ).
L
L
> L
<
(
1
3
.
1
1
0
)
F
r
om t
h
i
sa
n
dt
h
ea
b
ov
e
,t
h
ee
x
p
a
n
s
i
onoft
h
eGr
e
e
n
’
sf
u
n
c
t
i
on
si
nf
r
e
es
p
h
e
r
i
c
a
l
mu
l
t
i
p
ol
a
rwa
v
e
si
mme
d
i
a
t
e
l
yf
ol
l
ows
:
′
∗
G0(
r−r
)=k NL(
r
)
J
(
r
)
>
L
<
L
7
(
1
3
.
1
1
1
)
Ac
op
–ou
tp
h
r
a
s
ei
ft
h
e
r
ee
v
e
rwa
son
e
.I
tt
r
a
n
s
l
a
t
e
sa
s
:
b
e
c
a
u
s
et
h
a
t
’
st
h
ewa
yi
tt
u
r
n
sou
ta
tt
h
ee
n
d.
a
n
d
′
±
G (
r r)=
i
k
∓
± −
H(
r
L
L
∗
)
J(
r)
.
> L
<
(
1
3
.
1
1
2
)
Not
eWe
l
l
:Th
ec
omp
l
e
xc
on
j
u
g
a
t
i
onop
e
r
a
t
i
onu
n
d
e
rt
h
es
u
mi
sa
p
p
l
i
e
dt
ot
h
e
s
p
h
e
r
i
c
a
l
h
a
r
mo
n
i
c(
on
l
y
)
,
n
o
t
t
h
e
H
a
n
k
e
l
f
u
n
c
t
i
o
n
(
s
)
.
T
h
i
s
i
s
b
e
c
a
u
s
e
t
h
e
o
n
l
y
f
u
n
c
t
i
o
n
′∗
oft
h
ep
r
od
u
c
tYL(
r
ˆ
)
YL(
r
ˆ) i
st
or
e
c
on
s
t
r
u
c
tt
h
ePℓ(
Θ)v
i
at
h
ea
d
d
i
t
i
ont
h
e
or
e
mf
or
s
p
h
e
r
i
c
a
l
h
a
r
mon
i
c
s
.
St
u
d
yt
h
i
sp
oi
n
ti
nWy
l
dc
a
r
e
f
u
l
l
yony
ou
rown
.
Th
e
s
er
e
l
a
t
i
on
swi
l
l
a
l
l
owu
st
oe
x
p
a
n
dt
h
eHe
l
mh
ol
t
zGr
e
e
n
’
sf
u
n
c
t
i
on
se
x
a
c
t
l
yl
i
k
ewe
e
x
p
a
n
d
e
dt
h
eGr
e
e
n
’
sf
u
n
c
t
i
onf
ort
h
eL
a
p
l
a
c
e
/
Poi
s
s
one
qu
a
t
i
on
.Th
i
s
,
i
nt
u
r
n
,
wi
l
l
a
l
l
owu
s
t
op
r
e
c
i
s
e
l
ya
n
db
e
a
u
t
i
f
u
l
l
yr
e
c
on
s
t
r
u
c
tt
h
emu
l
t
i
p
ol
a
re
x
p
a
n
s
i
onoft
h
ev
e
c
t
orp
ot
e
n
t
i
a
l
,
a
n
d
h
e
n
c
et
h
eEMf
i
e
l
d
si
nt
h
ev
a
r
i
ou
sz
on
e
s
8
e
x
a
c
t
l
y.
Th
i
se
n
dsou
rb
r
i
e
fma
t
h
e
ma
t
i
c
a
lr
e
v
i
e
woff
r
e
es
p
h
e
r
i
c
a
lwa
v
e
sa
n
dwer
e
t
u
r
nt
o
t
h
ed
e
s
c
r
i
p
t
i
onofRa
d
i
a
t
i
on
.
13.
5 E
l
e
c
t
r
i
cDi
p
o
l
eRa
d
i
a
t
i
on
Nowt
h
a
tweh
a
v
et
h
a
tu
n
d
e
rou
rb
e
l
t
swec
a
na
d
d
r
e
s
st
h
emu
l
t
i
p
ol
a
re
x
p
a
n
s
i
onoft
h
e
v
e
c
t
orp
ot
e
n
t
i
a
li
n
t
e
l
l
i
g
e
n
t
l
y
.Tob
e
g
i
nwi
t
h
,wewi
l
lwr
i
t
et
h
ege
n
e
r
a
ls
ol
u
t
i
o
nf
ort
h
e
v
e
c
t
orp
ot
e
n
t
i
a
li
nt
e
r
msoft
h
emu
l
t
i
p
ol
a
re
x
p
a
n
s
i
onf
ort
h
eou
t
g
oi
n
gwa
v
eGr
e
e
n
’
s
f
u
n
c
t
i
onde
f
i
n
e
da
b
ov
e
:
∞
J
r
()
L
Ar
()=i
k
L
r
r
+
′
′(
∗
)3′
µ0J
r
()
HLr
() dr
′
′(
∗
)3′
r
()
J
r
() dr
+HLr
() µ0J
L
0
(
1
3
.
1
1
3
)
wh
e
r
e
,b
yc
on
v
e
n
t
i
on
,(
∗
)me
a
n
st
h
a
tt
h
eYL(
ˆ
r
)i
sc
on
j
u
g
a
t
e
db
u
tt
h
eb
e
s
s
e
l
/
n
e
u
ma
n
n
/
h
a
n
k
e
lf
u
n
c
t
i
oni
sn
o
t
.Th
i
si
sb
e
c
a
u
s
et
h
eon
l
yp
oi
n
toft
h
ec
on
j
u
g
a
t
i
oni
st
o
c
on
s
t
r
u
c
tPℓ(
Θ)f
r
om t
h
em–s
u
mf
ore
a
c
hℓv
i
at
h
ea
d
di
t
i
ont
h
e
or
e
mf
ors
p
h
e
r
i
c
a
l
+
−
h
a
r
mon
i
c
s
.Wec
e
r
t
a
i
n
l
yd
on
’
twa
n
tt
oc
h
a
n
g
eh i
n
t
oh,wh
i
c
hc
h
a
n
g
e
st
h
et
i
me
9
d
e
p
e
n
de
n
tb
e
h
a
v
i
oroft
h
es
ol
u
t
i
on.Not
et
h
a
tt
h
ei
n
t
e
g
r
a
lov
e
ra
l
ls
p
a
c
ei
sb
r
ok
e
nu
p
i
ns
u
c
hawa
yt
h
a
tt
h
eGr
e
e
n
’
sf
u
n
c
t
i
one
x
p
a
n
s
i
on
sa
b
ov
ea
l
wa
y
sc
on
v
e
r
g
e
.Th
i
s
s
ol
u
t
i
oni
se
x
a
c
te
v
e
r
y
wh
e
r
ei
ns
p
a
c
ei
n
c
l
u
d
i
n
gi
n
s
i
d
et
h
es
ou
r
c
ei
t
s
e
l
f!
Wec
a
nt
h
e
r
e
f
or
es
i
mp
l
i
f
you
rn
ot
a
t
i
onb
yd
e
f
i
n
i
n
gc
e
r
t
a
i
nf
u
n
c
t
i
on
soft
h
er
a
d
i
a
l
v
a
r
i
a
b
l
e
:
Ar
()= i
kC
L
+
L
r
)
Hr
().
(
r
)
J
r)+S (
(
L
L
(
1
3
.
1
1
4
)
8
We
l
l
,
i
nau
n
i
f
or
ml
yc
on
v
e
r
g
e
n
te
x
p
a
n
s
i
on
,
wh
i
c
hi
sk
i
n
dofe
x
a
c
t
,
i
nt
h
el
i
mi
tofa
ni
n
f
i
n
i
t
es
u
m.
I
nt
h
eme
a
n
t
i
me
,
i
ti
sad
a
mng
ooda
p
p
r
ox
i
ma
t
i
on
.
Us
u
a
l
l
y
.
9
Th
i
ss
u
g
g
e
s
t
st
h
a
tt
h
e
r
ea
r
es
omei
n
t
e
r
e
s
t
i
n
gc
on
n
e
c
t
i
on
sb
e
t
we
e
nt
h
ec
on
j
u
g
a
t
i
ons
y
mme
t
r
ya
n
dt
i
me
r
e
v
e
r
s
a
l
s
y
mme
t
r
y
.
Toob
a
dwewon
’
th
a
v
et
i
met
oe
x
p
l
or
et
h
e
m.
Youma
yo
ny
ou
rown
,
t
h
ou
g
h
.
I
nt
h
i
se
qu
a
t
i
on
,
∞
CL(
r
) =
SL(
r
) =
′
r
r
0
′(
∗
)3′
µ0J
r
()
HLr
() dr
′
′(
∗
)3′
µ0J
r
()
J
r
() dr
.
L
(
1
3
.
1
1
5
)
(
1
3
.
1
1
6
)
Cl
e
a
r
l
ySL(
0
)=0a
n
df
orr>d
,CL(
r
)=0
.Att
h
eor
i
g
i
nt
h
es
ol
u
t
i
oni
sc
omp
l
e
t
e
l
y
r
e
g
u
l
a
ra
n
ds
t
a
t
i
o
n
a
r
y
.Ou
t
s
i
det
h
eb
ou
n
di
n
gs
p
h
e
r
eoft
h
es
ou
r
c
edi
s
t
r
i
b
u
t
i
ont
h
e
s
ol
u
t
i
onb
e
h
a
v
e
sl
i
k
eal
i
n
e
a
rc
omb
i
n
a
t
i
onofou
t
g
oi
n
gs
p
h
e
r
i
c
a
lmu
l
t
i
p
ol
a
rwa
v
e
s
.
F
r
omn
owonwewi
l
lc
on
c
e
n
t
r
a
t
eont
h
el
a
t
t
e
rc
a
s
e
,
s
i
n
c
ei
ti
st
h
eon
er
e
l
e
v
a
n
tt
ot
h
e
z
on
e
s
.
13.
5.
1 Ra
d
i
a
t
i
o
no
u
t
s
i
d
et
h
es
o
u
r
c
e
Ou
t
s
i
d
et
h
eb
ou
n
d
i
n
gs
p
h
e
r
eoft
h
es
ou
r
c
e
,
+
Ar
()=i
k HLr
()
L
∞
0
′
′(
∗
)3′
µ0J
r
()
J
r
() dr
.
L
(
1
3
.
1
1
7
)
Atl
a
s
tweh
a
v
ema
d
ei
tt
oJ
a
c
k
s
on
’
se
qu
a
t
i
on9
.
1
1
,
b
u
tl
o
okh
owe
l
e
ga
n
to
u
ra
p
pr
o
a
c
h
wa
s
.I
n
s
t
e
a
dofaf
or
mt
h
a
ti
son
l
yv
a
l
i
di
nt
h
ef
a
rz
on
e
,wec
a
nn
ows
e
et
h
a
tt
h
i
si
sa
l
i
mi
t
i
n
gf
or
mofac
on
v
e
r
g
e
n
ts
ol
u
t
i
ont
h
a
twor
k
si
na
l
lz
on
e
s
,i
n
c
l
u
d
i
n
gi
n
s
i
det
h
es
ou
r
c
e
i
t
s
e
l
f
!
Th
ei
n
t
e
g
r
a
l
st
h
a
tg
oi
n
t
ot
h
eCL(
r
)a
n
dSL(
r
)ma
ywe
l
l
b
ed
a
u
n
t
i
n
gt
oap
e
r
s
ona
r
me
d
′
wi
t
hp
e
na
n
dp
a
p
e
r(
d
e
p
e
n
d
i
n
gonh
ow n
a
s
t
yJ
x
()i
s
)b
u
tt
h
e
ya
r
ev
e
r
yd
e
f
i
n
i
t
e
l
y
c
omp
u
t
a
b
l
ewi
t
hac
omp
u
t
e
r
!
Now,wemu
s
tu
s
es
e
v
e
r
a
li
n
t
e
r
e
s
t
i
n
gob
s
e
r
v
a
t
i
on
s
.F
i
r
s
tofa
l
l
,J
r
()g
e
t
ss
ma
l
l
L
r
a
p
i
d
l
yi
n
s
i
deda
sℓi
n
c
r
e
a
s
e
s(
b
e
y
on
dk
d
)
.Th
i
si
st
h
ea
n
gu
l
a
rmome
n
t
u
mc
u
t
–o
ffi
n
d
i
s
g
u
i
s
ea
n
dy
ous
h
ou
l
dr
e
me
mb
e
ri
t
.Th
i
sme
a
n
st
h
a
ti
fJ
(
r
)i
ss
e
n
s
i
b
l
yb
ou
n
d
e
d
,
t
h
e
′
i
n
t
e
g
r
a
l
ont
h
er
i
g
h
t(
wh
i
c
hi
sc
u
toffa
tr=d
)wi
l
l
g
e
ts
ma
l
l
f
or“
l
a
r
g
e
”ℓ
.I
nmos
tc
a
s
e
s
ofp
h
y
s
i
c
a
l
i
n
t
e
r
e
s
t
,
k
d<
<1b
yh
y
p
ot
h
e
s
i
sa
n
dwen
e
e
don
l
yk
e
e
pt
h
ef
i
r
s
tf
e
wt
e
r
ms(
!
)
.
I
np
r
a
c
t
i
c
a
l
l
ya
l
loft
h
e
s
ec
a
s
e
s
,t
h
el
owe
s
tor
d
e
rt
e
r
m(
ℓ=0
)wi
l
ly
i
e
l
da
ne
x
c
e
l
l
e
n
t
a
p
p
r
ox
i
ma
t
i
on
.
Th
i
st
e
r
mp
r
od
u
c
e
st
h
ee
l
e
c
t
r
i
cd
i
p
o
l
er
a
d
i
a
t
i
onf
i
e
l
d.
13.
5.
2 Di
p
o
l
eRa
d
i
a
t
i
o
n
L
e
tu
se
v
a
l
u
a
t
et
h
i
st
e
r
m.
I
ti
s(
c
.
f
.
J
9
.
1
3
)
:
i
kr
Ar
()=
µ0e
4
πr
r
′
3
′
J
r
(
)
d
r
0
(
1
3
.
1
1
8
)
√
∗
(
n
ot
e
:Y00(
ˆ
r
)=Y00(
ˆ
r
)=1
/4
π)
.I
fwei
n
t
e
g
r
a
t
et
h
i
st
e
r
mb
yp
a
r
t
s(
as
u
r
p
r
i
s
i
n
g
l
yd
i
ffic
u
l
t
c
h
or
et
h
a
twi
l
l
b
ea
ne
x
e
r
c
i
s
e
)a
n
du
s
et
h
ec
on
t
i
n
u
i
t
ye
qu
a
t
i
on
a
n
dt
h
ef
a
c
tt
h
a
tt
h
es
ou
r
c
ei
sh
a
r
mon
i
cweg
e
t
:
i
k
r
Ar
()=− i
µ0ω e
(
1
3
.
1
1
9
)
p
4
π
r
wh
e
r
e
p=r′
ρr
(′
)
d
3r
′(
1
3
.
1
2
0
)i
st
h
ee
l
e
c
t
r
i
cd
i
p
o
l
emo
me
n
t
(
s
e
eJ
4
.
8
)
.
Not
et
h
a
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p
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ob
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ort
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on
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ont
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se
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e
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t
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l
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l
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i
si
swon
d
e
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f
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l
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ys
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mp
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fon
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ywec
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e
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orp
ot
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l
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a
s
,
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o.
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e
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on
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u
c
tt
h
ee
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e
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t
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gn
e
t
i
cf
i
e
l
db
e
i
n
gr
a
d
i
a
t
e
da
wa
yf
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omt
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es
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e
f
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omt
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ee
x
p
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e
s
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i
on
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e
v
i
ou
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yg
i
v
e
n
a
n
d
Af
t
e
rat
r
e
me
n
dou
sa
mou
n
tofs
t
r
a
i
g
h
t
f
or
wa
r
db
u
tn
on
e
t
h
e
l
e
s
sd
i
ffic
u
l
ta
l
g
e
b
r
at
h
a
t
y
ouwi
l
l
doa
n
dh
a
n
di
nn
e
x
twe
e
k(
s
e
ep
r
ob
l
e
ms
)y
ouwi
l
l
ob
t
a
i
n
:
2
i
k
r
e
1
n×
p) r
H= 4π(
k
r
1− i
c
k
(
1
3
.
1
2
1
)
a
n
d
1
2
i
k
r
e
1
3
i
k
2
i
kr
n×
p)×n r +[
e(
13.
122)
E= 4πǫ0 k(
3
n(
np· )−
p] r − r
Th
ema
g
n
e
t
i
cf
i
e
l
di
sa
l
wa
y
st
r
a
n
s
v
e
r
s
et
ot
h
er
a
d
i
a
l
v
e
c
t
or
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l
e
c
t
r
i
cd
i
p
o
l
er
a
d
i
a
t
i
oni
s
t
h
e
r
e
f
or
ea
l
s
oc
a
l
l
e
dt
r
a
n
s
v
e
r
s
ema
gn
e
t
i
cr
a
di
a
t
i
on
.Th
ee
l
e
c
t
r
i
cf
i
e
l
di
st
r
a
n
s
v
e
r
s
ei
n
t
h
ef
a
rz
on
e
,
b
u
ti
nt
h
en
e
a
rz
on
ei
twi
l
l
h
a
v
eac
omp
on
e
n
t(
i
nt
h
e
pd
i
r
e
c
t
i
on
)t
h
a
ti
sn
ot
g
e
n
e
r
a
l
l
yp
e
r
p
e
n
d
i
c
u
l
a
rt
on
.
As
y
mp
t
ot
i
cp
r
op
e
r
t
i
e
si
nt
h
eZo
n
e
s
I
nt
h
en
e
a
rz
o
n
eweg
e
t
:
i
ωµ0
B =µ0H=
1
2
4
π(
nˆ×
p)r
1
1
(
1
3
.
1
2
3
)
3
3nˆ(
nˆp·)−
p]r
(
1
3
.
1
2
4
)
E = 4πǫ0 [
a
n
dc
a
nu
s
u
a
l
l
yn
e
g
l
e
c
tt
h
ema
g
n
e
t
i
cf
i
e
l
dr
e
l
a
t
i
v
et
ot
h
ee
l
e
c
t
r
i
cf
i
e
l
d(
i
ti
ss
ma
l
l
e
rb
ya
f
a
c
t
orofk
r<
<1
)
.Th
ee
l
e
c
t
r
i
cf
i
e
l
di
st
h
a
tofa“
s
t
a
t
i
c
”d
i
p
ol
e(
J
4
.
1
3
)os
c
i
l
l
a
t
i
n
g
h
a
r
mon
i
c
a
l
l
y
.
I
nt
h
ef
a
rz
on
eweg
e
t
:
2
c
kµ0
i
kr
e
B =µ0H=
4π (
n
ˆ×
p) r
i
c
(
1
3
.
1
2
5
)
E = k∇×B=c
(
1
3
.
1
2
6
)
B×n
ˆ.
Th
i
si
st
r
a
n
s
v
e
r
s
eEM r
a
di
a
t
i
on
.Ex
p
a
n
d
e
da
b
ou
ta
n
yp
oi
n
t
,i
tl
ook
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u
s
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i
k
eap
l
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n
e
wa
v
e(
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i
c
hi
sh
ow“
p
l
a
n
ewa
v
e
s
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r
eb
or
n
!
)
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r
emos
ti
n
t
e
r
e
s
t
e
d
,
a
sy
ouk
n
ow,
i
n
t
h
er
a
d
i
a
t
i
onz
on
ea
n
ds
owewi
l
l
f
oc
u
soni
tf
oramome
n
t
.
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n
e
r
gyr
a
d
i
a
t
e
db
yt
h
ed
i
p
o
l
e
Re
c
a
l
l
ou
rol
db
u
d
d
yt
h
ec
o
mp
l
e
xPoy
n
t
i
n
gv
e
c
t
o
rf
orh
a
r
mon
i
cf
i
e
l
d
s(
J
6
.
1
3
2
)
:
1
∗
S= 2ReE×H
.
(
1
3
.
1
2
7
)
Th
ef
a
c
t
orof1
/
2c
ome
sf
r
omt
i
mea
v
e
r
a
g
i
n
gt
h
ef
i
e
l
d
s
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i
si
st
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ee
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e
r
g
yp
e
ru
n
i
ta
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a
p
e
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n
i
tt
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met
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a
tp
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s
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e
sap
oi
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ti
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a
c
e
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i
n
dt
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et
i
mea
v
e
r
a
g
ep
owe
rp
e
rs
ol
i
d
a
n
g
l
e
,wemu
s
tr
e
l
a
t
et
h
en
or
ma
la
r
e
at
h
r
ou
g
hwh
i
c
ht
h
ee
n
e
r
g
yf
l
u
xp
a
s
s
e
st
ot
h
e
s
ol
i
da
n
g
l
e
:
2
d
An=rd
Ω
(
1
3
.
1
2
8
)
a
n
dp
r
oj
e
c
tou
tt
h
ea
p
p
r
op
r
i
a
t
ep
i
e
c
eofS,
i
.
e
.
—n·S.
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e
t(
wi
t
hµ=1
)
dP
1
2
∗
[
rn·(
E×H)
]
.
dΩ =2Re
(
1
3
.
1
2
9
)
wh
e
r
ewemu
s
tp
l
u
gi
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n
dHf
r
omt
h
ee
x
p
r
e
s
s
i
on
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b
ov
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ort
h
ef
a
rf
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e
l
d
.
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t
e
rab
u
n
c
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l
g
e
b
r
at
h
a
tI
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ms
u
r
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ouwi
l
l
e
n
j
oyd
oi
n
g
,
y
ouwi
l
l
ob
t
a
i
n
:
2
c
dP
dΩ
µ0
2
=32π
4
2
ǫ0k |
(
n×
p)×n|
.
(
1
3
.
1
3
0
)
Th
ep
ol
a
r
i
z
a
t
i
onoft
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er
a
d
i
a
t
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oni
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t
e
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e
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yt
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e
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t
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n
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i
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et
h
ea
b
s
ol
u
t
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a
l
u
es
i
g
n
s
.
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h
i
son
eme
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n
st
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a
ton
ec
a
np
r
oj
e
c
tou
te
a
c
hc
omp
on
e
n
tof
p (
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n
dh
e
n
c
et
h
er
a
d
i
a
t
i
on
)b
e
f
or
ee
v
a
l
u
a
t
i
n
gt
h
es
qu
a
r
ei
n
d
e
p
e
n
d
e
n
t
l
y
,
i
fs
o
d
e
s
i
r
e
d
.Not
et
h
a
tt
h
ed
i
ffe
r
e
n
tc
omp
on
e
n
t
sof
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e
e
dn
oth
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v
et
h
es
a
mep
h
a
s
e
(
e
l
l
i
p
t
i
c
a
l
p
ol
a
r
i
z
a
t
i
on
,
e
t
c
.
)
.
I
fa
l
l
t
h
ec
omp
on
e
n
t
sof
p(
i
ns
omec
oor
d
i
n
a
t
es
y
s
t
e
m)h
a
v
et
h
es
a
mep
h
a
s
e
,
t
h
e
n
p
n
e
c
e
s
s
a
r
i
l
yl
i
e
sa
l
on
gal
i
n
ea
n
dt
h
et
y
p
i
c
a
la
n
g
u
l
a
rd
i
s
t
r
i
b
u
t
i
oni
st
h
a
tof(
l
i
n
e
a
r
l
y
p
ol
a
r
i
z
e
d)di
p
ol
er
a
di
a
t
i
o
n
:
dP
dΩ
2
c
2
=32π
µ0
4
2
2
ǫ0 k p
|| si
nθ
(
1
3
.
1
3
1
)
wh
e
r
eθi
sme
a
s
u
r
e
db
e
t
we
e
npa
n
dn
.Wh
e
ny
oui
n
t
e
g
r
a
t
eov
e
rt
h
ee
n
t
i
r
es
ol
i
da
n
g
l
e
(
a
sp
a
r
tofy
ou
ra
s
s
i
g
n
me
n
t
)y
ouob
t
a
i
nt
h
et
ot
a
l
p
owe
rr
a
d
i
a
t
e
d
:
24
ck
P= 12π
4
µ0
2
||
ǫ0 p
(
1
3
.
1
3
2
)
Th
emos
ti
mp
or
t
a
n
tf
e
a
t
u
r
eoft
h
i
si
st
h
ek de
p
e
n
d
e
n
c
ewh
i
c
hi
s
,
a
f
t
e
ra
l
l
,
wh
yt
h
es
k
y
i
sb
l
u
e(
a
swes
h
a
l
l
s
e
e
,
n
e
v
e
rf
e
a
r
)
.
E
x
a
mp
l
e
:Ac
e
n
t
e
r
f
e
d
,
l
i
n
e
a
ra
n
t
e
n
n
a
I
nt
h
i
sa
n
t
e
n
n
a
,
d<
<λa
n
d
I
(
z
,
t
)=I
−i
ωt
e
.
1 2|z|
− d
0
(
1
3
.
1
3
3
)
F
r
omt
h
ec
on
t
i
n
u
i
t
ye
qu
a
t
i
on(
a
n
dal
i
t
t
l
es
u
b
t
l
eg
e
ome
t
r
y
)
,
′ −i
ωt
∇·
J=
∂ρ(
z
)
e
d
I
dz =−
∂t
′
=i
ωρ(
z
)
(
1
3
.
1
3
4
)
a
n
dwef
i
n
dt
h
a
tt
h
el
i
n
e
a
rc
h
a
r
g
ede
n
s
i
t
y(
p
a
r
t
i
c
i
p
a
t
i
n
gi
nt
h
eos
c
i
l
l
a
t
i
on
,wi
t
ha
p
r
e
s
u
me
dn
e
u
t
r
a
l
b
a
c
k
g
r
ou
n
d
)i
si
n
d
e
p
e
n
d
e
n
tofz
:
2
i
I
0
′
ρ(
z
)= ±ωd
(
1
3
.
1
3
5
)
wh
e
r
et
h
e+
/
−s
i
g
ni
n
di
c
a
t
e
st
h
eu
p
p
e
r
/
l
owe
rb
r
a
n
c
hoft
h
ea
n
t
e
n
n
aa
n
dt
h
e
′
me
a
n
st
h
a
twea
r
er
e
a
l
l
yt
r
e
a
t
i
n
gρ/
(
d
x
d
y
)(
wh
i
c
hc
a
n
c
e
l
st
h
er
e
l
a
t
e
dt
e
r
msi
nt
h
e
v
ol
u
mei
n
t
e
g
r
a
l
b
e
l
ow)
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a
nt
h
e
ne
v
a
l
u
a
t
et
h
ed
i
p
ol
emome
n
toft
h
ee
n
t
i
r
ea
n
t
e
n
n
a
f
ort
h
i
sf
r
e
qu
e
n
c
y
:
d
/
2
p
z=
i
I
d.
0
2ω
′
z
ρ(
z
)
dz=
−d
/
2
(
1
3
.
1
3
6
)
Th
ee
l
e
c
t
r
i
ca
n
dma
g
n
e
t
i
cf
i
e
l
dsf
orr>di
nt
h
ee
l
e
c
t
r
i
cd
i
p
ol
ea
p
p
r
ox
i
ma
t
i
ona
r
e
n
owg
i
v
e
nb
yt
h
ep
r
e
v
i
ou
s
l
yd
e
r
i
v
e
de
x
p
r
e
s
s
i
on
s
.Th
ea
n
g
u
l
a
rd
i
s
t
r
i
b
u
t
i
onofr
a
d
i
a
t
e
d
p
owe
ri
s
2
2
2
µ0
I
d
P
0
=
d
Ω
a
n
dt
h
et
ot
a
l
r
a
d
i
a
t
e
dp
owe
ri
s
2
128π
2
P=
(
k
d
)
s
i
n
θ
(
13
.
1
37
)
ǫ0
2
I
k
d)
0(
4
8
π
µ0
ǫ0
.
(
1
3
.
1
3
8
)
Re
ma
r
k
s
.F
orf
i
x
e
dc
u
r
r
e
n
tt
h
ep
owe
rr
a
d
i
a
t
e
di
n
c
r
e
a
s
e
sa
st
h
es
qu
a
r
eoft
h
ef
r
e
qu
e
n
c
y
(
a
tl
e
a
s
twh
e
nk
d<
<1
,i
.e
.–l
on
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v
e
l
e
n
g
t
h
sr
e
l
a
t
i
v
et
ot
h
es
i
z
eoft
h
ea
n
t
e
n
n
a
)
.Th
e
t
ot
a
lp
owe
rr
a
d
i
a
t
e
db
yt
h
ea
n
t
e
n
n
aa
p
p
e
a
r
sa
sa“
l
os
s
”i
n“
Oh
m’
sL
a
w”f
ort
h
ea
n
t
e
n
n
a
.
2
F
a
c
t
or
i
n
gou
tI
2
,t
h
er
e
ma
i
n
d
e
rmu
s
th
a
v
et
h
eu
n
i
t
sofr
e
s
i
s
t
a
n
c
ea
n
di
sc
a
l
l
e
dt
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e
0/
r
a
d
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a
t
i
onr
e
s
i
s
t
a
n
c
eoft
h
ea
n
t
e
n
n
a
:
Rrad=
2
(
k
d)
= 24π
2P
2
I
0
µ0
ǫ0
2
≈5
(
k
d
) oh
ms
)
(
1
3
.
1
3
9
)
wh
e
r
ewed
ot
h
el
a
t
t
e
rmu
l
t
i
p
l
i
c
a
t
i
ont
oc
on
v
e
r
tt
h
er
e
s
u
l
t
i
n
gu
n
i
t
st
ooh
ms
.Not
et
h
a
t
t
h
i
sr
e
s
i
s
t
a
n
c
ei
st
h
e
r
ef
orh
a
r
mon
i
cc
u
r
r
e
n
t
se
v
e
ni
ft
h
ec
on
du
c
t
i
v
i
t
yoft
h
eme
t
a
li
s
p
e
r
f
e
c
t
.Not
ef
u
r
t
h
e
rt
h
a
tb
yh
y
p
ot
h
e
s
i
st
h
i
se
x
p
r
e
s
s
i
onwi
l
lon
l
yb
ev
a
l
i
df
ors
ma
l
l
v
a
l
u
e
sofRr
.
a
d
Goodg
ol
l
y
,t
h
i
si
swon
d
e
r
f
u
l
.Weh
op
e
f
u
l
l
yr
e
a
l
l
yu
n
d
e
r
s
t
a
n
de
l
e
c
t
r
i
cdi
p
ol
e
r
a
d
i
a
t
i
ona
tt
h
i
sp
oi
n
t
.I
twou
l
db
et
r
u
l
ys
u
b
l
i
mei
fa
l
lr
a
d
i
a
t
or
swe
r
ed
i
p
ol
er
a
d
i
a
t
or
s
.
Ph
y
s
i
c
swou
l
db
es
oe
a
s
y
.Bu
t(
a
l
a
s
)s
ome
t
i
me
st
h
ec
u
r
r
e
n
td
i
s
t
r
i
b
u
t
i
onh
a
sn
oℓ=0
mo
me
n
ta
n
dt
h
e
r
ei
st
h
e
r
e
f
or
en
od
i
p
ol
et
e
r
m!I
nt
h
a
tc
a
s
ewemu
s
tl
ooka
tt
h
en
e
x
t
t
e
r
mors
oi
nt
h
emu
l
t
i
p
ol
a
re
x
p
a
n
s
i
on
s
L
e
s
ty
out
h
i
n
kt
h
a
tt
h
i
si
sawh
ol
l
yu
n
l
i
k
e
l
yoc
c
u
r
r
a
n
c
e
,
p
l
e
a
s
en
ot
et
h
a
tah
u
mb
l
e
l
oopc
a
r
r
y
i
n
gac
u
r
r
e
n
tt
h
a
tv
a
r
i
e
sh
a
r
mon
i
c
a
l
l
yi
son
es
u
c
hs
y
s
t
e
m.Sol
e
tu
sp
r
oc
e
e
d
t
o:
13.
6 Ma
gn
e
t
i
cDi
p
o
l
ea
n
dE
l
e
c
t
r
i
cQu
a
d
r
u
p
o
l
eRa
d
i
a
t
i
o
n
F
i
e
l
d
s
Th
en
e
x
tt
e
r
mi
nt
h
emu
l
t
i
p
ol
a
re
x
p
a
n
s
i
oni
st
h
eℓ=1t
e
r
m:
Ax
()=i
k
µ
0
1
+
h(
k
r
)
1
Y
1
,
m
(
ˆ
r
)
m=
−1
∞
′
′
J
(
x)
j
(
k
r
)
Y
1
0
∗ 3′
1
,
m
(
r̂)dx
′
(
1
3
.
1
4
0
)
Wh
e
ny
ou(
f
orh
ome
wor
k
,
ofc
ou
r
s
e
)
a
)m–s
u
mt
h
ep
r
od
u
c
toft
h
eYℓ,
s
m’
′
b
)u
s
et
h
es
ma
l
l
k
re
x
p
a
n
s
i
onf
orj
(
k
r
)i
nt
h
ei
n
t
e
g
r
a
l
a
n
dc
omb
i
n
ei
twi
t
ht
h
e
1
e
x
p
l
i
c
i
tf
or
mf
ort
h
er
e
s
u
l
t
i
n
gP1(
θ
)t
of
or
madotp
r
od
u
c
t
c
)c
a
n
c
e
l
t
h
e2
ℓ+1
’
s
d
)e
x
p
l
i
c
i
t
l
ywr
i
t
eou
tt
h
eh
a
n
k
e
l
f
u
n
c
t
i
oni
ne
x
p
on
e
n
t
i
a
l
f
or
m
y
ouwi
l
l
g
e
te
qu
a
t
i
on(
J
9
.
3
0
,
f
or–r
e
c
a
l
l
–d
i
s
t
r
i
b
u
t
i
on
swi
t
hc
omp
a
c
ts
u
p
p
or
t
)
:
i
kr
µ0e
πr
Ax
()= 4
1
k
r−i
∞
0
′
′ 3′
J
(
x)
(
n·x)
dx.
(
1
3
.
1
4
1
)
Ofc
ou
r
s
e
,
y
ouc
a
ng
e
ti
td
i
r
e
c
t
l
yf
r
omJ
9
.
9(
t
oal
owe
ra
p
p
r
ox
i
ma
t
i
on
)a
swe
l
l
,
b
u
t
t
h
a
td
oe
sn
o
ts
h
owy
ouwh
a
tt
odoi
ft
h
es
ma
l
lk
ra
p
p
r
ox
i
ma
t
i
oni
sn
otv
a
l
i
d(
i
ns
t
e
p2
a
b
ov
e
)a
n
di
tn
e
g
l
e
c
t
sp
a
r
toft
h
eou
t
g
oi
n
gwa
v
e
!
Th
e
r
ea
r
et
woi
mp
or
t
a
n
ta
n
di
n
de
p
e
n
d
e
n
tp
i
e
c
e
si
nt
h
i
se
x
p
r
e
s
s
i
on
.On
eoft
h
et
wo
′
p
i
e
c
e
si
ss
y
mme
t
r
i
ci
nJa
n
d
xa
n
dt
h
eot
h
e
ri
sa
n
t
i
s
y
mme
t
r
i
c(
g
e
tami
n
u
ss
i
g
nwh
e
n
t
h
ec
oor
d
i
n
a
t
es
y
s
t
e
mi
si
n
v
e
r
t
e
d
)
.An
yv
e
c
t
orqu
a
n
t
i
t
yc
a
nb
ed
e
c
omp
os
e
di
nt
h
i
s
ma
n
n
e
rs
ot
h
i
si
sav
e
r
yg
e
n
e
r
a
l
s
t
e
p
:
′
J
(
n·x)=
1
′
′
(
n·x)
J+(
n·J
)
x]
+
2[
1
′
x×J
)×n
.
2(
(
1
3
.
1
4
2
)
13.
6.
1 Ma
gn
e
t
i
cDi
p
o
l
eRa
d
i
a
t
i
on
L
e
t
’
sl
ooka
tt
h
ea
n
t
i
s
y
mme
t
r
i
cb
i
tf
i
r
s
t
,
a
si
ti
ss
ome
wh
a
ts
i
mp
l
e
ra
n
dwec
a
nl
e
v
e
r
a
g
e
ou
re
x
i
s
t
i
n
gr
e
s
u
l
t
s
.Th
es
e
c
on
dt
e
r
mi
st
h
ema
g
n
e
t
i
z
a
t
i
on(
d
e
n
s
i
t
y
)d
u
et
ot
h
ec
u
r
r
e
n
t
J
:
1
M= 2x
(×J
)
(
1
3
.
1
4
3
)
(
s
e
eJ
5
.
5
3
,
5
.
5
4
)s
ot
h
a
t
′ 3′
M(
x)
dx
m=
(
1
3
.
1
4
4
)
wh
e
r
e
mi
st
h
ema
g
n
e
t
i
cd
i
p
ol
emome
n
toft
h
e(
f
ou
r
i
e
rc
omp
on
e
n
tof
)t
h
ec
u
r
r
e
n
t
.
Con
s
i
d
e
r
i
n
gon
l
yt
h
i
sa
n
t
i
s
y
mme
t
r
i
ct
e
r
m,
wes
e
et
h
a
t
:
i
k
µ0
AM1x
()=
i
k
r
e
4π (
n×m) r
1
1− i
k
r
.
(
1
3
.
1
4
5
)
HMMMMMMM,(
y
ouh
a
db
e
t
t
e
rs
a
y
)
!Th
i
sl
ook
s“
j
u
s
tl
i
k
e
”t
h
ee
x
p
r
e
s
s
i
onf
ort
h
e
ma
g
n
e
t
i
cf
i
e
l
dt
h
a
tr
e
s
u
l
t
e
df
r
om t
h
ee
l
e
c
t
r
i
cd
i
p
ol
ev
e
c
t
orp
ot
e
n
t
i
a
l
.Su
r
ee
n
ou
g
h
,
wh
e
ny
ou(
f
orh
ome
wor
k
)c
r
a
n
kou
tt
h
ea
l
g
e
b
r
a
,
y
ouwi
l
l
s
h
ow
t
h
a
t
i
k
r
e
µ0
B= 4π
2
k(
n×m)×n
1 i
k
r +[
3
n
(
n·m)−m]
a
n
d
1 µ0
E=− 4π
2
i
k
r
e
ǫ0 k(
n×m) r
3
2
r −
r
i
k
r
e
(
1
3
.
1
4
6
)
1
1− i
k
r.
(
1
3
.
1
4
7
)
Cl
e
a
r
l
y
,
wed
on
’
tn
e
e
dt
od
i
s
c
u
s
st
h
eb
e
h
a
v
i
oroft
h
ef
i
e
l
d
si
nt
h
ez
on
e
ss
i
n
c
et
h
e
y
a
r
ec
omp
l
e
t
e
l
ya
n
a
l
og
ou
s
.Th
ee
l
e
c
t
r
i
cf
i
e
l
di
sa
l
wa
y
st
r
a
n
s
v
e
r
s
e
,a
n
dt
h
et
ot
a
lf
i
e
l
d
a
r
i
s
e
sf
r
omah
a
r
mon
i
cma
g
n
e
t
i
cd
i
p
ol
e
.F
ort
h
i
sr
e
a
s
on
,
t
h
i
sk
i
n
dofr
a
di
a
t
i
oni
sc
a
l
l
e
d
e
i
t
h
e
rma
gn
e
t
i
cd
i
p
o
l
e(
M1
)r
a
d
i
a
t
i
onort
r
a
n
s
v
e
r
s
ee
l
e
c
t
r
i
cr
a
d
i
a
t
i
on
.F
orwh
a
ti
t
’
s
wor
t
h
,
e
l
e
c
t
r
i
cd
i
p
ol
er
a
d
i
a
t
i
oni
sa
l
s
oc
a
l
l
e
d(
E
1
)r
a
d
i
a
t
i
on
.
Howe
v
e
r
,t
h
i
si
son
l
yONEp
a
r
toft
h
ec
on
t
r
i
b
u
t
i
onf
r
omℓ=1t
e
r
msi
nt
h
eGr
e
e
n
’
s
f
u
n
c
t
i
one
x
p
a
n
s
i
on
.
Wh
a
ta
b
ou
tt
h
eot
h
e
r(
s
y
mme
t
r
i
c
)p
i
e
c
e
?Oooo,
ou
c
h
.
13.
6.
2 E
l
e
c
t
r
i
cQu
a
d
r
u
p
ol
eRa
d
i
a
t
i
on
Nowl
e
t
’
st
ou
n
t
a
n
g
l
et
h
ef
i
r
s
t(
s
y
mme
t
r
i
c
)p
i
e
c
e
.Th
i
swi
l
lt
u
r
nou
tt
ob
ear
e
ma
r
k
a
b
l
y
u
n
p
l
e
a
s
a
n
tj
ob
.I
nf
a
c
ti
ti
smyn
e
f
a
r
i
ou
sa
n
ds
a
d
i
s
t
i
cp
l
a
nt
h
a
ti
tb
es
ou
n
p
l
e
a
s
a
n
tt
h
a
t
i
tp
r
op
e
r
l
ymot
i
v
a
t
e
sac
h
a
n
g
ei
na
p
p
r
oa
c
ht
oon
et
h
a
th
a
n
d
l
e
st
h
i
sn
a
s
t
yt
e
n
s
ors
t
u
ff
“
n
a
t
u
r
a
l
l
y
”
.
Weh
a
v
et
oe
v
a
l
u
a
t
et
h
ei
n
t
e
g
r
a
l
oft
h
es
y
mme
t
r
i
cp
i
e
c
e
.
Weg
e
t
:
′
1
′3′
i
ω x
[
(
n
ˆx)
J+(
n
ˆJ
x
)]
dx=
·
2
Th
es
t
e
p
si
n
v
ol
v
e
da
r
e
:
·
′
′
′ 3′
(
n
ˆx)
ρx
()
dx
(
1
3
.
1
4
8
)
·
− 2
a
)i
n
t
e
g
r
a
t
eb
yp
a
r
t
s(
wor
k
i
n
gt
oob
t
a
i
nd
i
v
e
r
g
e
n
c
e
sofJ
)
.
b
)c
h
a
n
g
i
n
g∇·Ji
n
t
oaρt
i
me
swh
a
t
e
v
e
rf
r
omt
h
ec
on
t
i
n
u
i
t
ye
qu
a
t
i
on(
f
ora
h
a
r
mon
i
cs
ou
r
c
e
)
.
c
)r
e
a
r
r
a
n
g
i
n
ga
n
dr
e
c
omb
i
n
i
n
g
.
Don
’
tf
or
g
e
tt
h
eb
ou
n
d
a
r
yc
on
d
i
t
i
ona
ti
n
f
i
n
i
t
y(
Ja
n
dρh
a
v
ec
omp
a
c
ts
u
p
p
or
t
)
!You
’
l
l
l
ov
ed
oi
n
gt
h
i
son
e
.
.
.
Th
ev
e
c
t
orp
ot
e
n
t
i
a
l
i
st
h
u
s
:
Ax
()=
E2
′
2
µ0ck
− 8π
i
kr
e
r
1
1
−
x
i
k
r
′
′
′ 3′
(
n
ˆx)
ρx
()
dx.
(
1
3
.
1
4
9
)
·
Not
et
h
a
t
xa
p
p
e
a
r
st
wi
c
eu
n
d
e
rt
h
ei
n
t
e
g
r
a
l
,a
n
dt
h
a
ti
t
sv
e
c
t
orc
h
a
r
a
c
t
e
rs
i
mi
l
a
r
l
y
′
a
p
p
e
a
r
st
wi
c
e
:on
c
ei
n
xi
t
s
e
l
fa
n
don
c
ei
ni
t
sp
r
oj
e
c
t
i
ononn
ˆ
.Th
ei
n
t
e
g
r
a
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5
9
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26
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ck
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(
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3
.
1
6
2
)
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a
r
t
i
c
u
l
a
r
,d
e
r
i
v
et
h
ec
ommu
t
a
t
i
onr
u
l
e
sf
ort
h
er
a
i
s
i
n
g
a
n
dl
owe
r
i
n
gop
e
r
a
t
or
sf
r
omt
h
ec
a
r
t
e
s
i
a
nc
ommu
t
a
t
i
onr
e
l
a
t
i
on
sf
orL
.F
r
omt
h
e
c
ommu
t
a
t
i
onr
u
l
e
sa
n
dL
e
r
i
v
et
h
e(
n
or
ma
l
i
z
e
d
)a
c
t
i
onofL
n
zY
ℓ
m =mY
ℓ
md
±o
Yℓ,
m.
k
)J
a
c
k
s
on
,
p
r
ob
l
e
ms9
.
2
,
9
.
3
,
9
.
4
Ch
a
p
t
e
r14
Ve
c
t
o
rMu
l
t
i
p
o
l
e
s
AsIn
ot
e
dj
u
s
ta
b
ov
e
,we
’
r
ea
l
r
e
a
d
yh
a
l
fwa
yt
h
r
ou
g
hJ
9
.
6
,wh
i
c
hi
smos
t
l
yt
h
er
e
v
i
e
wof
s
p
h
e
r
i
c
a
lb
e
s
s
e
l
,
n
e
u
ma
n
n
,
a
n
dh
a
n
k
e
lf
u
n
c
t
i
on
st
h
a
tweh
a
v
ej
u
s
th
a
d
.Th
er
e
ma
i
n
d
e
ri
sa
l
i
g
h
t
n
i
n
gr
e
v
i
e
wofs
c
a
l
a
rs
p
h
e
r
i
c
a
l
h
a
r
mon
i
c
s
.
Si
n
c
ewe
’
r
ea
b
ou
tt
oge
n
e
r
a
l
i
z
et
h
a
tc
on
c
e
p
t
,
we
’
l
l
qu
i
c
k
l
yg
oov
e
rt
h
eh
i
g
hp
a
r
t
s
.
14.
1 An
gu
l
a
rmome
n
t
u
ma
n
ds
p
h
e
r
i
c
a
l
h
a
r
mo
n
i
c
s
2
Th
ea
n
g
u
l
a
rp
a
r
toft
h
eL
a
p
l
a
c
eop
e
r
a
t
or∇ c
a
nb
ewr
i
t
t
e
n
:
1
2
r
1∂
2
1∂
∂
si
nθ∂θ si
nθ∂θ
2
2
2
n θ∂φ
+si
2
L
2
=− r
(
1
4
.
1
)
2
E
l
i
mi
n
a
t
i
n
g−
r(
t
os
ol
v
ef
ort
h
eL d
i
ffe
r
e
n
t
i
a
le
qu
a
t
i
on
)on
en
e
e
d
st
os
ol
v
ea
n
e
i
g
e
n
v
a
l
u
ep
r
ob
l
e
m:
2
Lψ=e
ψ
(
1
4
.
2
)
wh
e
r
eea
r
et
h
ee
i
g
e
n
v
a
l
u
e
s
,
s
u
b
j
e
c
tt
ot
h
ec
on
di
t
i
ont
h
a
tt
h
es
ol
u
t
i
onb
es
i
n
g
l
ev
a
l
u
e
d
onφ∈[
0
,
2
π)a
n
dθ∈[
0
,
π]
.
Th
i
se
qu
a
t
i
one
a
s
i
l
ys
e
p
a
r
a
t
e
si
nθ
,
φ.Th
eφe
qu
a
t
i
oni
st
r
i
v
i
a
l
–s
ol
u
t
i
on
sp
e
r
i
od
i
ci
nφ
a
r
ei
n
d
e
x
e
dwi
t
hi
n
t
e
g
e
rm.Th
eθe
qu
a
t
i
onon
eh
a
st
owor
ka
tab
i
t–t
h
e
r
ea
r
ec
on
s
t
r
a
i
n
t
s
ont
h
es
ol
u
t
i
on
st
h
a
tc
a
nb
eob
t
a
i
n
e
df
o
ra
n
yg
i
v
e
nm–b
u
tt
h
e
r
ea
r
ema
n
ywa
y
st
os
ol
v
ei
t
a
n
da
tt
h
i
sp
oi
n
ty
ous
h
ou
l
dk
n
owt
h
a
ti
t
ss
ol
u
t
i
on
sa
r
ea
s
s
o
c
i
a
t
e
dL
e
ge
n
d
r
ep
o
l
y
n
o
mi
a
l
s
Pℓ,
x
)wh
e
r
ex=c
osθ.
Th
u
s
m(
t
h
ee
i
g
e
n
s
ol
u
t
i
onb
e
c
ome
s
:
2
LYℓm=ℓ
(
ℓ+1
)
Yℓm
(
1
4
.
3
)
wh
e
r
eℓ=0
,
1
,
2
.
.
.a
n
dm=−
ℓ
,
−
ℓ+1
,
.
.
.
,
ℓ−1
,
ℓa
n
di
st
y
p
i
c
a
l
l
yor
t
h
on
or
ma
l
(
i
z
e
d)on
t
h
es
ol
i
da
n
g
l
e4
π.
1
7
7
Th
ea
n
g
u
l
a
rp
a
r
toft
h
eL
a
p
l
a
c
i
a
ni
sr
e
l
a
t
e
dt
ot
h
ea
n
gu
l
a
rmo
me
n
t
u
mofawa
v
ei
n
qu
a
n
t
u
mt
h
e
or
y
.
I
nu
n
i
t
swh
e
r
e=1
,
t
h
ea
n
g
u
l
a
rmome
n
t
u
mop
e
r
a
t
ori
s
:
1
x
(×∇)
L= i
(
1
4
.
4
)
a
n
d
2
2
2
2
L =L
x +L
y +L
z
(
1
4
.
5
)
Not
et
h
a
ti
na
l
loft
h
e
s
ee
x
p
r
e
s
s
i
on
sL
,L,L
e
t
c
.a
r
ea
l
lo
p
e
r
a
t
or
s
.Th
i
sme
a
n
st
h
a
t
z,
t
h
e
ya
r
ea
p
p
l
i
e
dt
ot
h
ef
u
n
c
t
i
on
sont
h
e
i
rr
i
g
h
t(
b
yc
on
v
e
n
t
i
on
)
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e
ny
ous
e
et
h
e
m
a
p
p
e
a
r
i
n
gb
yt
h
e
ms
e
l
v
e
s
,
r
e
me
mb
e
rt
h
a
tt
h
e
yon
l
yme
a
ns
ome
t
h
i
n
gwh
e
nt
h
e
ya
r
ea
p
p
l
i
e
d
,
s
o∇’
sou
tb
yt
h
e
ms
e
l
v
e
sont
h
er
i
g
h
ta
r
eok
.
Th
ezc
omp
on
e
n
tofLi
s
:
∂
L
i ∂φ
z=−
(
1
4
.
6
)
2
a
n
dwes
e
et
h
a
ti
nf
a
c
tYl
ms
a
t
i
s
f
i
e
st
h
et
woe
i
g
e
n
v
a
l
u
ee
qu
a
t
i
on
s
:
2
LYℓm=ℓ
(
ℓ+1
)
Yℓm
(
1
4
.
7
)
L
zY
ℓ
m =mY
ℓ
m
(
1
4
.
8
)
a
n
d
Th
eYl
m’
sc
a
n
n
otb
ee
i
g
e
n
s
ol
u
t
i
on
sofmor
et
h
a
non
eoft
h
ec
omp
on
e
n
t
sofLa
t
on
c
e
.Howe
v
e
r
,
wec
a
nwr
i
t
et
h
ec
a
r
t
e
s
i
a
nc
omp
on
e
n
t
sofLs
ot
h
a
tt
h
e
yf
or
ma
nf
i
r
s
t
r
a
n
kt
e
n
s
ora
l
ge
b
r
aofop
e
r
a
t
or
st
h
a
tt
r
a
n
s
f
or
mt
h
eYℓm,f
orag
i
v
e
nℓ
,a
mon
g
t
h
e
ms
e
l
v
e
s(
t
h
e
yc
a
n
n
o
tc
h
a
n
g
eℓ
,on
l
ymi
xm)
.Th
i
si
st
h
eh
op
e
f
u
l
l
yf
a
mi
l
i
a
rs
e
tof
e
qu
a
t
i
on
s
:
L
L
+ =L
x+i
y
(
1
4
.
9
)
L
L
− =L
x−i
y
(
1
4
.
1
0
)
L
0 =L
z
(
1
4
.
1
1
)
Th
e Ca
r
t
e
s
i
a
nc
omp
on
e
n
t
s ofL d
on
otc
ommu
t
e
.I
nf
a
c
t
,t
h
e
yf
or
m an
i
c
e
a
n
t
i
s
y
mme
t
r
i
cs
e
t
:
[
L
,
L
=i
ǫi
L
(
1
4
.
1
2
)
i
j]
j
k
k
wh
i
c
hc
a
nb
ewr
i
t
t
e
ni
nt
h
es
h
or
t
h
a
n
dn
ot
a
t
i
on
L×L=i
L
.
(
1
4
.
1
3
)
Con
s
e
qu
e
n
t
l
y
,t
h
ec
omp
on
e
n
t
se
x
p
r
e
s
s
e
da
saf
i
r
s
tr
a
n
kt
e
n
s
ora
l
s
odon
ot
c
ommu
t
ea
mon
gt
h
e
ms
e
l
v
e
s
:
[
L
,
L
]
=2
L
+
−
z
(
1
4
.
1
4
)
[
L
,
L
=∓L
±
z]
±
(
1
4
.
1
5
)
a
n
d
2
b
u
ta
l
l
t
h
e
s
ewa
y
sofa
r
r
a
n
g
i
n
gt
h
ec
omp
on
e
n
t
sofLc
ommu
t
ewi
t
hL:
2
[
L
,
L]
=0
i
(
1
4
.
1
6
)
a
n
dt
h
e
r
e
f
or
ewi
t
ht
h
eL
a
p
l
a
c
i
a
ni
t
s
e
l
f
:
2
2
[
∇,
L
]
=0
i
wh
i
c
hc
a
nb
ewr
i
t
t
e
ni
nt
e
r
msofL a
s
:
2
1∂
2
2
∇
(
1
4
.
1
7
)
2
L
2
r∂r (
r)− r
=
(
1
4
.
1
8
)
Ason
ec
a
ne
a
s
i
l
ys
h
ow e
i
t
h
e
rb
yc
on
s
i
de
r
i
n
gt
h
ee
x
p
l
i
c
ta
c
t
i
onoft
h
ea
c
t
u
a
l
d
i
ffe
r
e
n
t
i
a
lf
or
msont
h
ea
c
t
u
a
le
i
g
e
n
s
ol
u
t
i
on
sYℓm ormor
es
u
b
t
l
yb
yc
on
s
i
d
e
r
i
n
gt
h
e
a
c
t
i
onofL
nL
Yℓℓ(
a
n
ds
h
owi
n
gt
h
a
tt
h
e
yb
e
h
a
v
el
i
k
er
a
i
s
i
n
ga
n
dl
owe
rop
e
r
a
t
or
s
zo
±
f
orma
n
dp
r
e
s
e
r
v
i
n
gn
or
ma
l
i
z
a
t
i
on
)o
n
eob
t
a
i
n
s
:
L
Yℓm = (
ℓ−m)
(
ℓ+m+1
)Yℓ,
+
m+
1
(
1
4
.
1
9
)
ℓ+m)
(
ℓ−m+1
)Yℓ,
m−
1
= (
=mYℓm
(
1
4
.
2
0
)
(
1
4
.
2
1
)
L
−Y
ℓ
m
L
zY
ℓ
m
F
i
n
a
l
l
y
,n
ot
et
h
a
tLi
sa
l
wa
y
sor
t
h
og
on
a
lt
orwh
e
r
eb
ot
ha
r
ec
on
s
i
d
e
r
e
da
s
o
p
e
r
a
t
or
sa
n
dra
c
t
sf
r
omt
h
el
e
f
t
:
r·L=0
.
(
1
4
.
2
2
)
Youwi
l
ls
e
ema
n
yc
a
s
e
swh
e
r
ei
d
e
n
t
i
t
i
e
ss
u
c
ha
st
h
i
sh
a
v
et
ob
ewr
i
t
t
e
ndowni
na
p
a
r
t
i
c
u
l
a
ror
d
e
r
.
Be
f
or
eweg
oont
od
oamor
el
e
i
s
u
r
e
l
yt
ou
rofv
e
c
t
ors
p
h
e
r
i
c
a
lh
a
r
mon
i
c
s
,we
p
a
u
s
et
omot
i
v
a
t
et
h
ec
on
s
t
r
u
c
t
i
on
.
14.
2 Ma
gn
e
t
i
ca
n
dE
l
e
c
t
r
i
cMu
l
t
i
p
o
l
e
sRe
v
i
s
i
t
e
d
Asweh
a
v
en
ows
e
e
nr
e
p
e
a
t
e
d
l
yf
r
omCh
a
p
t
e
rJ
6on
,
i
nas
ou
r
c
ef
r
e
er
e
g
i
onofs
p
a
c
e
,
h
a
r
mon
i
ce
l
e
c
t
r
oma
g
n
e
t
i
cf
i
e
l
d
sa
r
ed
i
v
e
r
g
e
n
c
e
l
e
s
sa
n
dh
a
v
ec
u
r
l
sg
i
v
e
nb
y
:
∇×E =i
ωB=i
k
c
B
k
∇×B =−
icE
.
(
1
4
.
2
3
)
(
1
4
.
2
4
)
2
Byma
s
s
a
g
i
n
gt
h
e
s
eal
i
t
t
l
eb
i
t(
r
e
c
a
l
l
∇×(
∇×X)=∇(
∇·X)−∇Xa
n
d
∇·X=0f
orX=E
,
B)wec
a
ne
a
s
i
l
ys
h
owt
h
a
tb
ot
hEa
n
dBmu
s
tb
ed
i
v
e
r
g
e
n
c
e
l
e
s
s
s
ol
u
t
i
on
st
ot
h
eHHE:
2
2
(
∇ +k)
X=0
(
1
4
.
2
5
)
I
fwek
n
owas
ol
u
t
i
ont
ot
h
i
se
qu
a
t
i
onf
orX=Ewec
a
nob
t
a
i
nBf
r
omi
t
sc
u
r
l
f
r
omt
h
e
e
qu
a
t
i
ona
b
ov
e
:
i
B=− ω ∇×E
(
1
4
.
2
6
)
a
n
dv
i
c
ev
e
r
s
a
.Howe
v
e
r
,
t
h
i
si
sa
n
n
oy
i
n
gt
ot
r
e
a
td
i
r
e
c
t
l
y
,
b
e
c
a
u
s
eoft
h
ev
e
c
t
orc
h
a
r
a
c
t
or
ofEa
n
dBwh
i
c
hc
omp
l
i
c
a
t
et
h
ed
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m
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r
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(
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(
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4
.
7
2
)
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E
L.
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(
1
4
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3
)
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m
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h
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of
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i
r
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(
1
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5
)
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n
d
m
m
[
j
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j
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r
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r
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s
(
1
4
.
7
6
)
r
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L×A)=2
i
r
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ˆ
r×A)
(
1
4
.
7
7
)
r
ˆ×(
L×A)=L
(
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r·A)+i
r
ˆ×A.
(
1
4
.
7
8
)
a
n
d
Yous
h
ou
l
dg
e
t
m
m
[
j
(
j
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ℓ+1)+2]
r
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L·(
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(
1
4
.
7
9
)
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n
d
m
m
[
j
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j
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ℓ+1)
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(
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(
1
4
.
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0
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i
n
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os
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od
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c
t
,
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h
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c
a
l
a
re
qu
a
t
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on
:
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m
2
m
j
(
j
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(
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]
[
j
(
j
+1
)−ℓ
(
ℓ+1
)+2
]
(
ˆ
r·Y j
)=L(
ˆ
r·Y j
)
.
(
1
4
.
8
1
)
ℓ
ℓ
4[
m
Th
i
si
mp
l
i
e
st
h
a
t(
ˆ
r
·Yj
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sas
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r
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c
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l
h
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mon
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c
:
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h
a
ti
sac
on
s
t
a
n
t×
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m.
j
(
j
+1
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(
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2
j
(
j
+1
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(
ℓ+1
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(
k+1
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2
(
1
4
.
8
2
)
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i
sh
a
st
h
es
ol
u
t
i
on
s
a
)k=
j
(
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+
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j
(
j
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n
c
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n
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h
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m
m
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3
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4
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](
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(
1
4
.
8
5
)
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on
s
:
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1
m
−
j
r
ˆ+i
r
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]
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m
=− j
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j
+1
)[
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(
1
4
.
8
6
)
j
,
j
+
1
(
j
+1
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ˆ
r+i
r
ˆ×L
]
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m.
=− (
j
+1
)
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j
+1
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(
1
4
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7
)
j
,
j
−1
Y
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2
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j
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Y
j
,
j
−1
f d
f
2
j
+1 −j r +d
r
m
Y
j
,
j
+
1
.(
1
4
.
8
8
)
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ea
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t
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u
r
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15.
1 Th
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15.
1.
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r
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:
ML =
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f
(
k
r
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YL(
r
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)=
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m
f
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k
r
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l (
r
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5
.
2
)
ℓ
(
ℓ+1
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(
1
5
.
3
)
(
1
5
.
4
)
15.
1.
2 Th
e
i
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i
f
i
c
a
n
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t
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e
s
Th
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r
t
u
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n
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e
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t
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st
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tt
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u
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oma
t
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c
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l
l
y
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omp
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omp
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n
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si
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op
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t
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e
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ML =0
(
1
5
.
5
)
∇·
NL =0
(
1
5
.
6
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L
k
f
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k
r
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r
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L =i
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Al
s
o:
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ML
=−
i
k
NL
(
1
5
.
8
)
∇×
NL
k
ML
= i
= 0
(
1
5
.
9
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(
1
5
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1
0
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omp
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15.
1.
3 E
x
p
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ms
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i
n
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:
m
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(
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r
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Yℓℓ
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(
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5
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1
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L
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r
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m
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r
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k
r
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1
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(
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5
.
1
2
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(
1
5
.
1
3
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nd
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ffe
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e
n
t
i
a
l
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or
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m
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(
k
r
)
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1 d
(
1
5
.
1
4
)
m
(
k
r
f
)i
r
ˆ×Y ℓℓ −r
ˆℓ
(
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f
YL
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ℓ
NL = k
r d
(
k
r
)
1
d
m
L
L =
i
r
ˆ×f
Yℓℓ)−r
ˆ d
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(
k
r
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L
(
1
5
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5
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1
6
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15.
2 Gr
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t
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o
rt
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u
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l
mh
ol
t
ze
qu
a
t
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s
⇔
⇔
′
′
G±(
r
,
r
)=IG±(
r
,
r
)
(
1
5
.
1
7
)
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(
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e
r
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r
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t
h
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ti
s
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±
i
k
R
e
′
G±(
r
,
r
)=−
(
1
5
.
1
8
)
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′
f
orR=
|
r−r|
.
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e
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e
t
:
⇔
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′
(
r
,
r
)=
±
m
m∗ ′
h(
k
r )
Y (
ˆ
r
)
Y (
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r
)
j(
k
r)
i
k
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> ℓ
ℓ
j
,
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,
m
+
0∗
<
+
j
ℓ
j
ℓ
0∗
= ∓i
k
M L(
r
)
ML(
r
)+N L(
r
)
NL (
r
)+
>
<
>
<
L
+
0∗
L
r
)
L
r
)
L(
>
L (
<
(
1
5
.
1
9
)
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r
et
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ec
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ds
l
i
d
i
n
g
,
a
b
l
et
oa
p
p
l
yt
ot
h
eY j
(
r
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l
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l
e
i
t
h
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rt
e
r
mu
n
de
ra
ni
n
t
e
g
r
a
l
.
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on
oti
n
t
e
n
dt
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r
o
v
eak
e
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l
e
me
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h
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sa
s
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e
r
t
i
on(
t
h
a
tt
h
ep
r
od
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n
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l
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e
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d
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omp
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h
es
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h
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r
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c
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l
h
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mon
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e
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omt
h
eg
i
v
e
nor
t
h
on
or
ma
l
i
t
yr
e
l
a
t
i
on
.
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t
ht
h
e
s
er
e
l
a
t
i
on
si
nh
a
n
d,wee
n
dou
rma
t
h
e
ma
t
i
c
a
ld
i
g
r
e
s
s
i
oni
n
t
ov
e
c
t
or
s
p
h
e
r
i
c
a
lh
a
r
mon
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c
sa
n
dt
h
eHa
n
s
e
ns
ol
u
t
i
on
sa
n
dr
e
t
u
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nt
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h
el
a
n
dofmu
l
t
i
p
ol
a
r
r
a
d
i
a
t
i
on
.
15.
3 Mu
l
t
i
p
o
l
a
rRa
d
i
a
t
i
o
n
,
r
e
v
i
s
i
t
e
d
Wewi
l
ln
ow,a
tl
on
gl
a
s
t
,s
t
u
d
yt
h
ec
o
mp
l
e
t
er
a
d
i
a
t
i
onf
i
e
l
di
n
c
l
u
d
i
n
gt
h
es
c
a
l
a
r
,
l
on
g
i
t
u
d
i
n
a
l
,
a
n
dt
r
a
n
s
v
e
r
s
ep
a
r
t
s
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c
a
l
lt
h
a
twewi
s
ht
os
ol
v
et
h
et
woe
qu
a
t
i
on
s(
i
n
t
h
eL
or
e
n
t
zg
a
u
g
e
)
:
2
2
2
2
∇ +k
∇ +k
ρ
Φ(
x)= −ǫ0(
r
)
(
1
5
.
2
0
)
µ
J
x
()
0
Ax
()= −
(
1
5
.
2
1
)
wi
t
ht
h
eL
or
e
n
t
zc
on
d
i
t
i
on
:
1∂Φ
∇·
A+
2
c ∂t =0
(
1
5
.
2
2
)
wh
i
c
hi
sc
on
n
e
c
t
e
d(
a
swes
h
a
l
l
s
e
e
)t
ot
h
ec
on
t
i
n
u
i
t
ye
qu
a
t
i
onf
orc
h
a
r
g
ea
n
dc
u
r
r
e
n
t
.
Ea
n
dBa
r
en
ow (
a
su
s
u
a
l
)d
e
t
e
r
mi
n
e
df
r
om t
h
ev
e
c
t
orp
ot
e
n
t
i
a
lb
yt
h
ef
u
l
l
r
e
l
a
t
i
on
s
,
i
.
e
.
–wema
k
en
oa
s
s
u
mp
t
i
ont
h
a
twea
r
eou
t
s
i
d
et
h
er
e
g
i
onofs
ou
r
c
e
s
:
∂A
∂t
∇Φ−
=−
E
(
1
5
.
2
3
)
B
=∇×
(
1
5
.
2
4
)
A,
Us
i
n
gt
h
eme
t
h
od
sdi
s
c
u
s
s
e
db
e
f
o
r
e(
wr
i
t
i
n
gt
h
es
ol
u
t
i
ona
sa
ni
n
t
e
g
r
a
le
qu
a
t
i
on
,
b
r
e
a
k
i
n
gt
h
ei
n
t
e
g
r
a
lu
pi
n
t
ot
h
ei
n
t
e
r
i
ora
n
de
x
t
e
r
i
oroft
h
es
p
h
e
r
eofr
a
d
i
u
sr
,a
n
d
u
s
i
n
gt
h
ec
or
r
e
c
tor
d
e
roft
h
emu
l
t
i
p
ol
a
re
x
p
a
n
s
i
onoft
h
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e
e
n
’
sf
u
n
c
t
i
oni
nt
h
e
i
n
t
e
r
i
ora
n
de
x
t
e
r
i
orr
e
g
i
on
s
)wec
a
ne
a
s
i
l
ys
h
owt
h
a
tt
h
eg
e
n
e
r
a
ls
ol
u
t
i
ont
ot
h
eI
HE
’
s
a
b
ov
ei
s
:
Φ(
r)=i
k
e
x
t
i
n
t
+
p
r
)
J
r
()+p
r
)
HLr
()
L (
L
L (
(
1
5
.
2
5
)
L
wher
e
∞
e
x
t
p
r
)=
L (
r
r
i
n
t
p
r
)=
L (
0
+ ′ ∗ ′
′ 3′
h
k
r
)
YL(
r
ˆ)
ρ
r
()
dr
ℓ(
′ ∗ ′
′ 3′
j
(
k
r
)
YL(
r
ˆ)
ρr
()
dr
ℓ
(
1
5
.
2
6
)
(
1
5
.
2
7
)
Ou
t
s
i
d
et
h
e(
b
ou
n
d
i
n
gs
p
h
e
r
eoft
h
e
)s
ou
r
c
e
,
t
h
ee
x
t
e
r
i
orc
oe
ffic
i
e
n
ti
sz
e
r
oa
n
dt
h
e
i
n
t
i
n
t
e
r
i
orc
oe
ffic
i
e
n
ti
st
h
es
c
a
l
a
rmu
l
t
i
p
ol
emome
n
tp
(
∞)oft
h
ec
h
a
r
g
es
ou
r
c
e
L=p L
d
i
s
t
r
i
b
u
t
i
on
,
s
ot
h
a
t
:
Φ(
r)=
i
k
ǫ0
+
p
HLr
()
L
(
1
5
.
2
8
)
L
Th
i
si
sa
ni
mp
or
t
a
n
tr
e
l
a
t
i
ona
n
dwi
l
lp
l
a
ya
ns
i
g
n
i
f
i
c
a
n
tr
ol
ei
nt
h
ei
mp
l
e
me
n
t
a
t
i
onof
t
h
eg
a
u
g
ec
on
d
i
t
i
onb
e
l
ow.
Si
mi
l
a
r
l
ywec
a
nwr
i
t
et
h
ei
n
t
e
r
i
ora
n
de
x
t
e
r
i
ormu
l
t
i
p
ol
a
rmome
n
t
soft
h
ec
u
r
r
e
n
ti
n
t
e
r
msofi
n
t
e
g
r
a
l
sov
e
rt
h
ev
a
r
i
ou
sHa
n
s
e
nf
u
n
c
t
i
on
st
oob
t
a
i
nac
omp
l
e
t
e
l
yg
e
n
e
r
a
l
e
x
p
r
e
s
s
i
onf
ort
h
ev
e
c
t
orp
ot
e
n
t
i
a
lAr
()
.Tos
i
mp
l
i
f
yma
t
t
e
r
s
,
Ia
mg
oi
n
gt
oon
l
ywr
i
t
e
downt
h
es
ol
u
t
i
onob
t
a
i
n
e
dou
t
s
i
d
et
h
ec
u
r
r
e
n
tde
n
s
i
t
yd
i
s
t
r
i
b
u
t
i
on
,a
l
t
h
ou
g
ht
h
e
i
n
t
e
g
r
a
t
i
onv
ol
u
mec
a
ne
a
s
i
l
yb
es
p
l
i
ti
n
t
or
n
dr
i
e
c
e
sa
sa
b
ov
ea
n
da
ne
x
a
c
t
<a
>p
s
ol
u
t
i
onob
t
a
i
n
e
dona
l
l
s
p
a
c
ei
n
c
l
u
di
n
gi
n
s
i
det
h
ec
h
a
r
gedi
s
t
r
i
b
u
t
i
o
n
.
I
ti
s
:
+
Ar
()=i
k
µ0
+
mLMLr
()+n
NLr
()+l
L
r
()
L
L
L
(
1
5
.
2
9
)
L
wher
e
′
0 ′∗3′
′
0 ′∗3′
mL
=
J
r
()·MLr
()dr
n
L
=
J
r
()·NLr
()dr
l
L
=
J
r
()·L
()dr
Lr
′
0 ′∗3′
(
1
5
.
3
0
)
(
1
5
.
3
1
)
(
1
5
.
3
2
)
Not
ewe
l
lt
h
a
tt
h
ea
c
t
i
onoft
h
ed
otp
r
odu
c
twi
t
h
i
nt
h
ed
y
a
d
i
cf
or
mf
ort
h
eGr
e
e
n
’
s
f
u
n
c
t
i
on(
e
x
p
a
n
d
e
di
nHa
n
s
e
ns
ol
u
t
i
on
s
)r
e
d
u
c
e
st
h
edy
a
d
i
ct
e
n
s
ort
oav
e
c
t
ora
g
a
i
n
.
I
tt
u
r
n
sou
tt
h
a
tt
h
e
s
ef
ou
rs
e
t
sofn
u
mb
e
r
s
:p
,mL,n
,l
r
en
oti
n
d
e
p
e
n
de
n
t
.Th
e
y
L
L
La
a
r
er
e
l
a
t
e
db
yt
h
er
e
qu
i
r
e
me
n
tt
h
a
tt
h
es
ol
u
t
i
on
ss
a
t
i
s
f
yt
h
eL
o
r
e
n
t
zga
u
gec
o
n
d
i
t
i
on
,
wh
i
c
h
i
sac
o
n
s
t
r
a
i
n
tont
h
ea
d
mi
s
s
i
b
l
es
ol
u
t
i
on
s
.I
fwes
u
b
s
t
i
t
u
t
et
h
e
s
ef
or
msi
n
t
ot
h
eg
a
u
g
e
c
on
d
i
t
i
oni
t
s
e
l
fa
n
du
s
et
h
ed
i
ffe
r
e
n
t
i
a
lr
e
l
a
t
i
on
sg
i
v
e
na
b
ov
ef
ort
h
eHa
n
s
e
nf
u
n
c
t
i
on
st
o
s
i
mp
l
i
f
yt
h
er
e
s
u
l
t
s
,
weob
t
a
i
n
:
1∂
Φ
+
i
k
L
i
k
∇·
A+
i
ω
+
2
µ0l
∇·L
L
L
HL
− cǫ0p
L
+
L
2
c ∂t
+
l
∇·L
k
c
p
HL
L
L −i
L
2
{
l
p
}HL
L−c
L
L
+
=0
= 0
+
−
k
= 0
= 0
+
(
1
5
.
3
3
)
∗
wh
e
r
eweu
s
e
d∇·LL=i
k
HL i
nt
h
el
a
s
ts
t
e
p
.
I
fwemu
l
t
i
p
l
yf
r
omt
h
el
e
f
tb
yYℓ′
n
d
,
m′a
u
s
et
h
ef
a
c
tt
h
a
tt
h
eYLf
or
mac
omp
l
e
t
eor
t
h
on
or
ma
l
s
e
t
,
wef
i
n
dt
h
er
e
l
a
t
i
on
:
l
p
L−c
L=0
(
1
5
.
3
4
)
l
p
L=c
L
(
1
5
.
3
5
)
or
Th
i
st
e
l
l
su
st
h
a
tt
h
ee
ffe
c
toft
h
es
c
a
l
a
rmome
n
t
sa
n
dt
h
el
on
g
i
t
u
di
n
a
lmome
n
t
sa
r
e
c
on
n
e
c
t
e
db
yt
h
eg
a
u
g
ec
on
di
t
i
on
.
I
n
s
t
e
a
doff
ou
rr
e
l
e
v
a
n
tmome
n
t
sweh
a
v
ea
tmos
tt
h
r
e
e
.
I
nf
a
c
t
,
a
swewi
l
l
s
e
eb
e
l
ow,
weh
a
v
eon
l
yt
wo!
Re
c
a
l
l
t
h
a
tt
h
ep
ot
e
n
t
i
a
l
sa
r
en
otu
n
i
qu
e–t
h
e
yc
a
na
n
dd
ov
a
r
ya
c
c
or
d
i
n
gt
ot
h
eg
a
u
g
e
c
h
os
e
n
.Th
ef
i
e
l
d
s
,h
owe
v
e
r
,mu
s
tb
eu
n
i
qu
eorwe
’
dg
e
tdi
ffe
r
e
n
te
x
p
e
r
i
me
n
t
a
lr
e
s
u
l
t
si
n
d
i
ffe
r
e
n
tg
a
u
g
e
s
.
Th
i
swou
l
dob
v
i
ou
s
l
yb
eap
r
ob
l
e
m!
L
e
tu
st
h
e
r
e
f
or
ec
a
l
c
u
l
a
t
et
h
ef
i
e
l
d
s
.Th
e
r
ea
r
et
wowa
y
st
op
r
oc
e
e
d
.Wec
a
n
c
omp
u
t
eBd
i
r
e
c
t
l
yf
r
omv
A:
B =∇×
A
k
µ0
=i
k
µ0
=i
+
L
+
+
+
ml
(
−
i
k
NL)+n
(
i
k
ML)
l
L
2
0
=kµ
+
ml
(
∇×ML)+n
(
∇×NL)+l
(
∇×L
l
l
L)
+
+
mLNL −n
ML
L
L
(
1
5
.
3
6
)
E
∂
a
n
du
s
eAmp
e
r
e
’
sL
a
w,
∇×B=µ0ǫ0 ∂t =−
i
ωµ
ǫ0Et
of
i
n
dE
:
0
2
i
c
∇×B
k
c
E =
+
k
c
µ0
= i
+
L
mL(
∇×NL)−n
(
∇×ML)
L
1
+
µ0ǫ0 µ
0
=i
k
mL(
i
k
ML)−n
(
−
i
k
NL)
L
L
µ0
2
=−
k
+
+
M
ǫ0
L
2
=−
kZ0
+
N
mL
L+n
L
+
+
mLML +nLNL
L
µ0
wh
e
r
eZ0=
L
.
(
1
5
.
3
7
)
i
st
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l
i
mpe
da
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eoff
r
e
es
pa
c
e
,
a
r
oun
d377ohms
.
ǫ0
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c
a
l
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h
a
tt
h
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e
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et
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e
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h
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et
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h
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et
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on
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h
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i
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p
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n
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e
l
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l
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rp
ot
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t
i
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l
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e
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s
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a
t
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e
tu
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e
e
v
a
l
u
a
t
et
h
ee
l
e
c
t
r
i
cf
i
e
l
df
r
om:
∂A
∂
t
E =−
∇Φ−
i
k
∇ ǫ0
= −
i
k
=− ǫ0
2
k
= ǫ0
L
L
+
pLHL
+
+i
ωi
k
µ
0
1
L
+
p
L−cl
L−kZ
0
L L
+
(
Not
et
h
a
tweu
s
e
dω=k
ca
n
d∇H
ga
u
gec
on
di
t
i
on
:
+
−kµ0c
M
ml
L
+
=i
k
L.
)
L
L
l
p
L=c
L
M
2
l
LL
2
L
mLMLr
()+n
NLr
()+l
L
r
()
L
L
L
L
L
+
p
(
∇HL)−i
k
c
µ0ǫ0
L
+
+
N
L+n
L
L
ml
L
+
+
N
+n
L
L
+
(
1
5
.
3
8
)
L
F
r
omt
h
i
swes
e
et
h
a
ti
ft
h
e
(
1
5
.
3
9
)
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ss
a
t
i
s
f
i
e
d,
t
h
es
c
a
l
a
ra
n
dl
o
n
gi
t
u
d
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n
a
l
v
e
c
t
o
rp
a
r
t
soft
h
ee
l
e
c
t
r
i
cf
i
e
l
dc
a
n
c
e
l
e
x
a
c
t
l
y
!
Al
l
t
h
a
ts
u
r
v
i
v
e
sa
r
et
h
et
r
a
n
s
v
e
r
s
ep
a
r
t
s
:
2
+
+
E=−kZ0mLML +nLNL
(
1
5
.
4
0
)
L
a
sb
e
f
or
e
.Th
eL
or
e
n
t
zg
a
u
g
ec
on
d
i
t
i
oni
st
h
u
si
n
t
i
ma
t
e
l
yc
on
n
e
c
t
e
dt
ot
h
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a
n
i
s
h
i
n
g
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i
t
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di
n
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l
c
on
t
r
i
b
u
t
i
ont
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h
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l
d!
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s
on
ot
et
h
a
tt
h
ema
g
n
i
t
u
d
eofE
i
sg
r
e
a
t
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rt
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nt
h
a
tofBb
yc
,
t
h
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e
l
oc
i
t
yofl
i
g
h
t
.
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r
ei
n
t
e
r
e
s
t
e
d(
a
su
s
u
a
l
)mos
t
l
yi
nob
t
a
i
n
i
n
gt
h
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i
e
l
dsi
nt
h
ef
a
rz
on
e
,
wh
e
r
et
h
i
sa
l
r
e
a
dys
i
mp
l
ee
x
p
r
e
s
s
i
ona
t
t
a
i
n
sac
l
e
a
na
s
y
mp
t
ot
i
cf
or
m.Us
i
n
gt
h
ek
r→
∞f
or
moft
h
eh
a
n
k
e
l
f
u
n
c
t
i
on
,
i
k
r
−(
ℓ
+1
)
e
+
l
i
m h(
k
r
)=
k
r
→∞
ℓ
i
π
(
1
5
.
4
1
)
2
k
r
weob
t
a
i
nt
h
el
i
mi
t
i
n
gf
or
ms(
f
ork
r→ ∞)
:
i
k
r
−(
ℓ
+1
)
e
+
ML ∼
i
k
r
−
ℓ
e
NL∼
k
r
Y
−
i
)
Yℓℓ=(
ℓ+1
2
m
2
k
r
i
π
+
i
kr
ℓ+1e
m
i
π
k
r ℓℓ
ℓ
m
Y
2
ℓ
+1
ℓ
,
ℓ
−1
(
1
5
.
4
2
)
m
2
ℓ
+1 Yℓ,
ℓ
+1
+
(
1
5
.
4
3
)
Th
eb
r
a
c
k
e
ti
nt
h
es
e
c
on
de
qu
a
t
i
onc
a
nb
es
i
mp
l
i
f
i
e
d
,
u
s
i
n
gt
h
er
e
s
u
l
t
soft
h
et
a
b
l
e
I
h
a
n
de
do
u
tp
r
e
v
i
ou
s
l
y
.
Not
et
h
a
t
ℓ+1
2
ℓ
+1
ℓ
m
Y
ℓ
,
ℓ
−1
ℓ
+1
+ 2
m
m
Y
ℓ
,
ℓ
+1
m
π
−i
=i
(
r
ˆ×Yℓℓ)=−
e
(
r
ˆ×Yℓℓ)(
15.
44)
2
s
ot
h
a
t(
s
t
i
l
l
i
nt
h
ef
a
rz
on
el
i
mi
t
)
i
k
r
−(
ℓ
+1
)
e
i
kr
ℓ+1e
i
π
+
NL∼−
m
2
(
r
ˆ×Yℓℓ)=−
(
−
i
)
k
r
m
r
ˆ×Yℓℓ)
.
k
r(
(
1
5
.
4
5
)
L
e
tu
sp
a
u
s
et
oa
dmi
r
et
h
i
sr
e
s
u
l
tb
e
f
or
emos
e
y
i
n
gon
.
Th
i
si
sj
u
s
t
2
i
k
r
e
B=−
kµ0k
r
i
k
r
2
(
−
i
)
L
e
k
µ0 r
=−
ℓ
+
1
(
−
i
)
L
i
k
r
e
E=−
kZ0k
r
i
k
r
ℓ
+
1
L
e
k
Z0 r
=−
m
r Y
ℓ
+
1
(
−
i
)
ℓ
+
1
L
(
−
i
)
mL ˆ×
ℓ
ℓ
m
mLr
ˆ×Yℓℓ
Y
mL
m
+n
Y
L
m
+nLYℓℓ
m
ℓ
ℓ
ℓ
ℓ
(
1
5
.
4
6
)
m
−n
ˆ×Yℓℓ
L r
m
m
mLYℓℓ −nLrˆ×Yℓℓ
. (
1
5
.
4
7
)
I
fIh
a
v
ema
deas
ma
l
le
r
r
ora
tt
h
i
sp
oi
n
t
,
f
or
g
i
v
eme
.Cor
r
e
c
tme
,
t
oo.Th
i
si
sap
u
r
e
l
y
t
r
a
n
s
v
e
r
s
eou
t
g
oi
n
gs
p
h
e
r
i
c
a
l
wa
v
ewh
os
ev
e
c
t
orc
h
a
r
a
c
t
e
ri
sf
i
n
a
l
l
yt
r
a
n
s
l
u
c
e
n
t
,
i
fn
ot
t
r
a
n
s
p
a
r
e
n
t
.
Th
ep
owe
rf
l
u
xi
nt
h
eou
t
g
oi
n
gwa
v
ei
ss
t
i
l
l
n
ott
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a
s
yt
oe
x
p
r
e
s
s
,
b
u
ti
ti
sad
a
mn
s
i
g
h
te
a
s
i
e
rt
h
a
ni
twa
sb
e
f
or
e
.Atl
e
a
s
tweh
a
v
et
h
es
a
t
i
s
f
a
c
t
i
onofk
n
owi
n
gt
h
a
twe
c
a
ne
x
p
r
e
s
si
ta
sag
e
n
e
r
a
l
r
e
s
u
l
t
.
Re
c
a
l
l
i
n
g(
a
su
s
u
a
l
)
1
∗
S= 2Re
(
E×H)
(
1
5
.
4
8
)
a
n
dt
h
a
tt
h
ep
owe
rd
i
s
t
r
i
b
u
t
i
oni
sr
e
l
a
t
e
dt
ot
h
ef
l
u
xoft
h
ePoy
n
t
i
n
gv
e
c
t
ort
h
r
ou
g
ha
s
u
r
f
a
c
ea
tdi
s
t
a
n
c
eri
nad
i
ffe
r
e
n
t
i
a
l
s
ol
i
da
n
g
l
edΩ:
dP
1
2
∗
dΩ =2 Re
ˆ·(
E×H)
]
[
rn
(
1
5
.
4
9
)
weg
e
t
S=
2
k
′
2
ℓ
2
rZ0Re
∗
×
i−ℓ mL
′
L L
m
′
L
Y
m
ℓ
ℓ
m
−n
ˆ×Yℓℓ
Lr
∗
′
′
r
ˆ Ym∗ +n ′Y m∗
× ℓℓ
L
ℓℓ
′ ′
(
1
5
.
5
0
)
′ ′
(
Not
e
:Un
i
t
sh
e
r
en
e
e
dt
ob
er
e
c
h
e
c
k
e
d,b
u
tt
h
e
ya
p
p
e
a
rt
ob
ec
on
s
i
s
t
e
n
ta
tf
i
r
s
t
g
l
a
n
c
e
)
.
Th
i
si
sa
ne
x
t
r
e
me
l
yc
omp
l
i
c
a
t
e
dr
e
s
u
l
t
,b
u
ti
th
a
st
ob
e
,s
i
n
c
ei
te
x
p
r
e
s
s
e
st
h
e
mos
tge
n
e
r
a
lp
o
s
s
i
b
l
ea
n
gu
l
a
rd
i
s
t
r
i
b
u
t
i
o
no
fr
a
di
a
t
i
on(
i
nt
h
ef
a
rz
on
e
)
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ep
owe
r
d
i
s
t
r
i
b
u
t
i
onf
ol
l
owst
r
i
v
i
a
l
l
y
.Wec
a
n
,
h
owe
v
e
r
,
e
v
a
l
u
a
t
et
h
et
ot
a
lp
owe
rr
a
di
a
t
e
d
,
wh
i
c
h
i
sav
e
r
yu
s
e
f
u
l
n
u
mb
e
r
.
Th
i
swi
l
l
b
ea
ne
x
e
r
c
i
s
e
.
Youwi
l
l
n
e
e
dt
h
er
e
s
u
l
t
s
2
′
m∗
m
ˆ×Yℓ′ℓ′
dΩˆr·Yℓℓ ×r
=
′
m
m∗
dΩYℓℓ
·Yℓ′ℓ′
2
= δℓℓ′δmm′
and
2
(
1
5
.
5
1
)
′
m∗
m
Y
Ωˆ
r· ℓℓ ×Yℓ′ℓ′ =0
(
1
5
.
5
2
)
d
t
oe
v
a
l
u
a
t
et
y
p
i
c
a
l
t
e
r
ms
.
Us
i
n
gt
h
e
s
er
e
l
a
t
i
on
s
,
i
ti
sn
ott
ood
i
ffic
u
l
tt
os
h
owt
h
a
t
2
k
P= 2Z0
2
L
2
|
mL|+|
n
L|
(
1
5
.
5
3
)
wh
i
c
hi
st
h
es
u
moft
h
ep
owe
re
mi
t
t
e
df
r
oma
l
lt
h
ei
n
d
i
v
i
d
u
a
lmu
l
t
i
p
ol
e
s(
t
h
e
r
ei
sn
o
i
n
t
e
r
f
e
r
e
n
c
eb
e
t
we
e
nmu
l
t
i
p
ol
e
s
!
)
.
L
e
tu
se
x
a
mi
n
ee
.
g
.t
h
ee
l
e
c
t
r
i
cmu
l
t
i
p
ol
a
rmome
n
tn
os
e
eh
owi
tc
omp
a
r
e
st
o
Lt
t
h
eu
s
u
a
l
s
t
a
t
i
cr
e
s
u
l
t
s
.
St
a
t
i
cr
e
s
u
l
t
sa
r
eob
t
a
i
n
e
di
nt
h
ek→ 0(
l
on
gwa
v
e
l
e
n
g
t
h
)l
i
mi
t
.
ℓℓ
I
nt
h
i
sl
i
mi
te
.
g
.
j
(
k
r
)
∼k
r
a
n
d
:
ℓ
ℓ+1
n
c
L≈i
ℓ
k
ℓ
3
()
rYℓ,
r
ˆ
)
dr
ℓ (
2
ℓ+1
)
! ρr
m(
(
1
5
.
5
4
)
Th
edi
p
ol
et
e
r
mc
ome
sf
r
omℓ=1
.
F
oras
i
mp
l
ed
i
p
ol
e
:
n
1
,
m
√2
≈i
c 3k
3
ρr
Y1,
mdr
√
k
c2
≈i 3
3
4π e<r>
√
≈
k
c 6
i 3
6
π e<r>
i
e
≈−√ 6πω <r
¨>
(
1
5
.
5
5
)
2
wh
e
r
eweu
s
e<r
¨>
=−
ω <r>
.
I
nt
e
r
msoft
h
i
st
h
ea
v
e
r
a
g
ep
owe
rr
a
d
i
a
t
e
db
yas
i
n
g
l
ee
l
e
c
t
r
ondi
p
ol
ei
s
:
P=
1
2
e
πǫ0c3
2 6
wh
i
c
hc
omp
a
r
e
swe
l
l
wi
t
ht
h
eL
a
r
morF
or
mu
l
a
:
2
|
r
¨|
2
(
1
5
.
5
6
)
2
e
3
P= 3 4πǫ0c
2
|
r
¨
|
(
1
5
.
5
7
)
Th
el
a
t
t
e
ri
st
h
ef
or
mu
l
af
ort
h
ei
n
s
t
a
n
t
a
n
e
ou
sp
owe
rr
a
di
a
t
e
df
r
omap
oi
n
tc
h
a
r
g
ea
si
ti
s
a
c
c
e
l
e
r
a
t
e
d
.
E
i
t
h
e
rf
l
a
v
ori
st
h
ed
e
a
t
hk
n
e
l
l
ofc
l
a
s
s
i
c
a
l
me
c
h
a
n
i
c
s
–i
ti
sv
e
r
yd
i
ffic
u
l
tt
ob
u
i
l
damod
e
lf
oras
t
a
b
l
ea
t
omb
a
s
e
donc
l
a
s
s
i
c
a
lt
r
a
j
e
c
t
or
i
e
s
ofa
ne
l
e
c
t
r
ona
r
ou
n
dan
u
c
l
e
u
st
h
a
tdoe
sn
oti
n
v
ol
v
ea
c
c
e
l
e
r
a
t
i
onoft
h
ee
l
e
c
t
r
oni
n
qu
e
s
t
i
on
.
Wh
i
l
ei
ti
sn
ote
a
s
yt
os
e
e
,t
h
er
e
s
u
l
t
sa
b
ov
ea
r
ee
s
s
e
n
t
i
a
l
l
yt
h
os
eob
t
a
i
n
e
di
n
J
a
c
k
s
on(
J
9
.
1
5
5
)e
x
c
e
p
tt
h
a
t(
c
omp
a
r
i
n
ge
.
g
.J
9
.
1
1
9
,J
9
.
1
2
2
,a
n
dJ
9
1
.
1
6
5t
or
e
l
a
t
e
d
r
e
s
u
l
t
sa
b
ov
e
)J
a
c
k
s
o
n
’
sa
ℓ
,m)mome
n
t
sdi
ffe
rf
r
om t
h
eHa
n
s
e
nmu
l
t
i
p
ol
a
r
E
,
M(
mome
n
t
sb
yf
a
c
t
or
sofs
e
v
e
r
a
l
p
owe
r
sofk
.I
fon
ewor
k
sh
a
r
de
n
ou
g
h
,
t
h
ou
g
h
,
on
ec
a
n
s
h
owt
h
a
tt
h
er
e
s
u
l
t
sa
r
ei
de
n
t
i
c
a
l
,
a
n
de
v
e
nt
h
ou
g
hJ
a
c
k
s
on
’
sa
l
g
e
b
r
ai
smor
et
h
a
na
b
i
tEv
i
li
ti
swor
t
h
wh
i
l
et
od
ot
h
i
si
fon
l
yt
ov
a
l
i
d
a
t
et
h
er
e
s
u
l
t
sa
b
ov
e(
wh
e
r
er
e
c
a
l
l
t
h
e
r
eh
a
sb
e
e
nau
n
i
tc
on
v
e
r
s
i
ona
n
dh
e
n
c
et
h
e
ydon
e
e
dv
a
l
i
d
a
t
i
on
)
.
An
ot
h
e
ru
s
e
f
u
l
e
x
e
r
c
i
s
ei
st
or
e
c
ov
e
rou
rol
df
r
i
e
n
d
s
,
t
h
ed
i
p
ol
ea
n
dqu
a
d
r
u
p
ol
er
a
d
i
a
t
i
on
t
e
r
msofJ
9f
r
omt
h
ee
x
a
c
td
e
f
i
n
i
t
i
onoft
h
e
i
rr
e
s
p
e
c
t
i
v
emome
n
t
s
.On
emu
s
tma
k
et
h
el
on
g
wa
v
e
l
e
n
g
t
ha
p
p
r
ox
i
ma
t
i
onu
n
d
e
rt
h
ei
n
t
e
g
r
a
li
nt
h
ed
e
f
i
n
i
t
i
onoft
h
emu
l
t
i
p
ol
emome
n
t
s
,
i
n
t
e
g
r
a
t
eb
yp
a
r
t
sl
i
b
e
r
a
l
l
y
,
a
n
du
s
et
h
ec
on
t
i
n
u
i
t
ye
qu
a
t
i
on
.Th
i
si
squ
i
t
ed
i
ffic
u
l
t
,
a
si
tt
u
r
n
s
ou
t
,
u
n
l
e
s
sy
ouh
a
v
es
e
e
ni
tb
e
f
or
e
,
s
ol
e
tu
sl
ooka
ta
ne
x
a
mp
l
e
.L
e
tu
sa
p
p
l
yt
h
eme
t
h
od
s
weh
a
v
ed
e
v
e
l
op
e
da
b
ov
et
oob
t
a
i
nt
h
er
a
d
i
a
t
i
onp
a
t
t
e
r
nofad
i
p
ol
ea
n
t
e
n
n
a
,t
h
i
st
i
me
wi
t
h
o
u
ta
s
s
u
mi
n
gt
h
a
ti
t
’
sl
e
n
g
t
hi
ss
ma
l
lw.
r
.
t
.awa
v
e
l
e
n
g
t
h
.J
a
c
k
s
ons
ol
v
e
smor
eorl
e
s
s
t
h
es
a
mep
r
ob
l
e
mi
nh
i
ss
e
c
t
i
on9
.
1
2
,s
ot
h
i
swi
l
lp
e
r
mi
tt
h
ed
i
r
e
c
tc
omp
a
r
i
s
onoft
h
e
c
oe
ffic
i
e
n
t
sa
n
dc
on
s
t
a
n
t
si
nt
h
ef
i
n
a
le
x
p
r
e
s
s
i
on
sf
ort
ot
a
lr
a
d
i
a
t
e
dp
owe
rort
h
ea
n
g
u
l
a
r
d
i
s
t
r
i
b
u
t
i
onofp
owe
r
.
15.
4 AL
i
n
e
a
rCe
n
t
e
r
F
e
dHa
l
f
Wa
v
eAn
t
e
n
n
a
Su
p
p
os
ewea
r
eg
i
v
e
nac
e
n
t
e
r
f
e
ddi
p
ol
ea
n
t
e
n
n
awi
t
hl
e
n
g
t
hλ
/
2(
h
a
l
f
wa
v
ea
n
t
e
n
n
a
)
.
Wewi
l
la
s
s
u
mef
u
r
t
h
e
rt
h
a
tt
h
ea
n
t
e
n
n
ai
sa
l
i
g
n
e
dwi
t
ht
h
eza
x
i
sa
n
dc
e
n
t
e
r
e
dont
h
e
or
i
g
i
n
,
wi
t
hac
u
r
r
e
n
tg
i
v
e
nb
y
:
2πz
λ
I
=I
o
s
(
ωt
)c
os
0c
(
1
5
.
5
8
)
Not
et
h
a
ti
n“
r
e
a
ll
i
f
e
”i
ti
sn
ote
a
s
yt
oa
r
r
a
n
g
ef
orag
i
v
e
nc
u
r
r
e
n
tb
e
c
a
u
s
et
h
ec
u
r
r
e
n
t
i
n
s
t
a
n
t
a
n
e
ou
s
l
yde
p
e
n
d
sont
h
e“
r
e
s
i
s
t
a
n
c
e
”wh
i
c
hi
saf
u
n
c
t
i
onoft
h
er
a
d
i
a
t
i
onf
i
e
l
d
i
t
s
e
l
f
.Th
ec
u
r
r
e
n
ti
t
s
e
l
ft
h
u
sc
ome
sou
toft
h
es
ol
u
t
i
onofa
ne
x
t
r
e
me
l
yc
omp
l
i
c
a
t
e
d
b
ou
n
da
r
yv
a
l
u
ep
r
o
b
l
e
m.F
ora
t
omi
corn
u
c
l
e
a
rr
a
d
i
a
t
i
on
,h
owe
v
e
r
,t
h
e“
c
u
r
r
e
n
t
s
”a
r
e
g
e
n
e
r
a
l
l
yma
t
r
i
xe
l
e
me
n
t
sa
s
s
oc
i
a
t
e
dwi
t
ht
r
a
n
s
i
t
i
on
sa
n
dh
e
n
c
ea
r
ek
n
own
.
I
na
n
ye
v
e
n
t
,
t
h
ec
u
r
r
e
n
td
e
n
s
i
t
yc
or
r
e
s
p
on
d
i
n
gt
ot
h
i
sc
u
r
r
e
n
ti
s
J=z
ˆ
I
os
0c
f
orr≤λ
/
4a
n
d
2πr
δ(
1−|
c
osθ|
)
λ
2πrsi
nθ
(
1
5
.
5
9
)
2
J=0
(
1
5
.
6
0
)
f
orr>λ
/
4
.
Wh
e
nweu
s
et
h
eHa
n
s
e
nmu
l
t
i
p
ol
e
s
,t
h
e
r
ei
sl
i
t
t
l
ei
n
c
e
n
t
i
v
et
oc
on
v
e
r
tt
h
i
si
n
t
oa
f
or
m wh
e
r
ewei
n
t
e
g
r
a
t
ea
g
a
i
n
s
tt
h
ec
h
a
r
g
ede
n
s
i
t
yi
nt
h
ea
n
t
e
n
n
a
.I
n
s
t
e
a
dwec
a
n
e
a
s
i
l
ya
n
ddi
r
e
c
t
l
yc
a
l
c
u
l
a
t
et
h
emu
l
t
i
p
ol
emome
n
t
s
.
Th
ema
g
n
e
t
i
cmome
n
ti
s
0∗ 3
mL =
J·M Ldr
2
π
λ
/
4
m∗
I
0
π0
=2
0
m∗
0,
φ)+zˆ·Yℓℓ (
c
os
(
k
r
)
j
(
k
r
) zˆ·Yℓℓ (
ℓ
π,
φ) dφ(15dr.61)
(
wh
e
r
eweh
a
v
ed
on
et
h
ei
n
t
e
g
r
a
l
ov
e
rθ)
.
Now,
1
m
zˆ·Yℓℓ = ℓ
(
ℓ+1
)mYL
(
Wh
y
?Con
s
i
d
e
r(
z
ˆ·L
)
YL)
.
.
.
)a
n
dy
e
t
(
1
5
.
6
2
)
1
/
2
YL(
0
,
φ) = δm0
2
ℓ
+1
4π
(
1
5
.
6
3
)
2
ℓ
+1
ℓ
−
1
)δm0
YL(
π,
φ) = (
4π
1
/
2
.
(
1
5
.
6
4
)
Cons
e
que
nt
l
y
,
wec
a
nc
onc
l
ude(
mδm0=0)t
ha
t
mL=0
.
(
1
5
.
6
5
)
Al
lma
g
n
e
t
i
cmu
l
t
i
p
ol
emome
n
t
soft
h
i
sl
i
n
e
a
rdi
p
ol
ev
a
n
i
s
h
.Si
n
c
et
h
ema
g
n
e
t
i
c
mu
l
t
i
p
ol
e
ss
h
ou
l
db
ec
on
n
e
c
t
e
dt
ot
h
er
ot
a
t
i
on
a
lp
a
r
toft
h
ec
u
r
r
e
n
tde
n
s
i
t
y(
wh
i
c
hi
s
z
e
r
of
orl
i
n
e
a
rf
l
ow)t
h
i
ss
h
ou
l
dn
ots
u
r
p
r
i
s
ey
ou
.
Th
ee
l
e
c
t
r
i
cmome
n
t
sa
r
e
0∗ 3
n
L =
J·N Ldr
I
0
= 2π
−
2
π λ
/
4
0
ℓ
0
ℓ+1
c
os
(
k
r
)
m∗
2
ℓ
+1 j
ℓ
−1 (
k
r
)
m∗
zY
j
ℓ
+1
(
k
r
) ˆ·
2
ℓ
+1
m∗
zˆ·
Y
ℓ,
ℓ−1
m∗
(
0,
φ)+zˆ·Yℓ,
ℓ+1 (
π,
φ) d
φd
r
z Y
ℓ
,
ℓ
+1
(
0
,
φ)+
ˆ·
ℓ
,
ℓ
−1
(
π,
φ).
(
15.
66)
I
fwel
ooku
pt
h
ed
e
f
i
n
i
t
i
onoft
h
ev
.
s
.
h
.
’
sont
h
eh
a
n
d
ou
tt
a
b
l
e
,
t
h
ezc
omp
on
e
n
t
sa
r
e
g
i
v
e
nb
y
:
ℓ
m∗
zˆ·
Y
ℓ,
ℓ−1
m∗
zˆ·
Y
ℓ,
ℓ−1
m∗
zˆ·
Y
ℓ,
ℓ+1
4
π
(
0,
φ) =δm0
ℓ
−
1
)
(
π,
φ) =(
m0
−δ
ℓ+1
(
0,
φ) =−δm0
m∗
(
−
1
) − δm0
(
π,
φ) =−
s
ot
h
ee
l
e
c
t
r
i
cmu
l
t
i
p
ol
emome
n
t
sv
a
n
i
s
hf
orm=0
,
a
n
d
(
1
5
.
6
9
)
4π
(
1
5
.
7
0
)
λ
/
4
ℓ
(
ℓ+1
)
4
π(
2
ℓ+1
)
(
1
5
.
6
8
)
ℓ+1
ℓ,
ℓ+1
n
ℓ
,
0=I
0
δm0
4π
4
π
ℓ 1
zˆ·
Y
(
1
5
.
6
7
)
ℓ
1
ℓ
+
1
1+(
−
1
)
c
os
(
k
r
)j
(
k
r
)+j
(
k
r
)d
r
.
ℓ
−
1
ℓ
+
1
0
(
1
5
.
7
1
)
E
x
a
mi
n
i
n
gt
h
i
se
qu
a
t
i
on
,
wea
l
ls
e
et
h
a
ta
l
l
t
h
ee
v
e
nℓt
e
r
msv
a
n
i
s
h
!
Howe
v
e
r
,
t
h
e od
dℓ
,m = 0 t
e
r
ms d
on
otv
a
n
i
s
h
,
s
owec
a
n
’
tqu
i
ty
e
t
.Weu
s
et
h
e
f
ol
l
owi
n
gr
e
l
a
t
i
on
s
:
2
ℓ
+1
j
ℓ
−1+j
ℓ
+1=
ℓ
k
rj
(
1
5
.
7
2
)
(
t
h
ef
u
n
d
a
me
n
t
a
l
r
e
c
u
r
s
i
onr
e
l
a
t
i
on
)
,
c
os
(
k
r
)
n
(
k
r
)=−
0
(
t
r
u
ef
a
c
t
)a
n
d
z
)=
d
zf(
z
)
g′(
2
z
k
r
(
1
5
.
7
3
)
′
fg −f
g′
(
1
5
.
7
4
)
ℓ
ℓ
[
ℓ (
ℓ
′ ′
+1
)
−
ℓ
(
ℓ+1
)
]
ℓℓ
′
ℓ
ℓ
′
f
ora
n
yt
wos
p
h
e
r
i
c
a
lb
e
s
s
e
lt
y
p
ef
u
n
c
t
i
on
s(
av
a
l
u
a
b
l
et
h
i
n
gt
ok
n
owt
h
a
tf
ol
l
owsf
r
om
i
n
t
e
g
r
a
t
i
onb
yp
a
r
t
sa
n
dt
h
er
e
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r
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· L
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= 0
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5
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0
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ors
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e
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3
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i
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r
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s
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n
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c
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5
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9
3
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1
5
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4
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ts
u
b
s
t
i
t
u
t
eou
re
x
p
r
e
s
s
i
on
sf
orEa
n
dH:
2
+
kZ0
E= −
(
1
5
.
9
6
)
L
2
+
H=k
+
mLML +n
NL
L
+
mLNL −n
ML .
L
L
(
1
5
.
9
7
)
I
fwet
r
yt
ou
s
et
h
ea
s
y
mp
t
ot
i
cf
a
rf
i
e
l
dr
e
s
u
l
t
s
:
i
k
r
E
e
k
Z0 r
=−
ℓ
+
1
L
m
(
−
i
)
m
mLYℓℓ −nLr
ˆ×Yℓℓ
(
1
5
.
9
8
)
i
k
r
e
H
=−
k r
ℓ
+
1
m
(
−
i
)
L
m
mLr
ˆ×Yℓℓ
+nLYℓℓ
(
1
5
.
9
9
)
weget
:
∗
E H
×
=
2
kZ0
r
∗
m
2
L
r
ˆ
′
L
∗ m
×
r
ˆ
)+n ′Y
Ym′′′∗(
′
ℓ
−ℓ
i
L
ℓℓ
L
L
∗ m
r
ˆ
)
+m n ′Y (
ℓ
ℓ
×
L
(
r
ˆ
)
ℓℓ
m
r
ˆ
×
ℓ
ℓ
m ∗
Y ′′′ (
r
ˆ
)
× ℓℓ
r
ˆ
)
Ym′′′∗ (
ℓℓ
∗
m
−n
m L′ r
ˆ×Y ℓℓ(
r
ˆ
)
L
∗
∗
∗ m
′
×
′′
′
r
ˆ)
m m ′Y (
L
LL
m
r
ˆ
) n r
r
ˆ
)
m Y(
ˆ Y(
− L × ℓℓ
L ℓ
ℓ
′
L
= kZ0
2
r
m
′
ℓ
−ℓ
i
2
−n
nL′ r
ˆ×Y ℓℓ(
r
ˆ
)
L
′
m∗
×r
ˆ×Yℓ ′ℓ′ (
r
ˆ
)
′
m∗
×Yℓ ′ℓ′ (
r
ˆ
).
(
1
5
.
1
0
0
)
Wi
t
hs
omee
ffor
tt
h
i
sc
a
nb
es
h
ownt
ob
ear
a
d
i
a
lr
e
s
u
l
t–t
h
ePoy
n
t
i
n
gv
e
c
t
orp
oi
n
t
s
d
i
r
e
c
t
l
ya
wa
yf
r
om t
h
es
ou
r
c
ei
nt
h
ef
a
rf
i
e
l
dt
ol
e
a
di
n
gor
de
r
.Con
s
e
qu
e
n
t
l
y
,t
h
i
s
l
e
a
d
i
n
gor
d
e
rb
e
h
a
v
i
orc
on
t
r
i
b
u
t
e
sn
o
t
h
i
n
gt
ot
h
ea
n
g
u
l
a
rmome
n
t
u
mf
l
u
x
.Wemu
s
t
k
e
e
pa
tl
e
a
s
tt
h
el
e
a
d
i
n
gc
or
r
e
c
t
i
ont
e
r
mt
ot
h
ea
s
y
mp
t
ot
i
cr
e
s
u
l
t
.
I
ti
sc
on
v
e
n
i
e
n
tt
ou
s
ear
a
d
i
a
l
/
t
a
n
g
e
n
t
i
a
lde
c
omp
os
i
t
i
onoft
h
eHa
n
s
e
ns
ol
u
t
i
on
s
.
Th
eMLa
r
ec
omp
l
e
t
e
l
yt
a
n
g
e
n
t
i
a
l
(
r
e
c
a
l
l
r·ML=0
)
.
F
ort
h
eNLweh
a
v
e
:
1 d
ℓ
(
ℓ+1
)
m
r
f
(
k
r
)
)(
i
r
ˆ×Y ℓℓ(
r
ˆ)
)−r
ˆ
ℓ
NLr
()= k
rd
r(
f
l
l
(
k
r
)
YL(
r
ˆ
)
e
k
r
(
15.
101)
∗
Us
i
n
gou
rf
u
l
l
e
x
p
r
e
s
s
i
on
sf
orEa
n
dH:
2
E=
−
kZ0
H=
k
+
L
2
+
mLML +n
NL
L
+
(
15.
102)
+
mLNL −n
ML
L
(
15.
103)
L
wi
t
ht
h
i
sf
or
ms
u
b
s
t
i
t
u
t
e
df
orNLa
n
dt
h
eu
s
u
a
l
f
or
mf
orMLweg
e
t
:
L=
∗
(
E
×
H)
1Re r ×
2
c
4
kZ0
=−2c Re
+
L
′
L
m
(
k
r
)
Yℓℓ (
r
ˆ
)
r ×mLhℓ
+
+
1 d(
r
h(
k
r
)
)
h(
k
r
)
m
ℓ
(
r
ˆ
)
)−r
ˆ ℓ(
ℓ
ℓ
ℓ+1
)
ℓ
+
n
L
∗
m
×
k
r
d
r
(
i
r
ˆ×Y ′
−
m∗
1 d
(
r
h(
k
r
)
)
′
(i
r
ˆ Yℓ
′
′
(
r
ˆ
)
)
−r
ˆ
ℓ
− ×
k
r
d
r
−
L
m ∗
′
′
Y
′ ′
′
h(
k
r
)
ℓ
′
ℓ
(
ℓ+1
)
ℓ
′
YL(
r
ˆ
)
k
r
k
r
r
∗
Y(
r
ˆ
)
L
′
(
1
5
.
1
0
4
)
∗ −
k
r
) ℓℓ (
ℓ(
+nL h
ˆ
)
Al
l
t
h
ep
u
r
e
l
yr
a
d
i
a
l
t
e
r
msi
nt
h
eou
t
e
r
mos
t
unde
rt
hes
u
mdonotc
ont
r
i
but
e
t
ot
h
ea
n
g
u
l
a
rmome
n
t
u
mf
l
u
xd
e
n
s
i
t
y
.
Th
es
u
r
v
i
v
i
n
gt
e
r
msa
r
e
:
L=
−
k4Z0
Re
r
− 2
c
′
L L
∗
+n
m
L
n
−
n
−
1
+
m m∗ ′h(
k
r
)
× L L ℓ
+
h
ℓ
(
k
r
)
k
r
+
h
k
r
)
ℓ(
(
k
r
)
)
d
r
ℓ
(
ℓ+1
)
L
′
L L
×
k
r
(
r
ˆ
) r
ˆ
−
k
r
′
ℓ
ℓ(
ℓ +1
)
×
ℓℓ
′ ′
′
i
r
ˆ
1d(
r
hℓ(
k
r
)
)
−
∗
′
(
i
r
ˆ Y
ℓ
(
ℓ+1)
h(
k
r
)
(
ˆ
n
′
m
+
ℓ
r
Lk
′
m∗
L
d(
r
h
′
h(
k
r
)
m
′
ℓ(
ℓ +1
)ℓ
(
Y (
r
ˆ
) r
ˆ)
Y∗(rˆ)
× L
′ ′
k
r
ℓℓ
d
r
rY
×
′
m∗
′′
(
ˆ
)
ℓℓ
r
(
r
ˆ
−
h(
k
r
)
ℓ
k
r
′
× ×
Y
m
′
∗
ℓℓ
(
r
ˆ
)
)
′
∗
YL′(
r
ˆ
)
(
1
5
.
1
0
5
)
Th
el
owe
s
tor
d
e
rt
e
r
mi
nt
h
ea
s
y
mp
t
ot
i
cf
or
mf
ort
h
es
p
h
e
r
i
c
a
lb
e
s
s
e
lf
u
n
c
t
i
on
s
ma
k
e
sac
on
t
r
i
b
u
t
i
oni
nt
h
ea
b
ov
ee
x
p
r
e
s
s
i
on
s
.Af
t
e
ru
n
t
a
n
g
l
i
n
gt
h
ec
r
os
sp
r
odu
c
t
s
a
n
ds
u
b
s
t
i
t
u
t
i
n
gt
h
ea
s
y
mp
t
ot
i
cf
or
ms
,
weg
e
t
:
k
µ0Re
L=
mm
2
2r
∗
L
′
L
′
′ ′
L
L
m
ℓ−ℓ
i
Y∗′(
r
ˆ
)
Y
ℓ(
ℓ +1
)
′
L
Y (
r
ˆ
)
× ℓℓ
′
ℓ−ℓ
ℓ
(
ℓ+1
)
i
Y
+
n m
ℓ−ℓ
i
Y (
ℓ
(
ℓ+1
)
r
ˆ
)r
ˆ Ym′ ′∗ (
r
ˆ
)
× ℓℓ
L
L
L
L
(
r
ˆ
)r
ˆ
′
′
′
∗
′
m′∗
ℓ−ℓ
r
ˆ
)
Y
Y(
ℓ
(
ℓ+1
)i
′
+
n n′
L L
(
r
ˆ)
m
∗
nm ′
−L L
∗
ℓ
ℓ
L
(
r
ˆ
)
(
1
5
.
1
0
6
)
′′
ℓℓ
Th
ea
n
g
u
l
a
rmome
n
t
u
ma
b
ou
tag
i
v
e
na
x
i
se
mi
t
t
e
dp
e
ru
n
i
tt
i
mei
sob
t
a
i
n
e
db
y
s
e
l
e
c
t
i
n
gap
a
r
t
i
c
u
l
a
rc
omp
on
e
n
toft
h
i
sa
n
di
n
t
e
g
r
a
t
i
n
gi
t
sf
l
u
xt
h
r
ou
g
had
i
s
t
a
n
t
2
s
p
h
e
r
i
c
a
l
s
u
r
f
a
c
e
.F
ore
x
a
mp
l
e
,
f
ort
h
ez
c
omp
on
e
n
twef
i
n
d(
n
ot
i
n
gt
h
a
trc
a
n
c
e
l
sa
s
i
ts
h
ou
l
d)
:
dLz k
µ0
d
t = 2Re
′
z
ˆ·.
.
.s
i
n
(
θ)
dθ
d
φ
(
1
5
.
1
0
7
)
L
L
wh
e
r
et
h
eb
r
a
c
k
e
t
si
n
d
i
c
a
t
et
h
ee
x
p
r
e
s
s
i
ona
b
ov
e
.Wel
ooku
pt
h
ec
omp
on
e
n
t
soft
h
e
v
e
c
t
orh
a
r
mon
i
c
st
ol
e
tu
sd
ot
h
ed
otp
r
od
u
c
ta
n
df
i
n
d
:
m
m
z
ˆ·Yℓℓ
=
ℓ
(
ℓ+1
)Yℓ,m
(
15.
108)
ℓ+1
m
z
ˆ·(
r
ˆ×Yℓℓ)
ℓ
m
i
=−
ˆ·Yℓ,
ℓ
−1+
2
ℓ
+1z
= i
−
(
ℓ+1
)
(
ℓ −m ) Y
2
ℓ
(
2
ℓ
ˆ·Yℓℓ+1
2
ℓ
+1z
2
1
)
(
2
ℓ+1
)
−
m
2
2
[
(
ℓ+1
)−m ]
ℓ
ℓ
−1
,
m−
(
2
ℓ+1
)
(
2
ℓ+3
)
(
ℓ+1
)
k
µ0
d
t =2
Comp
a
r
et
h
i
st
o:
2
L
2
k
P= 2Z0
L
=k
µ0m
2
m|
mL|+|
n
|
L
2
t
e
r
mb
yt
e
r
m.
F
ore
x
a
mp
l
e
:
d
L
mL)
z(
ℓ
+
1
,
m
(
15.
109)
Doi
n
gt
h
ei
n
t
e
g
r
a
l
i
sn
ows
i
mp
l
e
,
u
s
i
n
gt
h
eor
t
h
on
or
ma
l
i
t
yoft
h
es
p
h
e
r
i
c
a
l
h
a
r
mon
i
c
s
.
On
eob
t
a
i
n
s(
a
f
t
e
rs
t
i
l
l
mor
ewor
k
,
ofc
ou
r
s
e
)
:
d
L
z
Y
2
|
mL|+|
n
L|
2
mL)=|
mL|
2 P(
(
1
5
.
1
1
0
)
(
1
5
.
1
1
1
)
(
1
5
.
1
1
2
)
(
wh
e
r
emi
nt
h
ef
r
a
c
t
i
oni
st
h
es
p
h
e
r
i
c
a
lh
a
r
mon
i
cm,n
ott
h
emu
l
t
i
p
ol
emL)
.I
not
h
e
r
wor
d
s
,f
orap
u
r
emu
l
t
i
p
ol
et
h
er
a
t
eofa
n
g
u
l
a
rmome
n
t
u
ma
b
ou
ta
n
yg
i
v
e
na
x
i
s
t
r
a
n
s
f
e
r
r
e
di
sm/
ωt
i
me
st
h
er
a
t
eofe
n
e
r
g
yt
r
a
n
s
f
e
r
r
e
d
,wh
e
r
em i
st
h
ea
n
g
u
l
a
r
mome
n
t
u
ma
l
i
g
n
e
dwi
t
ht
h
a
ta
x
i
s
.(
Not
et
h
a
ti
fwec
h
os
es
omeot
h
e
ra
x
i
swec
ou
l
d,
wi
t
he
n
ou
g
hwor
k
,
f
i
n
da
na
n
s
we
r
,
b
u
tt
h
ea
l
ge
br
ai
son
l
ys
i
mp
l
ea
l
on
gt
h
ez
a
x
i
sa
st
h
e
mu
l
t
i
p
ol
e
swe
r
eor
i
g
i
n
a
l
l
yde
f
i
n
e
dwi
t
ht
h
e
i
rmi
n
de
xr
e
f
e
r
r
e
dt
ot
h
i
sa
x
i
s
.Al
t
e
r
n
a
t
i
v
e
l
y
wec
ou
l
dr
ot
a
t
ef
r
a
me
st
oa
l
i
g
nwi
t
ht
h
en
e
wdi
r
e
c
t
i
ona
n
dd
ot
h
ee
n
t
i
r
ec
omp
u
t
a
t
i
on
ov
e
r
.
)
Th
i
si
squ
i
t
ep
r
of
ou
n
d
.I
fwei
n
s
i
s
t
,
f
ore
x
a
mp
l
e
,
t
h
a
te
n
e
r
g
yb
et
r
a
n
s
f
e
r
r
e
di
nu
n
i
t
s
ofω,
t
h
e
na
n
g
u
l
a
rmome
n
t
u
mi
sa
l
s
ot
r
a
n
s
f
e
r
r
e
di
nu
n
i
t
sofm!
15.
7 Co
n
c
l
u
d
i
n
gRe
ma
r
k
sAb
o
u
tMu
l
t
i
p
ol
e
s
Th
e
r
ea
r
es
t
i
l
lma
n
y
,
ma
n
yt
h
i
n
g
swec
ou
l
ds
t
u
d
yc
on
c
e
r
n
i
n
gmu
l
t
i
p
ol
e
sa
n
dr
a
d
i
a
t
i
on
.
F
ore
x
a
mp
l
e
,
weh
a
v
en
oty
e
tdon
eama
g
n
e
t
i
cl
oopa
n
t
e
n
n
a
,
b
u
td
oi
n
gon
es
h
ou
l
dn
ow
b
es
t
r
a
i
g
h
t
f
or
wa
r
d(
t
oob
t
a
i
nama
g
n
e
t
i
cdi
p
ol
er
a
d
i
a
t
i
onf
i
e
l
dt
ol
e
a
di
n
gor
de
r
)
.
Hmmm,
s
ou
n
d
sl
i
k
eah
ome
wor
kore
x
a
mp
r
ob
l
e
mt
ome
.
.
.
St
i
l
l
,
I
h
op
et
h
a
tt
h
i
sh
a
sl
e
f
ty
ouwi
t
he
n
ou
g
hf
u
n
da
me
n
t
a
l
st
h
a
ty
ou
:
a
)Un
d
e
r
s
t
a
n
db
e
s
s
e
l
f
u
n
c
t
i
on
s
;
b
)Un
d
e
r
s
t
a
n
ds
p
h
e
r
i
c
a
l
h
a
r
mon
i
c
s
;
c
)Un
d
e
r
s
t
a
n
da
tl
e
a
s
ts
ome
t
h
i
n
ga
b
ou
tv
e
c
t
ors
p
h
e
r
i
c
a
l
h
a
r
mon
i
c
s
;
d
)Kn
owwh
a
ta“
mu
l
t
i
p
ol
a
re
x
p
a
n
s
i
on
”i
s
;
e
)Kn
owh
owt
oe
x
p
a
n
dav
a
r
i
e
t
yofi
mp
or
t
a
n
tGr
e
e
n
’
sf
u
n
c
t
i
on
sf
orv
e
c
t
ora
n
d
s
c
a
l
a
rHe
l
mh
ol
t
ze
qu
a
t
i
on
s(
i
n
c
l
u
d
i
n
gt
h
ePoi
s
s
one
qu
a
t
i
on
)
.
f
)Kn
ow h
ow t
of
or
mu
l
a
t
ea
ni
n
t
e
g
r
a
le
qu
a
t
i
ons
ol
u
t
i
ont
ot
h
e
s
ed
i
ffe
r
e
n
t
i
a
l
e
qu
a
t
i
on
sb
a
s
e
dont
h
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e
e
n
’
sf
u
n
c
t
i
on
,a
n
da
tl
e
a
s
tf
or
ma
l
l
ys
ol
v
ei
tb
y
p
a
r
t
i
t
i
on
i
n
gt
h
ei
n
t
e
g
r
a
l
i
n
t
odoma
i
n
sofc
on
v
e
r
g
e
n
c
e
.
g
)Kn
owh
owt
od
e
s
c
r
i
b
et
h
ee
l
e
c
t
r
oma
g
n
e
t
i
cf
i
e
l
da
tav
a
r
i
e
t
yofl
e
v
e
l
s
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e
s
el
e
v
e
l
s
h
a
db
e
t
t
e
ri
n
c
l
u
d
et
h
ee
l
e
me
n
t
a
r
yde
s
c
r
i
p
t
i
onoft
h
eE
1
,
E
2
,
a
n
dM1“
s
t
a
t
i
c
”l
e
v
e
l
sa
s
we
l
la
se
n
ou
g
hk
n
owl
e
d
g
et
ob
ea
b
l
et
odoi
tc
or
r
e
c
t
l
yf
ore
x
t
e
n
d
e
ds
ou
r
c
e
sor
s
ou
r
c
e
swh
e
r
eh
i
g
h
e
ror
de
rmome
n
t
sa
r
ei
mp
or
t
a
n
t
,
a
tl
e
a
s
ti
fy
ou
rl
i
f
eorj
oborn
e
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t
p
a
p
e
rd
e
p
e
n
doni
t
.
h
)Ca
np
a
s
sp
r
e
l
i
ms
.
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fy
ouf
e
e
lde
f
i
c
i
e
n
ti
na
n
yoft
h
e
s
ea
r
e
a
s
,Ir
e
c
omme
n
dt
h
a
ty
out
a
k
et
h
et
i
met
o
r
e
v
i
e
wa
n
dl
e
a
r
nt
h
ema
t
e
r
i
a
l
a
g
a
i
n
,
c
a
r
e
f
u
l
l
y
.Th
i
sh
a
sb
e
e
nt
h
emos
ti
mp
or
t
a
n
tp
a
r
tof
t
h
ec
ou
r
s
ea
n
di
st
h
eon
et
h
i
n
gy
ous
h
ou
l
dn
otf
a
i
l
t
ot
a
k
eou
tofh
e
r
ewi
t
hy
ou
.
I
h
op
ey
ouh
a
v
ee
n
j
oy
e
di
t
.
15.
8 Ta
b
l
eofPr
o
p
e
r
t
i
e
so
fVe
c
t
orHa
r
mo
n
i
c
s
a
)Ba
s
i
cDe
f
i
n
i
t
i
on
s
Y
m
ℓ
ℓ
Yℓ,m
ℓ
(
ℓ+1
)
1
m
Y
L
1
=
m
ℓ
r
ˆ+i
r
ˆ×LYℓ,
m
=− ℓ
(
2
ℓ+1
) −
1
ℓ
ℓ
+1
ℓ+1
)
r
ˆ+i
r
ˆ×LYℓ,
m
=− (
ℓ+1
)
(
2
ℓ+1
) (
ℓ
ℓ
−1
Y
b
)Ei
g
e
n
v
a
l
u
e
s(
j
,
ℓ
,
ma
r
ei
n
t
e
g
r
a
l
)
:
2
m
2
m
m
JYj
(
j
+1
)
Yj
ℓ =j
ℓ
m
LYj
(
ℓ+1
)
Yj
ℓ =ℓ
ℓ
m
m
ℓ
= mYj
JzYjℓ
c
)Pr
oj
e
c
t
i
v
eOr
t
h
on
or
ma
l
i
t
y
:
′
m∗
m
Yjℓ·Yj′ℓ′
d
Ω=δjj′δ
ℓ
ℓ
′δ
mm′
d
)Comp
l
e
xCon
j
u
g
a
t
i
on
:
m ∗
ℓ
+
1
−j
Y j
−
1
)
ℓ =(
m −m
(
−
1
)Y j
ℓ
e
)Add
i
t
i
onTh
e
or
e
m(
L
CBn
ot
e
sc
or
r
u
p
t–t
h
i
sn
e
e
d
st
ob
ec
h
e
c
k
e
d
)
:
m
Y
j
ℓ
∗
′
m′
·Yj
′ℓ
′
=
m+
1
(
−
1
)
4
π(
2
n+1
)
n
′′
j
ℓ
j
ℓ;
n
)
Y
′W(
Cℓℓ′nCjj′n
0
0
0
′
(
2
ℓ+1
)
(
2
ℓ+1
)
(
2
j+1
)
(
2
j
+1
)
0
,
−m,
m
′
n
,
(
m −m)
f
)F
orFa
n
yf
u
n
c
t
i
onofron
l
y
:
m
∇·(
Yℓℓ F) =
m
0
ℓ
F
d
F
∇·(
Yℓℓ−1F) =
2
ℓ
+1 (
r Yℓ,m
ℓ
−
1
) r− d
ℓ+1
F
d
F
∇·(
Yℓℓ+1F) =
2
ℓ
+1 (
r Yℓ,m
ℓ
+2
) r− d
m
×
g
)Di
t
t
o:
ℓ+1
m
i
∇×(
Y
F
)=
ℓ
ℓ
F
d
F
2
ℓ
+
1 (
r
ℓ
+1
) r +d
ℓ+1
F d
F
m
ℓ
m
Y
ℓ
ℓ
−1
m
+
F
d
F
2
ℓ
+1 −
r
ℓr +d
i
∇×(
Yℓℓ−1F)=− 2
ℓ
+1 (
ℓ
−
1
) r− d
r Yℓℓ
m
ℓ
F d
F m
i
∇×(
Yℓℓ+1F)=
2
ℓ
+
1 (
r
ℓ
+2
) r −d
Y
ℓ
ℓ
h
)Th
i
sp
u
t
st
h
eVSHsi
n
t
ov
e
c
t
orf
or
m:
2ℓ
(
ℓ
+
1)
−
Yℓ,
m−1
(
ℓ
+
m)
(
ℓ
−m+1
)
Y
ℓ
ℓ
m
√
=
ℓ
,
m
ℓ
(
ℓ
+
1
)Y
m
(
ℓm)
(
ℓ
+m+1
)
−
2
ℓ
(
ℓ
+1
)
Yℓ,
m+1
2ℓ
(
2ℓ−1)
Yℓ−1,
m−1
(
ℓ
+
m−1
)
(
ℓ
+
m)
=
ℓ
ℓ1
m
−
ℓ
(
2
ℓ
−
1)
Yℓ 1,
m
−
(
ℓ
−m)
(
ℓ
+
m)
Y
(
ℓm
1
)
(
ℓ
− −
−
2
ℓ
(
2
ℓ
−1
)
m)
Yℓ
1
,
m+1
−
(
ℓ
−m+
1
)
(
ℓ
−m+
2
)
2
(
ℓ
+
1)
(
2
ℓ
+3
)
m
Y
=
ℓ
ℓ
+1
m+1
)
(
ℓ
+
m+1
)
(
ℓ
−
(
ℓ
+1
)
(
2
ℓ
+3
)
(
ℓ
+m+2
)
(
ℓ
+
m+
1)
Yℓ+1,m−1
Yℓ+1,m
Yℓ+1,
m+1
2
(
ℓ
+
1)
(
2
ℓ
+3
)
i
)Ha
n
s
e
nMu
l
t
i
p
ol
ePr
op
e
r
t
i
e
s
∇·
ML=0∇·
NL=
0
∇·L
L
∇×
ML
∇×
NL
= i
k
f
(
k
r
)
YL(
r
ˆ
)
ℓ
∇×
L
L
m
Y
ℓ
ℓ
+1
= −
i
k
NL
= i
k
ML
= 0
j
)Ha
n
s
e
nMu
l
t
i
p
ol
eE
x
p
l
i
c
i
tF
or
ms
m
ML =f
(
k
r
)
Yℓℓ
ℓ
ℓ+1
NL =
L
L =
2
ℓ
+1f
ℓ
−1
ℓ
ℓ
−1
2
ℓ
+1f
ℓ
m
(
k
r
)
Y
ℓ
,
ℓ
−1
m
(
k
r
)
Y
ℓ
,
ℓ
−1
−
+
m
k
r
)
Yℓ,
2
ℓ
+1f
ℓ
+1
ℓ
+
1 (
ℓ+1
2
ℓ
+1f
ℓ
+
1
m
ML = f
(
k
r
)
Y ℓℓ
ℓ
NL
L
L
m
(
k
r
)
Y
1 d
m
(
k
r
f
)
i
r
ˆ
×
Y
ˆℓ
(
ℓ+1
)
f
YL
ℓ
ℓ
k
r
d
(
k
r
)
=
ℓ
ℓ −r
1
d
m
i
r
ˆ×f
Yℓℓ)−r
ˆ d
ℓ
r(
(
k
r
)f
ℓY
L
= ℓ
(
ℓ+1
)k
ℓ
,
ℓ
+1
Ch
a
p
t
e
r16
Op
t
i
c
a
l
Sc
a
t
t
e
r
i
n
g
16.
1 Ra
d
i
a
t
i
o
nRe
a
c
t
i
onofaPol
a
r
i
z
a
b
l
eMe
d
i
u
m
Us
u
a
l
l
y
,
wh
e
nwec
on
s
i
d
e
rop
t
i
c
a
l
s
c
a
t
t
e
r
i
n
g
,
wei
ma
g
i
n
et
h
a
tweh
a
v
eamo
n
o
c
h
r
oma
t
i
c
p
l
a
n
ewa
v
ei
n
c
i
d
e
n
tu
p
onap
o
l
a
r
i
z
a
b
l
eme
d
i
u
me
mb
e
d
d
e
di
n(
f
ort
h
es
a
k
eofa
r
g
u
me
n
t
)
f
r
e
es
p
a
c
e
.
Th
et
a
r
ge
twei
ma
g
i
n
ei
sa“
p
a
r
t
i
c
l
e
”ofs
omes
h
a
p
ea
n
dh
e
n
c
ei
s
ma
t
h
e
ma
t
i
c
a
l
l
ya(
s
i
mp
l
y
)c
on
n
e
c
t
e
dd
oma
i
nwi
t
hc
omp
a
c
ts
u
p
p
or
t
.
Th
ep
i
c
t
u
r
ewemu
s
t
d
e
s
c
r
i
b
ei
st
h
u
s
Th
ei
n
c
i
de
n
twa
v
e(
i
nt
h
ea
b
s
e
n
c
eoft
h
et
a
r
g
e
t
)i
st
h
u
sap
u
r
ep
l
a
n
ewa
v
e
:
E
i
n
c
H
ˆ
i
kn
=eˆ0E0e
0
·
r
i
n
c
ˆ
/
Z0.
0×E
i
n
c
=n
Th
ei
n
c
i
de
n
twa
v
ei
n
d
u
c
e
sat
i
mede
p
e
n
d
e
n
t p
ol
a
r
i
z
a
t
i
on
(
1
6
.
1
)
(
1
6
.
2
)
d
e
n
s
i
t
yi
n
t
ot
h
e
2
1
1
me
di
u
m.I
fwei
ma
g
i
n
e(
n
otu
n
r
e
a
s
on
a
b
l
y
)t
h
a
tt
h
et
a
r
g
e
ti
sap
a
r
t
i
c
l
eora
t
ommu
c
h
s
ma
l
l
e
rt
h
a
nawa
v
e
l
e
n
g
t
h
,t
h
e
nwec
a
nd
e
s
c
r
i
b
et
h
ef
i
e
l
dr
a
d
i
a
t
e
df
r
om i
t
si
n
d
u
c
e
d
d
i
p
ol
emome
n
ti
nt
h
ef
a
rz
on
ea
n
dd
i
p
o
l
ea
p
p
r
ox
i
ma
t
i
on(
s
e
ee
.
g
.
4
.
1
2
2
)
:
E
s
c
H
i
kr
e
1
2
(
n
ˆ×
p)×n
ˆ−n
ˆ×
m/
c
}
= 4πǫ0 k r {
(
1
6
.
3
)
ˆ×E
/
Z0.
s
c
= n
(
1
6
.
4
)
k0
k
,
whi
l
ee
ˆ
e
ˆa
r
et
hepol
a
r
i
z
a
t
i
onof
0,
n
dn
ˆ= k
I
nt
h
e
s
ee
x
p
r
e
s
s
i
on
s
,
n
ˆ 0=k0 a
t
h
ei
n
c
i
d
e
n
ta
n
ds
c
a
t
t
e
r
e
dwa
v
e
s
,
r
e
s
p
e
c
t
i
v
e
l
y
.
Wea
r
ei
n
t
e
r
e
s
t
e
di
nt
h
er
e
l
a
t
i
v
ep
owe
rd
i
s
t
r
i
b
u
t
i
oni
nt
h
es
c
a
t
t
e
r
e
df
i
e
l
d(
wh
i
c
h
s
h
ou
l
db
ep
r
op
or
t
i
on
a
lt
ot
h
ei
n
c
i
d
e
n
tf
i
e
l
di
nawa
yt
h
a
tc
a
nb
ema
d
ei
n
d
e
p
e
n
d
e
n
tof
i
t
sma
g
n
i
t
u
d
ei
nal
i
n
e
a
rr
e
s
p
on
s
e
/
s
u
s
c
e
p
t
i
b
i
l
i
t
ya
p
p
r
ox
i
ma
t
i
on
)
.Th
ep
owe
rr
a
d
i
a
t
e
d
i
nd
i
r
e
c
t
i
onn
ˆwi
t
hp
ol
a
r
i
z
a
t
i
one
ˆi
sn
e
e
de
dp
e
ru
n
i
ti
n
t
e
n
s
i
t
yi
nt
h
ei
n
c
i
d
e
n
twa
v
e
s
c
wi
t
hn
ˆ
,
e
ˆ
.
Th
i
squ
a
n
t
i
t
yi
se
x
p
r
e
s
s
e
da
s
0
0
2
dσ(
nˆ,eˆ,nˆ0,eˆ0)=r
d
Ω
∗
1
e
2
Z0
ˆ·
1
2
Z0
∗
e
E
2
s
c
2
E
ˆ
0·
(
1
6
.
5
)
i
n
c
[
On
eg
e
t
st
h
i
sb
yc
o
n
s
i
d
e
r
i
n
gt
h
ep
owe
rd
i
s
t
r
i
b
u
t
i
on
:
dP
1
dΩ
∗
2
nˆ·
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i
x
e
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n
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n
gi
nt
h
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e
c
t
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ona
ts
p
e
e
dv
)
.
Th
e
n
′
′
F
¨
¨
j=mx
j=mx
j=F
j
(
1
7
.
5
)
orNe
wt
o
n
’
sL
a
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r
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r
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a
n
twi
t
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tt
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nt
r
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n
s
f
or
ma
t
i
o
n
.
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t
∂
∂
∂x =∂x′
1
1
1∂
+
v∂t
′
(
1
7
.
6
)
a
n
ds
o
2
2
∂x
∂x
2
2
∂x
2
2
2
∂x
3
2 ∂
+ v2 ∂t2 + v∂x′ ∂t
1
′
′
1
∂2
=
=
(
1
7
.
8
)
′
2
(
1
7
.
9
)
∂x
3
2
(
1
7
.
7
)
′
2
∂x
2
2
∂
2
∂
∂
2
∂t
Th
u
si
f
2
1∂
′
2
=
1
∂
∂
2
2
∂
∂
′
2
= ∂t .
(
1
7
.
1
0
)
2
1∂
2
2
2∂
t
∇ −c
t
h
e
n
∇
′
2
2
1∂2
−
2∂
c
t
ψ=0
2
1∂ψ
ψ= −
2
v ∂t
′
(
1
7
.
1
1
)
2
2 ∂ψ
=0
∂t
−
2 v∂x′
′
1
(
1
7
.
1
2
)
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omb
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h
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oor
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n
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t
eor
i
g
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n
s
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e
a
c
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ni
t
sownf
r
a
me
:
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2 2
2
′2
′
2 ′
2
′
2
(
c
t
)−(
x+y +z)=0
(
1
7
.
1
3
)
a
n
d
(
c
t
)−(
x +y +z)=0
(
1
7
.
1
4
)
a
r
es
i
mu
l
t
a
n
e
o
u
sc
on
s
t
r
a
i
n
t
sont
h
ee
qu
a
t
i
on
s
.
Mos
tg
e
n
e
r
a
l
l
y
,
2
2
2
2
2
(
c
t
)−(
x +y +z)=λ
′2
′
2
′
2
′
2
(
c
t
)−(
x +y +z)
(
1
7
.
1
5
)
wh
e
r
e
,
λ
(
v
)d
e
s
c
r
i
b
e
sap
os
s
i
b
l
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h
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n
g
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c
a
l
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e
t
we
e
nt
h
ef
r
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me
s
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n
s
i
s
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h
a
t
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t
h
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oor
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et
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or
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t
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omog
e
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n
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r
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cb
e
t
we
e
nt
h
ef
r
a
me
s,
t
h
e
n
λ
=
1
.
(
1
7
.
1
6
)
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I
fwer
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l
a
xt
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sr
e
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e
me
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oor
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t
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ou
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r
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t
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r
e
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h
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or
ma
l
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ou
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r
e
s
u
l
t
s
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e
tu
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e
f
i
n
e
x
0 =c
t
(
1
7
.
1
7
)
x
1
=x
(
1
7
.
1
8
)
x
2 =y
(
1
7
.
1
9
)
x
3 =z
(
1
7
.
2
0
)
(
x4 =i
c
tMi
n
k
ows
k
i
me
t
r
i
c
)
(
1
7
.
2
1
)
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e
nwen
e
e
dal
i
n
e
a
rt
r
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n
s
f
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ma
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h
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oor
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n
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t
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h
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tmi
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e
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n
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t
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n
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h
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e
c
t
i
onofvi
ns
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c
hawa
yt
h
a
tt
h
el
e
n
g
t
h
2
2
2
2
2
s =(
x
)−(
x
0
1 +x
2 +x
3)
(
1
7
.
2
2
)
i
sc
on
s
e
r
v
e
da
n
dt
h
a
tg
oe
si
n
t
ot
h
eGa
l
l
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nt
r
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n
s
f
or
ma
t
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ona
sv→ 0
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fwec
on
t
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n
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e
t
oa
s
s
u
met
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a
tvi
si
nt
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r
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c
t
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on
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t
h
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sl
e
a
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ot
h
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o
r
e
n
t
zt
r
a
n
s
f
or
ma
t
i
o
n
:
′
x
0
′
x
1
′
x
2
′
x
3
(
x
x
)
0−β
1
=γ
(
1
7
.
2
3
)
(
x
x
)
=γ
1−β
0
(
1
7
.
2
4
)
=x
2
(
1
7
.
2
5
)
=x
3
(
1
7
.
2
6
)
′
wh
e
r
ea
tx
,
1=0
v
x
t→ β=
1=v
c.
(
1
7
.
2
7
)
Th
e
n
2
l
e
a
dst
o
′
2
s
s=
2
2
2 2
2
22 2
(
1
7
.
2
8
)
2
x
x
+γβ(
x
0 −x
1 =γ(
0 −x
1)
1 −x
0)
or
2
2
(
1
7
.
2
9
)
γ(
1−β)=1
(
1
7
.
3
0
)
±
1
(
1
7
.
3
1
)
s
o
γ=
2
1
−
β
wh
e
r
ewec
h
oos
et
h
e+s
i
g
nb
yc
on
v
e
n
t
i
on
.
Th
i
sma
k
e
sγ
(
0
)=+1
.
F
i
n
a
l
l
y
,
1
γ
(
v
)=
(
1
7
.
3
2
)
2
v
1− c2
a
swea
l
l
k
n
owa
n
dl
ov
e
.
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l
e
tmer
e
mi
n
dy
out
h
a
twh
e
nv<
<c
,
2
1v
2
γ
(
v
)=1+ 2 c +
.
.
.
(
1
7
.
3
3
)
v
t
ol
owe
s
ts
u
r
v
i
v
i
n
gor
d
e
ri
n c.Aswes
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l
ls
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e
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h
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ce
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t
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e
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t
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x
0
′
(
x
x
0+β
1)
=γ
(
1
7
.
3
4
)
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′
(
x
x
=γ
1+β
0)
′
=x
2
′
3
=x
x
1
x
2
x
3
(
1
7
.
3
5
)
(
1
7
.
3
6
)
(
1
7
.
3
7
)
wh
i
c
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sp
e
r
f
e
c
t
l
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mme
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r
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c
,
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t
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v
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ta
p
p
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r
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h
a
twh
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c
hf
r
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mei
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t“
r
e
s
t
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n
d
wh
i
c
hi
smov
i
n
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t
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e
ma
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c
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l
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tl
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s
t
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t
t
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p
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c
t
i
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e
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F
i
n
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l
l
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i
fwel
e
t
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(
1
7
.
3
8
)
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t
r
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r
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i
r
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c
t
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n
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h
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v
eb
u
tt
ou
s
edotp
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od
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c
t
st
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l
i
g
nt
h
ev
e
c
t
or
t
r
a
n
s
f
or
ma
t
i
one
qu
a
t
i
on
swi
t
ht
h
i
sd
i
r
e
c
t
i
on
:
′
x
βx
)
·
x =x+ γ−1(
βx
)
β
0
=γ
(
x
(
1
7
.
3
9
)
0−
′
2
β
·
γ
βx
−
(
1
7
.
4
0
)
0
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h
i
n
kt
h
a
ty
ous
h
ou
l
dp
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et
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r
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r
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l
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ffic
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h
o
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h
a
tt
h
i
sr
e
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e
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et
h
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or
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b
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h
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t
d
i
r
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c
t
i
ona
n
da
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r
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i
t
r
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r
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r
a
n
s
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e
r
s
ed
i
r
e
c
t
i
on
.
Sol
u
t
i
on
:Not
et
h
a
t
β·x
)
β
x= (
(
1
7
.
4
1
)
β2
L
or
e
n
t
zt
r
a
n
s
f
or
mi
ta
c
c
or
d
i
n
gt
ot
h
i
sr
u
l
ea
n
don
eg
e
t
s(
b
yi
n
s
p
e
c
t
i
on
)
′
x =γ
(
x−βx
)
0
(
1
7
.
4
2
)
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a
son
es
h
ou
l
d
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ex0t
r
a
n
s
f
or
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ou
s
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i
n
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l
l
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t
h
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h
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omp
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n
t
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e
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r
i
b
u
t
i
onf
r
omγ
.
Th
a
ti
s
,
′
x=(
x
+γ
x−x−γ
βx
⊥+x)
0
(
1
7
.
4
3
)
(
r
e
c
on
s
t
r
u
c
t
i
n
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er
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s
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l
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e
c
t
l
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i
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e
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e
r
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t
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r
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et
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r
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me
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e
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ta
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n
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e
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t
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s
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t
h
el
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mi
t
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n
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t
r
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t
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r
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h
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n
s
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t
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l
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p
p
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twi
t
h
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s
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d
e
r
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n
gt
h
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n
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t
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a
l
f
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l
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I
ti
se
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s
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os
e
ef
r
omt
h
ede
f
i
n
i
t
i
ona
b
ov
et
h
a
t
2
22
(
γ −γβ)=1
.
(
1
7
.
4
4
)
Th
er
a
n
g
eofβ i
sd
e
t
e
r
mi
n
e
db
yt
h
er
e
qu
i
r
e
me
n
tt
h
a
tt
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et
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n
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or
ma
t
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on b
e
n
on
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n
g
u
l
a
ra
n
di
t
ss
y
mme
t
r
y
:
0≤β<1 s
ot
h
a
t1≤γ<∞.
(
1
7
.
4
5
)
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i
n
ka
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ou
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u
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c
t
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o
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st
h
a
t“
n
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t
u
r
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y
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g
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t
h
e
ya
r
e
:
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os
h ξ−s
i
nh ξ=1
(
1
7
.
4
6
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wher
e
−
ξ
β =t
a
n
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os
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= 2(
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= 2(
∈ [
0
,
∞)
.
(
1
7
.
4
7
)
(
1
7
.
4
8
)
(
1
7
.
4
9
)
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ep
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r
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me
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e
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ra
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e
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me
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on
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e
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n
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e
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et
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er
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t
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ni
ma
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n
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g
l
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.
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c
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t
et
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on
,
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=x coshξ xs
i
n
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−
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(
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7
.
5
0
)
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i
n
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os
hξ
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=−
(
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7
.
5
1
)
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=x
(
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7
.
5
2
)
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a
ti
st
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e4×4t
r
a
n
s
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o
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rd
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ort
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e
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ookl
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t
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rd
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.
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3 4Ve
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r
s
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swed
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tt
h
a
tme
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n
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t
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a
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yt
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ot
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t
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on
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p
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c
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c
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1
2
3
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b
i
t
r
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r
y4
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e
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r
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A2,
A3)
.
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r
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t
r
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oor
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n
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t
e
s
,
t
h
e
n
A′
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(
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β A)
− ·
(
1
7
.
5
3
)
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′
A
′
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(
A−βA0)
=γ
(
1
7
.
5
4
)
=A⊥
(
1
7
.
5
5
)
a
n
d
2
2
2
2
2
A =A0 −(
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=
2
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A
(
1
7
.
5
6
)
i
sa
ni
n
v
a
r
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omp
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d
s
!
Ama
z
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rf
r
i
e
n
d
s
!
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t
ou
n
dy
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e
i
g
h
b
or
s
!
Sh
owt
h
a
t
A′ B′ A′
0 0−
′
·
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A B
·
0 0−
(
1
7
.
5
7
)
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sa
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n
v
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r
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i
si
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e
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ome
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k
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i
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u
r
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.
1
:
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eL
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g
h
tCo
n
e
:
Pa
s
t
,
n
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f
u
t
u
r
e
,
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e
wh
e
r
e
.
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v
e
n
t
s
.
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ewor
l
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i
n
e
.
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s
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rt
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n
t
s
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n
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ei
n
v
a
r
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a
n
ti
n
t
e
r
v
a
l
2
2
2
2
S12 =c(
t
)−|
x
|
1−t
2
1−x
2
(
1
7
.
5
8
)
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h
e
nweh
a
v
ea
2
t
i
me
l
i
k
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e
p
a
r
a
t
i
o
nS12 >0
2
2
2
⇒ c(
t
)>|
x
|.
1−t
2
1−x
2
Bot
he
v
e
n
t
sa
r
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n
s
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5
9
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d
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x
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7
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6
6
)
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x
1
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x
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7
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6
7
)
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d
x
3
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g
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te
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ed
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ffe
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et
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n
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i
r
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t
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h
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e
r
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ou
l
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t
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n
s
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or
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i
k
e
α
β
F =
α
β
∂x¯ ∂x¯
γ
δ
γ δF .
∂x ∂
x
(
1
8
.
5
)
Si
mi
l
a
r
l
y
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weh
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v
e
c
ov
a
r
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n
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e
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o
r
so
fr
a
n
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(
1
8
.
6
)
a
n
d
mi
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e
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e
n
s
o
r
sofr
a
n
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α δ
α
γ
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β
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¯
(
1
8
.
7
)
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e
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t
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y
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c
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r
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ore
x
a
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nc
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t
r
u
c
tas
e
c
on
dr
a
n
kt
e
n
s
orb
y
:
α
β
αβ
F =AB
(
1
8
.
8
)
wh
e
r
eαa
n
dβr
u
nov
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rt
h
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l
lr
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a
tt
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i
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ema
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c
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.
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oc
c
u
r
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mp
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n
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e
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i
oni
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e
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h
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ra
l
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on
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a
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ewr
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ti
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y
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c
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l
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nt
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e
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on
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n
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si
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tr
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v
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r
ed
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t
h
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t
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c
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l
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y
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t
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ona
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l
l
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n
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h
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l
dr
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t
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r
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et
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a
da
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ou
tge
o
me
t
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l
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b
r
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s
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l
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f
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l
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yg
e
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r
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z
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n
gt
h
en
ot
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omp
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u
mb
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st
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omp
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p
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b
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t
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a
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yd
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on
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i
l
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n
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i
l
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t
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r
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j
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t
s
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na
d
d
i
t
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ont
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t
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di
n
gt
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er
a
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koft
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t
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or
mi
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a
di
c
,
t
r
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a
d
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t
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er
a
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r
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t
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c
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i
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wot
e
n
s
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st
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er
e
s
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l
tofs
e
t
t
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n
g
t
wooft
h
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n
d
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t
y
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c
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t
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ns
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mma
t
i
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e
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h
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r
e
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a
n
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e
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i
sr
e
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u
c
e
st
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er
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n
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t
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x
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r
e
s
s
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e
l
a
t
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v
et
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h
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t
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on
s
t
i
t
u
e
n
t
s
,
h
e
n
c
et
h
et
e
r
m“
c
on
t
r
a
c
t
i
on
”
.
Ane
x
p
r
e
s
s
i
onc
a
nb
ec
on
t
r
a
c
t
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dov
e
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e
v
e
r
a
lc
omp
on
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n
t
sa
tat
i
mewh
e
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n
g
a
l
g
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b
r
as
os
e
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on
dr
a
n
kt
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s
or
sc
a
nb
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on
t
r
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c
t
e
dt
of
or
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s
c
a
l
a
r
,
f
ore
x
a
mp
l
e
,
or
t
h
i
r
dr
a
n
kt
e
n
s
or
sc
a
nb
ec
on
t
r
a
c
t
e
dt
of
i
r
s
t
.
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rf
a
mi
l
i
a
rn
ot
i
onofmu
l
t
i
p
l
y
i
n
gav
e
c
t
orb
yama
t
r
i
xt
op
r
od
u
c
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e
c
t
ori
np
r
op
e
r
t
e
n
s
orl
a
n
g
u
a
g
ei
st
of
or
mt
h
eou
t
e
rp
r
od
u
c
toft
h
ema
t
r
i
x(
s
e
c
on
dr
a
n
kt
e
n
s
or
)a
n
dt
h
e
v
e
c
t
or(
f
i
r
s
tr
a
n
kt
e
n
s
or
)
,
s
e
tt
h
er
i
g
h
t
mos
ti
n
di
c
e
st
ob
ee
qu
a
l
a
n
ds
u
mov
e
rt
h
a
ti
n
d
e
x
t
op
r
od
u
c
et
h
er
e
s
u
l
t
i
n
gf
i
r
s
tr
a
n
kt
e
n
s
or
.
He
n
c
ewed
e
f
i
n
eou
rs
c
a
l
a
rp
r
o
d
u
c
tt
ob
et
h
ec
o
n
t
r
a
c
t
i
onofac
ov
a
r
i
a
n
ta
n
d
c
on
t
r
a
v
a
r
i
a
n
tv
e
c
t
or
.
α
B·
A=
BαA
(
1
8
.
9
)
Not
et
h
a
tI
’
v
ei
n
t
r
od
u
c
e
das
or
tof“
s
l
o
p
p
y
”c
on
v
e
n
t
i
ont
h
a
tas
i
n
g
l
equ
a
n
t
i
t
yl
i
k
eBorA
c
a
nb
eaf
ou
r
v
e
c
t
ori
nc
on
t
e
x
t
.Cl
e
a
r
l
yt
h
ee
x
p
r
e
s
s
i
onont
h
er
i
g
h
ts
i
d
ei
sl
e
s
s
a
mb
i
g
u
ou
s
!
Then:
γ
′′
α
∂x ∂
x
¯
δ
δ
B·
A = ∂x¯α Bγ ∂x
A
γ
=
∂x
δ
A
γ
δB
∂x
δ
=δγδBγA
δ
B·
A
=BδA =
(
1
8
.
1
0
)
a
n
dt
h
ede
s
i
r
e
di
n
v
a
r
i
a
n
c
ep
r
op
e
r
t
yi
sp
r
ov
e
d.
Hmmm,
t
h
a
twa
sp
r
e
t
t
ye
a
s
y
!
Ma
y
b
et
h
e
r
ei
ss
ome
t
h
i
n
gt
ot
h
i
sn
ot
a
t
i
ont
h
i
n
ga
f
t
e
ra
l
l
!
18.
3 Th
eMe
t
r
i
cTe
n
s
o
r
Th
es
e
c
t
i
ona
b
ov
ei
ss
t
i
l
lv
e
r
yg
e
n
e
r
i
ca
n
dl
i
t
t
l
eofi
td
e
p
e
n
dsonwh
e
t
h
e
rt
h
et
e
n
s
or
sa
r
e
t
h
r
e
eorf
ou
rort
e
nd
i
me
n
s
i
on
a
l
.
Wen
o
wn
e
e
dt
oma
k
et
h
e
mwor
kf
ort
h
es
p
e
c
i
f
i
cg
e
ome
t
r
y
wea
r
ei
n
t
e
r
e
s
t
e
di
n
,
wh
i
c
hi
son
ewh
e
r
ewewi
l
lu
l
t
i
ma
t
e
l
yb
es
e
e
k
i
n
gt
r
a
n
s
f
or
ma
t
i
on
st
h
a
t
p
r
e
s
e
r
v
et
h
ei
n
v
a
r
i
a
n
ti
n
t
e
r
v
a
l
:
2
02
12
22
32
(
d
s
)=(
d
x)−(
d
x)−(
d
x)−(
dx)
(
1
8
.
1
1
)
a
st
h
i
si
st
h
eon
et
h
a
td
i
r
e
c
t
l
ye
n
c
od
e
sa
ni
n
v
a
r
i
a
n
ts
p
e
e
dofl
i
g
h
t
.
2
2
F
r
om t
h
i
sp
oi
n
ton
,wemu
s
tb
ec
a
r
e
f
u
ln
ott
oc
on
f
u
s
ex·x=x a
n
dx =y
,e
t
c
.
Con
t
r
a
v
a
r
i
a
n
ti
n
d
i
c
e
ss
h
ou
l
db
ec
l
e
a
rf
r
omc
on
t
e
x
t
,a
ss
h
o
u
l
db
ep
owe
r
s
.Tos
i
mp
l
i
f
yl
i
f
e
,
a
l
g
e
b
r
a
i
c
a
l
l
yi
n
di
c
e
sa
r
ea
l
wa
y
sg
r
e
e
k(
4
–v
e
c
t
or
)orr
oma
ni
t
a
l
i
c(
3
–v
e
c
t
or
)wh
i
l
ep
owe
r
s
a
r
es
t
i
l
l
p
owe
r
sa
n
dh
e
n
c
ea
r
eg
e
n
e
r
a
l
l
yi
n
t
e
g
e
r
s
.
µ
L
e
tu
swr
i
t
et
h
i
si
nt
e
r
msofon
l
yc
on
t
r
a
v
a
r
i
a
n
tp
i
e
c
e
sd
x.Th
i
sr
e
qu
i
r
e
st
h
a
twe
i
n
t
r
od
u
c
ear
e
l
a
t
i
v
emi
n
u
ss
i
gnwh
e
nc
on
t
r
a
c
t
i
n
gou
tt
h
ec
o
mp
on
e
n
t
soft
h
e
s
p
a
t
i
a
lp
a
r
to
ft
h
ed
i
ffe
r
e
n
t
i
a
lo
n
l
y
.Wec
a
nmos
te
a
s
i
l
ye
n
c
od
et
h
i
sr
e
qu
i
r
e
me
n
ti
n
t
oa
s
p
e
c
i
a
l
ma
t
r
i
x(
t
e
n
s
or
)c
a
l
l
e
dt
h
eme
t
r
i
ct
e
n
s
ora
s
:
2
α β
(
d
s
)=g
xd
x
α
βd
(
1
8
.
1
2
)
Th
et
e
n
s
orgob
v
i
ou
s
l
ys
a
t
i
s
f
i
e
st
h
ef
ol
l
owi
n
gp
r
op
e
r
t
y
:
g
α
β=g
βα
(
1
8
.
1
3
)
(
t
h
a
ti
s
,i
ti
ss
y
mme
t
r
i
c
)b
e
c
a
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onwi
t
hr
e
s
p
e
c
tt
oac
o
v
a
r
i
a
n
tv
e
c
t
orc
oor
d
i
n
a
t
et
r
a
n
s
f
or
ms
l
i
k
eac
on
t
r
a
v
a
r
i
a
n
tv
e
c
t
orop
e
r
a
t
o
r
.Th
i
sa
l
s
of
ol
l
owsf
r
om t
h
ea
b
ov
eb
yu
s
i
n
gt
h
e
me
t
r
i
ct
e
n
s
or
,
∂
∂
β
x .
∂x
α =g
α
β ∂
(
1
8
.
2
4
)
I
ti
st
e
d
i
ou
st
owr
i
t
eou
ta
l
loft
h
ep
i
e
c
e
sofp
a
r
t
i
a
lde
r
i
v
a
t
i
v
e
sw.
r
.
t
.v
a
r
i
ou
s
c
omp
on
e
n
t
s
,
s
owe(
a
su
s
u
a
l
,
b
e
i
n
gt
h
el
a
z
ys
or
t
st
h
a
twea
r
e
)i
n
t
r
odu
c
ea
“
s
i
mp
l
i
f
y
i
n
g
”n
ot
a
t
i
on
.
I
td
oe
s
,
t
oo,
a
f
t
e
ry
oug
e
tu
s
e
dt
oi
t
.
α
∂
∂
∂
0
∂x ,
α =(
−
∇)
= ∂x
∂
∂
α
(
1
8
.
2
5
)
0
∂α = ∂x
∂x ,
+∇)
.
=(
(
1
8
.
2
6
)
Not
et
h
a
tweh
a
v
ec
l
e
v
e
r
l
yi
n
d
i
c
a
t
e
dt
h
ec
o/
c
on
t
r
an
a
t
u
r
eoft
h
ev
e
c
t
orop
e
r
a
t
or
sb
y
t
h
ep
l
a
c
e
me
n
toft
h
ei
n
d
e
xont
h
eb
a
r
ep
a
r
t
i
a
l
.
Wec
a
n
n
otr
e
s
i
s
twr
i
t
i
n
gdownt
h
e4–di
v
e
r
ge
n
c
eofa4–v
e
c
t
or
:
0
0
∂A
α
α
∂Aα=∂
A=
α
1 ∂A
0
∂x +
c ∂
∇·
A=
t+
∇·
A
(
1
8
.
2
7
)
wh
i
c
hl
ook
sal
o
tl
i
k
eac
on
t
i
n
u
i
t
ye
qu
a
t
i
onorac
e
r
t
a
i
nwe
l
l
–k
n
owng
a
u
g
ec
on
d
i
t
i
on
.
µ
(
Me
d
i
da
t
eonj
u
s
twh
a
tA wou
l
dn
e
e
dt
ob
ef
ore
i
t
h
e
roft
h
e
s
ee
qu
a
t
i
on
st
ob
er
e
a
l
i
z
e
d
a
saf
ou
r
s
c
a
l
a
r
)
.
Hmmmmmm,
I
s
a
y
.
E
v
e
nmor
ee
n
t
e
r
t
a
i
n
i
n
gi
st
h
e4
–L
a
p
l
a
c
i
a
n
,
c
a
l
l
e
dt
h
eD’
L
a
mb
e
r
t
i
a
nop
e
r
a
t
or
:
∂2
α
02
2
∂ = ∂x −∇
α
✷ =∂
(
1
8
.
2
8
)
2
1∂
2 2
2
(
1
8
.
2
9
)
= c ∂t −∇
wh
i
c
hj
u
s
th
a
p
p
e
n
st
ob
et
h
e(
n
e
g
a
t
i
v
eoft
h
e
)wa
v
eo
p
e
r
a
t
or
!Hmmmmmmmm!By
s
t
r
a
n
g
ec
oi
n
c
i
d
e
n
c
e
,c
e
r
t
a
i
nob
j
e
c
t
sofg
r
e
a
ti
mp
or
t
a
n
c
ei
ne
l
e
c
t
r
od
y
n
a
mi
c
s“
j
u
s
t
h
a
p
p
e
n
”t
ob
eL
or
e
n
t
zs
c
a
l
a
r
s
!Re
me
mb
e
rt
h
a
tId
i
ds
a
ya
b
ov
et
h
a
tp
a
r
toft
h
ep
oi
n
tof
i
n
t
r
odu
c
i
n
gt
h
i
sl
ov
e
l
yt
e
n
s
orn
ot
a
t
i
onwa
st
oma
k
et
h
ev
a
r
i
ou
st
r
a
n
s
f
or
ma
t
i
on
a
l
s
y
mme
t
r
i
e
sofp
h
y
s
i
c
a
lqu
a
n
t
i
t
i
e
sma
n
i
f
e
s
t
,a
n
dt
h
i
sa
p
p
e
a
r
st
ob
et
r
u
ewi
t
ha
v
e
n
g
e
a
n
c
e
!
Th
a
twa
st
h
e“
e
a
s
y
”p
a
r
t
.I
twa
sa
l
lg
e
ome
t
r
y
.Nowweh
a
v
et
od
ot
h
eme
s
s
yp
a
r
t
a
n
dd
e
r
i
v
et
h
ei
n
f
i
n
i
t
e
s
i
ma
l
t
r
a
n
s
f
or
ma
t
i
on
st
h
a
tl
e
a
v
es
c
a
l
a
r
si
nt
h
i
sme
t
r
i
ci
n
v
a
r
i
a
n
t
.
18.
4 Ge
n
e
r
a
t
o
r
so
ft
h
eL
o
r
e
n
t
zGr
o
u
p
L
e
t
0
x
x=
1
x
3
x
2
x
(
1
8
.
3
0
)
b
eac
ol
u
mnv
e
c
t
or
.No
t
et
h
a
twen
ol
on
g
e
ri
n
d
i
c
a
t
eav
e
c
t
orb
yu
s
i
n
gav
e
c
t
ora
r
r
ow
a
n
d/
orb
ol
d
f
a
c
e–t
h
os
ea
r
er
e
s
e
r
v
e
df
ort
h
es
p
a
t
i
a
l
p
a
r
toft
h
ef
ou
r
v
e
c
t
oron
l
y
.Th
e
na
“
ma
t
r
i
x
”s
c
a
l
a
rp
r
od
u
c
ti
sf
or
me
di
nt
h
eu
s
u
a
l
wa
yb
y
(
a
,
b
)=a
b
˜
(
1
8
.
3
1
)
wh
e
r
ea
˜i
st
h
e(
r
owv
e
c
t
or
)t
r
a
n
s
p
os
eofa
.
Th
eme
t
r
i
xt
e
n
s
ori
sj
u
s
tama
t
r
i
x
:
g=0 −
1 0 0
10
0 0
00
0
1
(
1
8
.
3
2
)
00
−
1 0
−
andg
2
⇔
=I.
F
i
n
a
l
l
y
,
0
g
x= x
− x1
x
−x
2
=x0 .
(
1
8
.
3
3
)
x
1
2
x
3
x
3
−
I
nt
h
i
sc
omp
a
c
tn
ot
a
t
i
onwede
f
i
n
et
h
es
c
a
l
a
rp
r
od
u
c
ti
nt
h
i
sme
t
r
i
ct
ob
e
α
β
α
a·b=(
a
,
g
b
)=(
g
a
,
b
)=a
g
b
˜=ag
.
α
βb =ab
α
(
1
8
.
3
4
)
Wes
e
e
kt
h
es
e
t(
g
r
o
u
p
,
weh
op
e
)ofl
i
n
e
a
rt
r
a
n
s
f
or
ma
t
i
on
st
h
a
tl
e
a
v
e
s(
x
,
g
x
)=x·x
i
n
v
a
r
i
a
n
t
.Si
n
c
et
h
i
si
st
h
e“
n
or
m”(
s
qu
a
r
e
d
)ofaf
ou
rv
e
c
t
or
,t
h
e
s
ea
r
e“
l
e
n
g
t
h
p
r
e
s
e
r
v
i
n
g
”t
r
a
n
s
f
or
ma
t
i
on
si
nt
h
i
sf
ou
rd
i
me
n
s
i
on
a
lme
t
r
i
c
.Th
a
ti
s
,wewa
n
ta
l
l
ma
t
r
i
c
e
sAs
u
c
ht
h
a
t
′
(
1
8
.
3
5
)
′′
(
1
8
.
3
6
)
x=Ax
l
e
a
v
e
st
h
en
or
mofxi
n
v
a
r
i
a
n
t
,
′ ′
x·x=x
˜g
x=x
g
x
˜=x·x
or
˜
x
˜
Ag
Ax=x
g
x
˜
or
˜
Ag
A=g
.
Cl
e
a
r
l
yt
h
i
sl
a
s
tc
on
d
i
t
i
o
ni
ss
u
ffic
i
e
n
tt
oe
n
s
u
r
et
h
i
sp
r
op
e
r
t
yi
nA.
Now,
˜
2
t|
g|
(
de
t|
A|
) =de
t|
g|
d
e
tAg
A =de
(
1
8
.
3
7
)
(
1
8
.
3
8
)
(
1
8
.
3
9
)
wh
e
r
et
h
el
a
s
te
qu
a
l
i
t
yi
sr
e
qu
i
r
e
d.
Bu
tde
t|
g
|
=−
1=0
,
s
o
d
e
t|
A|
=±
1
(
1
8
.
4
0
)
i
sac
o
n
s
t
r
a
i
n
tont
h
ea
l
l
owe
dma
t
r
i
c
e
s(
t
r
a
n
s
f
or
ma
t
i
on
s
)A.
Th
e
r
ea
r
et
h
u
st
wo
c
l
a
s
s
e
soft
r
a
n
s
f
or
ma
t
i
on
swec
a
nc
on
s
i
d
e
r
.
Th
e
p
r
op
e
rL
o
r
e
n
t
zt
r
a
n
s
f
o
r
ma
t
i
o
n
swi
t
hd
e
t|
A|
=+
1
;
a
n
d
i
mp
r
op
e
rL
o
r
e
n
t
zt
r
a
n
s
f
or
ma
t
i
o
n
swi
t
hd
e
t|
A|
=±
1
.
Pr
op
e
rL
.T.
’
sc
on
t
a
i
nt
h
ei
de
n
t
i
t
y(
a
n
dt
h
u
sc
a
nf
or
m ag
r
ou
pb
yt
h
e
ms
e
l
v
e
s
)
,b
u
t
i
mp
r
o
p
e
rL
.T.
’
sc
a
nh
a
v
ee
i
t
h
e
rs
i
g
noft
h
ed
e
t
e
r
mi
n
a
n
t
.Th
i
si
sas
i
g
n
a
lt
h
a
tt
h
e
me
t
r
i
cwea
r
eu
s
i
n
gi
s“
i
n
d
e
f
i
n
i
t
e
”
.Twoe
x
a
mp
l
e
sofi
mp
r
op
e
rt
r
a
n
s
f
or
ma
t
i
on
st
h
a
t
i
l
l
u
s
t
r
a
t
et
h
i
sp
oi
n
ta
r
es
p
a
t
i
a
l
i
n
v
e
r
s
i
on
s(
wi
t
hde
t|
A|=−
1
)a
n
dA=−
I(
s
p
a
c
ea
n
dt
i
me
i
n
v
e
r
s
i
on
,
wi
t
hd
e
t|
A|
=+
1
)
.
I
nv
e
r
yg
e
n
e
r
a
l
t
e
r
ms
,
t
h
ep
r
op
e
rt
r
a
n
s
f
or
ma
t
i
on
sa
r
et
h
ec
on
t
i
n
u
ou
s
l
yc
on
n
e
c
t
e
don
e
s
t
h
a
tf
or
maL
i
eg
r
ou
p
,
t
h
ei
mp
r
op
e
ron
e
si
n
c
l
u
deon
eormor
ei
n
v
e
r
s
i
on
sa
n
da
r
en
ote
qu
a
l
t
ot
h
ep
r
odu
c
tofa
n
yt
wop
r
op
e
rt
r
a
n
s
f
or
ma
t
i
on
s
.Th
ep
r
op
e
rt
r
a
n
s
f
or
ma
t
i
on
sa
r
ea
s
u
b
g
r
ou
poft
h
ef
u
l
lg
r
ou
p—t
h
i
si
sn
ott
r
u
eoft
h
ei
mp
r
op
e
ron
e
s
,wh
i
c
h
,a
mon
got
h
e
r
t
h
i
n
g
s
,
l
a
c
kt
h
ei
d
e
n
t
i
t
y
.Wi
t
ht
h
i
si
nmi
n
d,
l
e
tu
sr
e
v
i
e
wt
h
ep
r
op
e
r
t
i
e
sofi
n
f
i
n
i
t
e
s
i
ma
l
l
i
n
e
a
r
t
r
a
n
s
f
or
ma
t
i
on
s
,p
r
e
p
a
r
a
t
or
yt
od
e
d
u
c
i
n
gt
h
ep
a
r
t
i
c
u
l
a
ron
e
st
h
a
tf
or
mt
h
eh
omog
e
n
e
ou
s
L
or
e
n
t
zg
r
ou
p
.
18.
4.
1 I
n
f
i
n
i
t
e
s
i
ma
l
Tr
a
n
s
f
o
r
ma
t
i
on
s
Wes
e
e
k(
L
i
e
)g
r
ou
p
sofc
on
t
i
n
ou
sl
i
n
e
a
rt
r
a
n
s
f
or
ma
t
i
on
s
,
′
x=Tax
(
1
8
.
4
1
)
or
′
µ
µ
x =f (
x
;
a
)
(
1
8
.
4
2
)
f
orµ=1
,
2
,
...n
.Wer
e
q
u
i
r
et
h
a
tt
h
ea=a
,
...,
a
r
err
e
a
ln
u
mb
e
r
s(
p
a
r
a
me
t
e
r
s
)t
h
a
t
1
ra
c
h
a
r
a
c
t
e
r
i
z
et
h
et
r
a
n
s
f
or
ma
t
i
on
.
rmu
s
tb
emi
n
i
ma
l
(
“
e
s
s
e
n
t
i
a
l
”
)
.
E
x
a
mp
l
e
soft
r
a
n
s
f
o
r
ma
t
i
on
sofi
mp
or
t
a
n
c
ei
np
h
y
s
i
c
s(
t
h
a
ty
ous
h
ou
l
da
l
r
e
a
d
yb
e
f
a
mi
l
i
a
rwi
t
h
)i
n
c
l
u
d
e
′
x =Tdx
=x+d
(
1
8
.
4
3
)
wh
e
r
ed=(
d
,...,dn)
.Th
i
si
st
h
e(
np
a
r
a
me
t
e
r
)t
r
a
n
s
l
a
t
i
ong
r
ou
pi
nnd
i
me
n
s
i
on
s
.
1
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om c
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t
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t
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1
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p
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t
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os
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.
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.
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n
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8
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5
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s
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ma
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n
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r
:
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ti
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e
f
ta
sa
ne
x
e
r
c
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s
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r
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f
yt
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a
t.
.
.
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t
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f
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omb
e
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or
e
.
·
Now,weh
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v
ee
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t
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u
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n
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n
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i
n
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ei
td
e
p
e
n
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eor
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rt
h
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ep
e
r
f
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me
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n
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v
e
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s
e
,
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h
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od
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c
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oos
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se
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n
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l
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n
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ft
h
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oos
t
sa
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nt
h
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a
me
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i
r
e
c
t
i
on
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ewor
s
tp
a
r
t
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r
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st
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e
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r
ai
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l
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s
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r
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r
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l
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h
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ors
ome
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et
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c
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l
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g
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ma
t
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c
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g
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t
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ot
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or
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b
l
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i
si
sk
n
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Th
oma
sp
r
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on
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18.
5 Thoma
sPr
e
c
e
s
s
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o
n
Wemu
s
tb
e
g
i
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rd
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t
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l
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th
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p
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h
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ge
µ= s
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0
3
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e
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l
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c
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t
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c
e
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c
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t
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n
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e
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l
t
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t
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e
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pe
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e
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r
a
c
t
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on
:
ge
UAZ=
−
2mcs·B
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8
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1
0
4
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sc
or
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e
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t
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e
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l
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ef
oramome
n
tofr
e
l
i
g
i
ou
ss
i
l
e
n
c
ea
n
dc
on
t
e
mp
l
a
t
eag
r
e
a
twon
d
e
rof
n
a
t
u
r
e
.
Th
i
si
st
h
es
c
i
e
n
t
i
s
t
’
sv
e
r
s
i
onof“
p
r
a
y
e
ri
ns
c
h
ool
”
.
18.
7 Th
eTr
a
n
s
f
o
r
ma
t
i
o
no
fE
l
e
c
t
r
o
ma
gn
e
t
i
cF
i
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d
s
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h
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tweh
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v
et
h
i
si
nh
a
n
d,wec
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ne
a
s
i
l
ys
e
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owt
ot
r
a
n
s
f
or
mt
h
ee
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c
t
r
i
ca
n
d
ma
g
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e
t
i
cf
i
e
l
d
swh
e
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oos
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r
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me
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r
s
e
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h
a
td
oe
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otg
u
a
r
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n
t
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et
h
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tt
h
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r
e
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u
l
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l
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es
i
mp
l
e
.
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β
′
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on
v
e
r
tF f
r
o
mKt
oK,
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s
tc
on
t
r
a
c
ti
t
si
n
d
i
c
e
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t
ht
h
et
r
a
n
s
f
or
ma
t
i
on
t
e
n
s
or
s
,
′
α ′
β
∂x ∂x
γ
δ
′
α
β
F= γ δF .
(
1
8
.
1
6
6
)
∂x ∂
x
Not
et
h
a
ts
i
n
c
eAi
sal
i
n
e
a
rt
r
a
n
s
f
or
ma
t
i
on
:
α
Aγ=
′
α
∂
x
(
1
8
.
1
6
7
)
γ
∂x
(
wh
e
r
eIh
a
v
ede
l
i
b
e
r
a
t
e
l
yi
n
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e
r
t
e
das
p
a
c
et
odi
ffe
r
e
n
t
i
a
t
et
h
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i
r
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ti
n
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e
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r
om t
h
e
s
e
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on
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a
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t
et
h
i
si
nt
e
r
msoft
h
ec
omp
on
e
n
t
sofAa
s
:
α
′
α
β =A F γδA β
F
γ
α
δ
γ
δ˜
β
=AγF A δ
(
1
8
.
1
6
8
)
or(
i
nac
ompr
e
s
s
e
dn
ot
a
t
i
on
)
:
′
˜
F=AF
A
(
1
8
.
1
6
9
)
Th
i
si
sj
u
s
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p
e
c
i
f
i
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s
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r
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r
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lt
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t
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n
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or
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t
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n
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n
s
orb
yc
on
t
r
a
c
t
i
n
gi
ta
p
p
r
op
r
i
a
t
e
l
ywi
t
he
a
c
hi
n
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e
x
.
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a
wi
nou
rd
i
s
c
u
s
s
i
onofTh
oma
sp
r
e
c
e
s
s
i
on
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l
lh
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c
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ort
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r
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r
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t
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h
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t
h
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os
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r
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et
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r
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t
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on
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e
t
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ss
e
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h
i
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s
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c
a
l
l
t
h
a
t
Af
orap
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nt
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r
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t
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h
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t
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a
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r
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t
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t
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n
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f
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a
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r
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n
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t
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h
ed
i
a
g
on
a
l
a
n
d−
γ
βont
h
ec
or
n
e
r
s
)
.
Th
u
s
:
s
o:
′
0
1
F
′
E
0 0
1 1
1
2E
1
2 2E
γ c
= −
−γβ c
1
−c
0 1
0 1
= A0F A1 +A1F A1
′
2 22
E1 =(
γ +γβ)
E1
′
E1 = E
1
(
1
8
.
1
7
0
)
Not
et
h
a
tweh
a
v
ee
x
t
r
a
c
t
e
dt
h
eor
d
i
n
a
r
yc
a
r
t
e
s
i
a
nc
omp
on
e
n
t
sofEa
n
dBf
r
omFa
f
t
e
r
t
r
a
n
s
f
or
mi
n
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t
.Il
e
a
v
et
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er
e
s
toft
h
e
mt
owor
kou
ty
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r
s
e
l
f
.Yous
h
ou
l
db
ea
b
l
et
o
s
h
owt
h
a
t
:
′
E
1
′
E
2
′
E
3
′
B1
′
B2
′
B3
1
=E
(
1
8
.
1
7
1
)
(
E
B3)
=γ
2−β
(
1
8
.
1
7
2
)
(
E
B2)
=γ
3+β
(
1
8
.
1
7
3
)
=B1
(
1
8
.
1
7
4
)
(
B2+βE
)
=γ
3
(
1
8
.
1
7
5
)
(
B3−βE2)
=γ
(
1
8
.
1
7
6
)
Th
ec
omp
on
e
n
toft
h
ef
i
e
l
d
si
nt
h
ed
i
r
e
c
t
i
onoft
h
eb
oos
ti
su
n
c
h
a
n
g
e
d,t
h
e
p
e
r
p
e
n
d
i
c
u
l
a
rc
omp
on
e
n
t
soft
h
ef
i
e
l
da
r
emi
x
e
d(
a
l
mos
ta
si
ft
h
e
ywe
r
es
p
a
c
e
–t
i
me
p
i
e
c
e
s
)b
yt
h
eb
oos
t
.I
fy
ouu
s
ei
n
s
t
e
a
dt
h
eg
e
n
e
r
a
lf
or
mofAf
orab
oos
ta
n
de
x
p
r
e
s
s
t
h
ec
omp
on
e
n
t
si
nt
e
r
msofdotp
r
od
u
c
t
s
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ous
h
ou
l
da
l
s
os
h
ow t
h
a
tt
h
eg
e
n
e
r
a
l
t
r
a
n
s
f
or
ma
t
i
oni
sg
i
v
e
nb
y
:
′
E
′
B
2
γ
=γ
(
E
+
β×
B)
−
γ+1β(
β·E
)
γ
(
1
8
.
1
7
7
)
2
=γ
(
B−
β×
E
)
−
γ+1β(
β·B)
.
(
1
8
.
1
7
8
)
Ap
u
r
e
l
ye
l
e
c
t
r
i
corma
g
n
e
t
i
cf
i
e
l
di
non
ef
r
a
mewi
l
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h
u
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t
u
r
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t
r
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r
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et
h
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r
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l
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h
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r
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t
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h
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omp
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ore
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e
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y
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c
a
u
s
et
h
ee
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a
t
i
on
sa
b
ov
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l
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a
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omemi
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t
u
r
ef
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n
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n
n
a
t
u
r
ea
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on
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t
r
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i
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e
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ou
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i
e
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t
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ont
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l
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ots
p
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a
l
u
a
b
l
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l
a
s
st
i
meont
h
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s
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h
owe
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e
r
.
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n
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t
e
a
dwewi
l
le
n
dt
h
i
s
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f
t
e
ra
l
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r
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t
h
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t
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c
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l
/
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ome
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c
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s
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u
c
et
h
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ov
a
r
i
a
n
td
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n
a
mi
c
s of r
e
l
a
t
i
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s
t
i
cp
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r
t
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c
l
e
si
n(
a
s
s
u
me
df
i
x
e
d
)
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l
e
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t
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oma
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n
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cf
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s
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a
p
t
e
r19
Re
l
a
t
i
v
i
s
t
i
cDy
n
a
mi
c
s
19.
1 Co
v
a
r
i
a
n
tF
i
e
l
dTh
e
or
y
Wea
r
ei
n
t
e
r
e
s
t
e
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nd
e
d
u
c
i
n
gt
h
ed
y
n
a
mi
c
sofp
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n
tc
h
a
r
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e
dp
a
r
t
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c
l
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si
n“
g
i
v
e
n
”(
i
.
e
.
—
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i
x
e
d
)e
l
e
c
t
r
oma
g
n
e
t
i
cf
i
e
l
ds
.
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l
r
e
a
d
y“
k
n
ow”t
h
ea
n
s
we
r
,
i
ti
sg
i
v
e
nb
yt
h
ec
ov
a
r
i
a
n
t
f
or
mofNe
wt
on
’
sl
a
w,
t
h
a
ti
s
:
α
dp
dτ
α
=m
dU
q
= FαβUβ.
dτ
c
(
1
9
.
1
)
F
r
omt
h
i
swec
a
nf
i
n
dt
h
e4
–a
c
c
e
l
e
r
a
t
i
on
,
α
dU
dτ
=
q
α
β
F Uβ
mc
(
1
9
.
2
)
wh
i
c
hwec
a
ni
n
t
e
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r
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t
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i
np
r
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of
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e4
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r
a
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e
c
t
or
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h
ep
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r
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e
s
t
i
on
.
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v
e
r
,
t
h
i
si
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otu
s
e
f
u
lt
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s
.Re
a
lp
h
y
s
i
c
i
s
t
sd
o
n
’
tu
s
eNe
wt
on
’
sl
a
wa
n
y
mor
e
.
Th
i
si
sn
ot
h
i
n
ga
g
a
i
n
s
tNe
wt
on
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ti
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h
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e
e
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l
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r
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n
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e
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s
f
or
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l
a
t
i
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n
a
mi
c
si
nor
de
rt
oc
on
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r
u
c
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mt
h
e
or
y(
ore
v
e
na
ne
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e
g
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n
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l
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s
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lt
h
e
or
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)
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rf
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r
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h
or
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e
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or
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l
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et
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st
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r
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n
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mi
l
t
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qu
a
t
i
on
sofmo
t
i
ont
of
ou
rd
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me
n
s
i
on
s
.
19.
1.
1 Th
eBr
u
t
eF
o
r
c
eWa
y
Re
c
a
l
lt
h
a
tt
h
eL
a
g
r
a
n
g
i
a
np
a
t
ht
ot
h
ed
y
n
a
mi
c
sofap
a
r
t
i
c
l
e(
wh
i
c
hi
smos
te
a
s
i
l
yma
de
c
ov
a
r
i
a
n
t
,
s
i
n
c
ei
tu
s
e
s(
q(
t
)
,
q˙
(
t
)
,
t
)a
si
t
sv
a
r
i
a
b
l
e
s
)i
sb
a
s
e
dont
h
e
Ac
t
i
on
t
1
A=
L
(
q(
t
)
,
q˙
(
t
)
,
t
)
d
t
.
t
0
(
1
9
.
3
)
2
7
7
Byr
e
qu
i
r
i
n
gt
h
a
tAb
ea
ne
x
t
r
e
mu
ma
saf
u
n
c
t
i
o
n
a
l
oft
h
es
y
s
t
e
mt
r
a
j
e
c
t
or
y
,
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t
a
i
n
t
h
eE
u
l
e
r
–L
a
gr
a
n
gee
q
u
a
t
i
o
n
s
d
∂L
∂L
dt ∂qi̇ − ∂qi
=0
.
(
1
9
.
4
)
Th
e
s
ea
r
ee
qu
i
v
a
l
e
n
tt
oNe
wt
on
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sl
a
wf
ors
u
i
t
a
b
l
ed
e
f
i
n
i
t
i
on
sofLa
n
dt
h
ef
or
c
e
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e
s
i
mp
l
e
s
twa
yt
oma
k
et
h
i
sr
e
l
a
t
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s
t
i
ci
st
oe
x
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n
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i
c
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l
l
yn
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e
s
s
a
r
y
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ec
omp
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er
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l
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t
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s
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cL
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r
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orac
h
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r
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e
dp
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r
t
i
c
l
ei
st
h
u
s
2
u
2
q
1− +u A qΦ.
2 c· −
c
L=−
mc
(
1
9
.
1
1
)
I
ts
h
ou
l
dt
a
k
ey
oua
b
ou
ton
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ou
rt
os
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a
tt
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sy
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e
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t
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v
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c
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or
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n
t
zf
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el
a
w.
Th
ef
r
e
ep
a
r
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i
c
l
ep
a
r
ti
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v
i
ou
s
,
t
h
ee
l
e
c
t
r
i
cf
i
e
l
di
sob
v
i
ou
s
.
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l
l
h
a
v
et
owor
kab
i
t
,
u
s
i
n
g
d
∂
d
t = ∂t+
u·∇
(
1
9
.
1
2
)
t
os
qu
e
e
z
e−
u×(
∇×A)ou
toft
h
er
e
ma
i
n
d
e
r
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u
g
g
e
s
tt
h
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ty
ous
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mp
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e
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omeof
y
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e
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ora
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e
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i
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l
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e
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a
s
s
i
g
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me
n
t
,
s
of
e
e
l
f
r
e
et
os
t
a
r
t
.
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ec
a
n
on
i
c
a
lmome
n
t
u
mPc
on
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u
g
a
t
et
ot
h
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os
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oor
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n
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t
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t
a
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e
d
(
a
su
s
u
a
l
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r
om
∂L
q
Pi= ∂u
mu
.
i =γ
i+ cAi
(
1
9
.
1
3
)
Th
i
sr
e
s
u
l
t
,
P=
p
q
+A
c
(
1
9
.
1
4
)
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wh
e
r
e
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st
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er
e
l
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t
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v
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s
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or
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me
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c
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c
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c
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l
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m”
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l
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on
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c
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l
mome
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t
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mv
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a
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1
9
.
1
5
)
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eb
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cr
e
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l
th
e
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te
l
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mi
n
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t
e
ui
nf
a
v
orofAa
n
dP.
Not
et
h
a
t
c
P−qA
u=
q
P− cA
2
(
1
9
.
1
6
)
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+m c
(
s
ome
t
h
i
n
gt
h
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ti
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h
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et
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t
r
a
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t
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or
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r
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t
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e
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et
e
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i
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g
e
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a
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ns
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h
a
tt
h
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mi
l
t
on
i
a
ni
s
:
2
24
H= (
c
P−qA)+m c +qΦ=W.
(
1
9
.
1
7
)
F
r
omt
h
i
sr
e
s
u
l
t
,
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mi
l
t
on
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t
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el
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et
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ti
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l
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t
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t
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on
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k
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l
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et
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n
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r
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t
u
a
l
l
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l
ec
h
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n
g
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nt
h
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ou
r
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e
c
t
ormome
n
t
u
m:
2
2
24
α
(
W−qΦ)−(
c
P−qA)=m c =pp
α
(
wh
i
c
hh
a
st
h
eu
s
u
a
l
f
or
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f
E
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.
1
8
)
1
q
W−e
Φ)
,
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= c(
c A
(
1
9
.
1
9
)
)
.
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oma
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st
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n
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h
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i
t
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c
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.
19.
1.
2 Th
eE
l
e
ga
n
tWa
y
Wec
anwr
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t
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e
epa
r
t
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ngonl
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al
a
rr
e
duc
t
i
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ui
t
a
bl
e
4
–v
e
c
t
o
r
s
:
mc
α
γ UαU
L
f
r
e
e=−
2−
1
(
1
9
.
2
0
)
.
Th
ea
c
t
i
o
ni
st
h
u
s
(
wh
i
c
hi
ss
t
i
l
l mc γ )
−
τ
1
A=−
mc
τ
0
α
UαU dτ
.
(
1
9
.
2
1
)
Th
ev
a
r
i
a
t
i
on
sont
h
i
sa
c
t
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s
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ec
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r
r
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ts
u
b
j
e
c
tt
ot
h
ec
on
s
t
r
a
i
n
t
α
2
(
1
9
.
2
2
)
UαU =c
wh
i
c
hs
e
v
e
r
e
l
yl
i
mi
t
st
h
ea
l
l
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ds
ol
u
t
i
on
s
.
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i
t
et
h
i
sa
s
α
d(
UαU )
d
τ
α
dUα Uα +UαdU
dτ
d
τ
α
dUα αβ
dU
α
g
U
+
U
α
g βα
dτ
dτ
β
dU
Uβ dτ
Now,
α
UαU d
τ=
α
dU
+Uαdτ
α
dU
2Uαdτ
α
dU
Uαdτ
d
τ
= 0
= 0
= 0
= 0
= 0
α
β
g d
x
d
x
α
β
α
d
x
x
αd
= 0
d
τ=
d
τ
(
1
9
.
2
3
)
(
1
9
.
2
4
)
wh
i
c
hi
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ni
n
f
i
n
i
t
e
s
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ma
l
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t
hi
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el
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t
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ote
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t
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r
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t
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a
t
i
n
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r
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e
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ot
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c
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l
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t
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u
ti
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h
e
r
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s
ea
r
b
i
t
r
a
r
y
.
Th
e
n
s
1
A=−
mc
g
dx
x
αd
β
αβ
s
0
d
sds d
s
.
(
1
9
.
2
5
)
Wea
r
ec
l
e
a
r
l
yma
k
i
n
gp
r
og
r
e
s
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v
et
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f
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c
t
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d
e
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n
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n
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a
b
l
e
.
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k
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i
t
t
l
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a
p
p
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e
r
,
n
ot
et
h
a
tt
h
i
sh
a
sn
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ott
h
ef
or
mof
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(
1
9
.
2
6
)
A= L
d
s
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wh
e
r
eLi
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l
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r“
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r
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i
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h
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ep
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v
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a
s
h
e
dt
h
ea
n
n
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i
n
g
−1
γ
.
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fwen
owd
ot
h
ec
a
l
c
u
l
u
sofv
a
r
i
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t
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on
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h
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e
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l
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a
t
i
on
s
i
nf
o
u
rd
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me
n
s
i
on
s
:
d
d
s
˜
∂
d
L
dx
α
ds
α
−∂
˜
L=0
(
1
9
.
27
)
(
f
orα=0
,
4
)
.
Ap
p
l
y
i
n
gt
h
e
mt
ot
h
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a
n
g
r
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ni
nt
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c
t
i
on
,
t
h
e
yt
u
r
nou
tt
ob
e
:
1
2
δβdx
βd
x
δ
d∂ g
mc
ds
∂
d
s d
s
dx
α
d
s
=0
(
1
9
.
2
8
)
α
α
dx
mcd
ds
ds
2d
s
+ dx
ds
ds
β
dx
βd
x
=0
(
1
9
.
2
9
)
=0
.
(
1
9
.
3
0
)
α
d
x
d
mcds
dsds
ds
β
dx
βd
x
Th
i
ss
t
i
l
ld
oe
sn
oth
a
v
et
h
ec
on
s
t
r
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i
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ta
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t
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et
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e
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on
s
t
r
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n
t
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f
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n
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t
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ns
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c
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h
a
tt
h
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on
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t
r
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i
mu
l
t
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n
e
ou
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l
y
s
a
t
i
s
f
i
e
d
:
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dx
x
αd
d
sds
d
d
τ
α
d
s= c
dτ
=
2
c
d
α
d
x
xd
s
αd
d
s ds
(
1
9
.
3
1
)
(
wh
i
c
hr
e
qu
i
r
e
sb
ot
hd
s=dτa
n
dUαU =c)
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l
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e
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l
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s
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1
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3
2
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c
m
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x
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s
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1
9
.
3
3
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x
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1
9
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3
4
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.
3
5
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1
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3
6
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x
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d
s
q
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1
9
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3
7
)
(
1
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3
8
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1
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ome
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2
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q
α
α
x
(
1
9
.
4
3
)
α
P=
− ∂ ds =mU + cA
α
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t
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1
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4
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1
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4
5
)
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s
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6
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d
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1
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.
4
7
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n
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h
a
t
:
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α
α
q α
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p =mU =P −
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(
1
9
.
4
9
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(
1
9
.
5
0
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1
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h
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1
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n
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e
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emor
emon
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u
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h
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a
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ti
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t
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nt
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omeofi
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n
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r
er
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b
l
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orwor
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ou
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c
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n
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n
c
l
i
n
a
t
i
on
s
,
a
n
da
b
i
l
i
t
i
e
s
,
ony
ou
rown
.
19.
4 Th
eSy
mme
t
r
i
cSt
r
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s
sTe
n
s
o
r
I
ma
g
i
n
eab
i
gb
l
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l
l
y
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ma
g
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d
e
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ewh
ol
et
h
i
n
gwi
g
g
l
e
sa
n
d
di
s
t
or
t
s
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st
h
ef
or
c
eofy
ou
rp
ok
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c
t
sont
h
ee
n
t
i
r
eb
l
obofj
e
l
l
y
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ema
t
h
e
ma
t
i
c
a
l
me
c
h
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n
i
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mt
h
a
td
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r
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sh
owy
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t
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u
t
e
di
sc
a
l
l
et
h
es
t
r
e
s
st
e
n
s
o
roft
h
e
ma
t
e
r
i
a
l
.
I
tt
e
l
l
sh
owe
n
e
r
g
ya
n
dmome
n
t
u
ma
r
ec
on
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e
c
t
e
db
yt
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eme
di
u
mi
t
s
e
l
f
.
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es
a
mec
on
c
e
p
tc
a
nb
eg
e
n
e
r
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l
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z
e
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me
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on
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lme
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e
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et
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e“
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e
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ss
p
a
c
et
i
mei
t
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l
f
.
L
e
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sn
ows
t
u
dywh
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ta
ne
l
e
c
t
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oma
g
n
e
t
i
cs
t
r
e
s
s
t
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n
s
ori
s
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a
n
dh
owi
tr
e
l
a
t
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st
oe
l
e
c
t
r
o
ma
g
n
e
t
i
c“
p
ok
e
s
”
.
Re
c
a
l
l
t
h
a
t
∂L
p
i=
(
1
9
.
7
7
)
∂qi̇
i
st
h
ec
a
n
on
i
c
a
l
mome
n
t
u
mc
or
r
e
s
p
o
n
d
i
n
gt
ot
h
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a
r
i
a
b
l
eqi
L
a
g
r
a
n
g
i
a
n
.
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eHa
mi
l
t
on
i
a
ni
sg
i
v
e
n
,
i
nt
h
i
sc
a
s
e
,
b
y
H=pi
qi̇−L
i
na
na
r
b
i
t
r
a
r
y
(
1
9
.
7
8
)
i
a
su
s
u
a
l
.I
f∂L
/
∂t=0t
h
e
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ec
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ns
h
owt
h
a
t∂H/
∂
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orf
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rd
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on
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l
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r
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e
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r
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mi
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n
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mi
l
t
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e
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s
t
e
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s
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ts
h
ou
l
dt
r
a
n
s
f
or
ml
i
k
et
h
ez
e
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ot
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omp
on
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n
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ou
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e
c
t
or
.
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u
s
,
s
i
n
c
e
3
H= Hdx
4
(
1
9
.
7
9
)
3
a
n
ddx=d
x
dx
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h
e
nHmu
s
tt
r
a
n
s
f
or
ml
i
k
et
h
et
i
mec
omp
on
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n
tofas
e
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on
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a
n
k
0
t
e
n
s
or
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e
f
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et
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mi
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on
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t
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nt
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r
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h
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a
g
r
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n
g
i
a
nd
e
n
s
i
t
yLofa
f
i
e
l
d
,
t
h
e
n
H
= ∂
k
∂L
∂φk
∂φk
∂t
∂t −L
.
(
1
9
.
8
0
)
We
l
l
,
g
r
e
a
t
!
Th
ef
i
r
s
tf
a
c
t
ori
nt
h
es
u
mi
st
h
ec
on
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u
g
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t
emome
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t
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t
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n
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on
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st
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e
n
e
r
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n
c
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s
tt
r
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n
s
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o
r
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ome
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h
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n
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t
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r
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v
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t
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v
et
h
e
r
e
,
i
n
s
t
e
a
d
.
Wet
r
y
β
Tαβ=
α
β
.
∂L ∂ φk−g L
k
(
1
9
.
8
1
)
∂(
∂αφk)
Th
i
si
sc
a
l
l
e
dt
h
ec
a
n
on
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c
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ls
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r
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st
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h
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t
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d
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n
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st
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t
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r
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yt
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e
c
t
r
oma
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n
e
t
i
cf
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l
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s
f
or
m.
Wh
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ti
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h
i
st
e
n
s
or
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ti
s
,
i
nf
a
c
t
,
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g
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yn
on
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r
i
v
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l
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eb
e
s
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nd
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ot
et
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t
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t
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ra
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l
)
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h
e
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n
s
h
owt
h
a
t
0
03
T dx=
1
2
2
(
ǫ0E +
12 3
B)
dx=E
f
i
e
l
d
µ0
(
1
9
.
8
2
)
a
n
d
1
3
i
E×H)
dx=c
Pf
c (
i
i
e
l
d
0
i3
3
T dx=ǫ0c(
E×B)
dx=
i
(
1
9
.
8
3
)
wh
i
c
ha
r
et
h
e“
u
s
u
a
l
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x
p
r
e
s
s
i
on
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ort
h
ee
n
e
r
g
ya
n
dmome
n
t
u
moft
h
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r
e
ef
i
e
l
d
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l
e
a
s
ti
fI
g
ott
h
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h
a
n
g
et
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u
n
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t
sr
i
g
h
t
.
.
.
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t
,
y
oumi
g
h
ta
s
k
,
i
st
h
i
sg
oodf
or
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l
l
,
a
s
i
d
ef
r
omt
h
i
sc
or
r
e
s
p
on
d
a
n
c
e(
wh
i
c
h
i
sf
u
l
l
ofh
ol
e
s
,
b
yt
h
ewa
y
)
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wec
a
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t
et
h
ee
n
e
r
g
y
–mome
n
t
u
mc
on
s
e
r
v
a
t
i
onl
a
w
α
β
∂αT =0
.
(
1
9
.
8
4
)
Th
i
si
sp
r
ov
e
ni
nJ
a
c
k
s
on
,
wi
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s
c
u
s
s
i
onofs
omeofi
t
ss
h
or
t
c
omi
n
g
s
.
On
eoft
h
e
s
ei
st
h
a
ti
ti
sn
ots
y
mme
t
r
i
c
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i
sc
r
e
a
t
e
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ffic
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e
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on
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i
d
e
r
t
h
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n
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u
l
a
rmome
n
t
u
mc
a
r
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e
db
yt
h
ef
i
e
l
d.Si
n
c
et
h
ea
n
g
u
l
a
rmome
n
t
u
md
e
n
s
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t
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s
i
mp
or
t
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n
twh
e
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r
e
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ep
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ot
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c
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th
a
v
equ
a
n
t
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z
e
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g
u
l
a
r
mome
n
t
a
)
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i
ti
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t
h
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i
l
et
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on
s
t
r
u
c
tt
h
es
y
mme
t
r
i
cs
t
r
e
s
st
e
n
s
or
1 αβ
α
β α
µ
λ
β
µλ
Θ =g F
F +
g F
F
µλ
µλ
(
1
9
.
8
5
)
4
i
nt
e
r
msofwh
i
c
hwec
a
nc
or
r
e
c
t
l
yc
on
s
t
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r
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n
tg
e
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r
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l
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t
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e
r
g
y
mome
n
t
u
mc
on
s
e
r
v
a
t
i
onl
a
w
α
β
∂αΘ =0
(
1
9
.
8
6
)
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n
dt
h
ea
n
g
u
l
a
rmome
n
t
u
mt
e
n
s
or
γ−
β
Mαβγ=Θαβx
Θαγx
(
1
9
.
8
7
)
wh
i
c
hi
st
h
e
r
e
f
or
ec
on
s
e
r
v
e
d
.Th
i
sf
or
m oft
h
es
t
r
e
s
st
e
n
s
orc
a
na
l
s
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ed
i
r
e
c
t
l
y
c
ou
p
l
e
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os
ou
r
c
et
e
r
ms
,
r
e
s
u
l
t
i
n
gi
nt
h
ec
ov
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r
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n
tf
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h
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ke
n
e
r
g
yt
h
e
or
e
mf
or
t
h
ec
omb
i
n
e
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y
s
t
e
mo
fp
a
r
t
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c
l
e
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n
df
i
e
l
d
s
.
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i
si
sa
b
ou
ta
l
l
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l
s
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ou
tt
h
i
sa
tt
h
i
st
i
me
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r
e
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l
i
z
et
h
a
ti
ti
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n
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t
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s
f
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n
da
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e
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fweh
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don
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es
e
me
s
t
e
rt
og
e
t
h
e
r
,
wec
ou
l
dd
oi
tp
r
op
e
r
l
y
,
b
u
twe
d
on
’
t
.
Th
e
r
e
f
or
e
,
i
ti
sont
o
19.
5 Co
v
a
r
i
a
n
tGr
e
e
n
’
sF
u
n
c
t
i
o
n
s
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u
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st
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1
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ore
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i
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e
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z z −κ
=
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e
00
0
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k
0 −κ
0k
−
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2
2
+Cdz z −κ
(
1
9
.
9
9
)
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e
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,
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h
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n
t
e
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ory
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Th
u
s
:
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k y
e
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e
2 2
k
0 k
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0 −κ
= d
z z −κ
0
0
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2
=
2
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z
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0
e
l
i
m(
−
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πi
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s
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κ−i
ǫ)
)
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z+(
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e
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i
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e
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κ
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κ + −
s
i
n
(
κ
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0
κ
2
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(
1
9
.
1
0
0
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nt
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n
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t
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n
(
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D(
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π)
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e
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0 R
′
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x
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D(
x x)= θ
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r −
(
1
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0
3
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(
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0
4
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om F
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θ
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5
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A(
x
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(
1
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.
1
0
6
)
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n
d
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′α ′
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(
1
9
.
1
0
7
)
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r
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0
8
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e
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(
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(
1
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t
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on
:
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✷A =µ
J
0
a
r
e
0
′
′
0
(
2
0
.
1
)
0
(
x −x)δ(
x x
′
0
D(
x x)= θ
R)
− −
4πR
r −
(
2
0
.
2
)
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n
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on
swe
r
e
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α
A(
x
)=Ai
x
)+µ
n(
0
4′
′α ′
4′
′α ′
dxDr
(
x−x)
J(
x)
(
2
0
.
4
)
a
n
d
α
α
A(
x
)=Aout(
x
)+µ
0
dxDa(
x−x)
J(
x)
.
(
2
0
.
5
)
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ort
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t
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h
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t
y
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(
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)
]
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p
U·(
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c
R(
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=γ
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e
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t
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o
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n
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e
r
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(
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/
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s
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a
l
.
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a
l
l
t
h
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tA=(
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c
,
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u
s
:
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A(
x
)=
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γ
c
(
2
0
.
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0
)
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R(
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t
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n
d
e
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t
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A=
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t
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1
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t
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er
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t
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l
a
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l
y
Ax
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t
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=
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e
t
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π γ
µ
e
β
0
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2
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π ǫ0 R(
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e
t
wh
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r
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g
a
i
nt
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n
g
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s
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l
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t
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e
t
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r
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me
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r
t
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c
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l
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tb
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h
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e
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a
t
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s
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cf
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nt
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e
l
i
mi
t|
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ng
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tt
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s
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t
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c
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e
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v
a
t
i
v
e
s
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l
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t
u
r
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t
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h
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n
t
e
g
r
a
l
f
or
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e
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a
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e
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n
c
t
i
on
s
.
eµ0c
∂A = 2π
αβ
β
α
2
dτU (
τ)
θ[
x
τ)
]
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0−r
0(
wh
e
r
e
α
d
α
∂δ[
f]
=∂f·
[
x−r
(
τ)
]
d
τ
α
(
20.
23)
d
d
fδ[
f·
d
τδ[
f] =∂f· d
f]
.
(
20.
24)
2
Ag
a
i
n
,
wel
e
tf=[
x−r
(
τ)
].
Th
e
n
α
α
(
x−r
)
∂δ[
f]
=
−U
(
x
d δ[
f]
(
20.
25)
r
)d
τ
· −
Th
i
si
si
n
s
e
r
t
e
di
n
t
ot
h
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x
p
r
e
s
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i
ona
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n
di
n
t
e
g
r
a
t
e
db
yp
a
r
t
s
:
e
µ0c
∂αAβ =−
2
π
β
dτU (
τ)
θ[
x
α
x−r
)
r(
τ)
](
0− 0
U
(
x
dδ[
f]
r
)d
τ
α · −
)
d β
(
x−r
2
U
(
τ
)
dτ
θ[
x
r (
τ)
]
δ(
[
x
r
(
τ
(
20
)
]
)
.
.
2
6)
−
−
0
d
τ
U (
x r
)
0
2
π
· −
Th
e
r
ei
sn
oc
on
t
r
i
b
u
t
i
onf
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omt
h
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n
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t
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onb
e
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ed
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r
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t
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t
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u
n
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t
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u
n
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t
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t
ht
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es
a
mea
r
g
u
me
n
t
s
= eµ0c
d
(
x
(
τ)
)=δ[
x
(
τ)
]
d
τθ
0 −r
0
0 −r
0
(
2
0
.
2
7
)
2
wh
i
c
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on
s
t
r
a
i
n
st
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rd
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l
t
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u
n
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ont
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R)
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i
son
l
yg
e
t
sac
on
t
r
i
b
u
t
i
ona
t
R=0(
ont
h
ewor
l
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or
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e
d
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c
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l
s
k
i
p
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03
(
n
ˆ−β)
n
ˆ×(
n
ˆ−β)×β
+e
˙
e
µ0
E
x
(,
t
)=
r
e
t
2
4
πc
2
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πc
γ(
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3
r
e
t
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0
.
2
9
)
a
n
d
1
Bx
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t
)= c (
n
ˆ×E
)
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0
.
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0
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r
r
g
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e
l
d
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a
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t
h
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E =
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×
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e
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e
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e
t
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or
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S= µ0(
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e
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11
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3
|
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2
2
1
e
2
23
= 16πǫ0R c |
nˆ×(
nˆ×cβ)
| nˆ
2
e
1
˙2
2 23
nˆ×(
nˆ×v)
|nˆ
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(
2
0
.
3
4
)
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l
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y
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h
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ne
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e
2 3 2 2
||s
i
n(
Θ)
= 16πǫ0c v
2
3
˙
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0
.
3
5
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n
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i
n
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h
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l
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ra
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,
s
ot
h
a
t
2
e
|
v
˙
|
2.
3
πǫ0c
P= 6
(
2
0
.
3
6
)
Th
i
si
st
h
eL
a
r
morf
or
mu
l
af
ort
h
ep
owe
rr
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di
a
t
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r
oman
on
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e
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r
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e
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i
r
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a
c
t
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ta
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n
dc
on
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e
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i
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oor
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n
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e
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e
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t
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or
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a
s
t
,
wec
on
v
e
r
tti
n
t
o
τ:
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P=
e
1
3 2
6πǫ0c m
2
=
e
23
6πǫ0m c
2
d
(
mv)
dt
d(
mv)
γ
d
τ
2
e
23
2
pd 2
2
1−β) dτ
= 6πǫ0m c (
2
2
2
e
pd
1d
E
23
= 6πǫ0m c
d
τ
2
τ
− cd
2
e
23
=−6πǫ0m c
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dpαdp
d
τ d
τ
(
2
0
.
3
7
)
2
Th
i
sc
a
nb
ewr
i
t
t
e
non
emor
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y
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s
u
b
s
t
i
t
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t
i
n
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t
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n
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t
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u
et
oL
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n
a
r
d
:
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e
˙2
6
3
˙
2
(
β×
β)
P= 6πǫ0c γ [
(
β) −
]
(
2
0
.
3
8
)
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r
ea
l
l
b
e
t
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t
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el
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or
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.
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a
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et
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s“
r
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i
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i
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e
a
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e
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h
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th
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h
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l
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r
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nt
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t
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on
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h
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e
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n
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ee
a
s
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s
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r
yt
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a
l
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n
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u
me
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c
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i
a
t
i
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e
a
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et
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c
c
ou
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tf
ort
h
e“
mi
s
s
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n
ge
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g
y
”
.
Th
i
swa
st
h
ea
p
p
r
oa
c
ht
a
k
e
nb
yAb
r
a
h
a
ma
n
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or
e
n
t
zma
n
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sa
g
o.
21.
2 Ra
d
i
a
t
i
o
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c
t
i
ona
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dE
n
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r
gyCon
s
e
r
v
a
t
i
o
n
Wek
n
owt
h
a
t
˙
Ft
ot=mv
(
2
1
.
7
)
i
s(
n
on
r
e
l
a
t
i
v
i
s
t
i
c
)Ne
wt
on
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s2
n
dL
a
wf
orac
h
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r
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e
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a
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i
c
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i
n
ga
c
c
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l
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r
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t
e
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ya(
f
or
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h
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n
t
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t
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v
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e
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tt
h
es
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c
t
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ee
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e
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lf
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e(
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n
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s
a
c
c
e
l
e
r
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t
i
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ti
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l
s
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d
i
a
t
i
n
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o
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ra
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h
et
ot
a
l
r
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e
:
2
2
P(
t
)= 2 e v ˙
2
3
c4
πǫ0c
=
2mr
e
v
3 c
2̇
2̇
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r
= mτ
(
2
1
.
8
)
(
t
h
eL
a
r
mo
rf
or
mu
l
a
)
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e
s
ea
r
et
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rt
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t
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l
y
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n
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g
l
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t
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n
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et
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nt
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h
e
r
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v
e
r
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nor
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e
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o
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or
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t
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e
a
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h
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on
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t
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n
e
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y
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t
h
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yt
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ee
x
t
e
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lf
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emu
s
te
qu
a
lt
h
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n
c
r
e
a
s
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nk
i
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r
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l
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e
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t
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n
t
ot
h
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i
e
l
d
.
E
n
e
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g
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on
s
e
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v
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t
i
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ort
h
i
ss
y
s
t
e
ms
t
a
t
e
st
h
a
t
:
Wext=E
e+E
f
(
2
1
.
9
)
ort
h
et
ot
a
lwor
kd
on
eb
yt
h
ee
x
t
e
r
n
a
lf
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emu
s
te
qu
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lt
h
ec
h
a
n
g
ei
nt
h
et
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a
le
n
e
r
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y
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h
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h
a
r
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e
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a
r
t
i
c
l
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e
l
e
c
t
r
on
)p
l
u
st
h
ee
n
e
r
g
yt
h
a
ta
p
p
e
a
r
si
nt
h
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i
e
l
d
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fwe
r
e
a
r
r
a
n
g
et
h
i
st
o:
Wext−E
f=E
e
(
2
1
.
1
0
)
a
n
dc
on
s
i
d
e
rt
h
ee
l
e
c
t
r
o
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l
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e
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on
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l
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det
h
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h
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r
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s
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n
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h
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on
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yt
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or
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ed
e
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er
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h
er
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t
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o
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o
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o
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e
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d
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u
s(
r
e
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t
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n
gNe
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e
c
on
dl
a
wi
nt
e
r
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i
sf
or
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e
)
:
F F
e
x
t+ r
a
d
F
r
a
d
˙
=mv
˙
=mv −Fext
(
2
1
.
1
1
)
d
e
f
i
n
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st
h
er
a
d
i
a
t
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rf
oru
st
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e
ti
n
t
o“
t
r
ou
b
l
e
”.
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l
dl
i
k
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n
e
r
g
yt
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ec
on
s
e
r
v
e
d(
a
si
n
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c
a
t
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da
b
ov
e
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h
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er
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yt
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er
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ne
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t
e
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a
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h
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h
a
r
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c
c
e
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e
r
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t
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l
dl
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et
h
i
sf
or
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et
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a
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nt
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e
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lf
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ev
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n
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s
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h
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t
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t
h
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ne
x
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e
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n
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l
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g
e
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ta
c
t
i
n
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e
m.
2
Tr
ou
b
l
es
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c
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r
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p
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b
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i
f
t
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n
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ms
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l
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pb
yt
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e
i
rownme
t
a
p
h
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c
a
l
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oot
s
t
r
a
p
s
.
.
.
2
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l
dl
i
k
et
h
er
a
d
i
a
t
e
dp
o
we
rt
ob
ep
r
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on
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et
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n
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et
h
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or
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i
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h
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e
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toft
h
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g
noft
h
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h
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r
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e
.
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i
n
a
l
l
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n
tt
h
ef
or
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et
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n
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ol
v
et
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c
h
a
r
a
c
t
e
r
i
s
t
i
ct
i
me
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wh
e
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e
e
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t
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me
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b
l
e
.
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e
t
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ss
t
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r
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t
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r
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e
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n
tt
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yr
a
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t
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db
ys
ome“
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ou
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h
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r
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on
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n
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e
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e
r
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er
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e
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h
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r
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v
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se
qu
a
t
i
on
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e
t
’
ss
t
a
r
tb
ye
x
a
mi
n
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u
s
tt
h
e
r
e
a
c
t
i
onf
or
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ea
n
dt
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er
a
d
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a
t
e
dp
owe
r
,
t
h
e
n
,
a
n
ds
e
tt
h
et
ot
a
lwor
kdon
eb
yt
h
eon
et
o
e
qu
a
l
t
h
et
ot
a
l
e
n
e
r
g
yr
a
d
i
a
t
e
di
nt
h
eot
h
e
r
,
ov
e
ras
u
i
t
a
b
l
et
i
mei
n
t
e
r
v
a
l
:
t
2
t
1
t
2
t
2
Fr
·d
adv
t=−
t
1
Pd
t=−
t
1
˙ ˙
mτvr ·
vdt
(
2
1
.
1
2
)
f
ort
h
er
e
l
a
t
i
onb
e
t
we
e
nt
h
er
a
t
e
s
,wh
e
r
et
h
emi
n
u
ss
i
g
ni
n
d
i
c
a
t
e
st
h
a
tt
h
ee
n
e
r
g
yi
s
r
e
mov
e
df
r
omt
h
es
y
s
t
e
m.
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a
ni
n
t
e
g
r
a
t
et
h
er
i
g
h
th
a
n
ds
i
d
eb
yp
a
r
t
st
oob
t
a
i
n
t
2
t
2
¨
˙
t
2
mτv
·dt−mτr
v
(v
·)|
1
rv
t
Fr
·d
t= t1
a
dv
(
2
1
.
1
3
)
F
i
n
a
l
l
y
,
t
h
emot
i
oni
s“
p
e
r
i
od
i
c
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n
dweon
l
ywa
n
tt
h
er
e
s
u
l
tov
e
rap
e
r
i
od
;
we
t
1
˙
c
a
nt
he
r
e
f
or
epi
c
kt
hee
ndpoi
nt
ss
uc
ht
ha
t
vv
·
t
2
t
1
=0
.
Th
u
sweg
e
t
¨
Fr
v
· d
a
d−mτ
r v
t=0
.
(
2
1
.
1
4
)
On
e(
s
u
ffic
i
e
n
tb
u
tn
otn
e
c
e
s
s
a
r
y
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yt
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n
s
u
r
et
h
a
tt
h
i
se
qu
a
t
i
onb
es
a
t
i
s
f
i
e
di
s
t
ol
e
t
¨
Fr
v
a
d=mτ
r
Th
i
st
u
r
n
sNe
wt
on
’
sl
a
w(
c
or
r
e
c
t
e
df
orr
a
d
i
a
t
i
onr
e
a
c
t
i
on
)i
n
t
o
F
(
2
1
.
1
5
)
˙
e
x
t= mv −Fr
a
d
˙ ¨
( −τ
v
r)
=mv
(
2
1
.
1
6
)
Th
i
si
sc
a
l
l
e
dt
h
eAb
r
a
h
a
m–L
o
r
e
n
t
ze
qu
a
t
i
onofmo
t
i
ona
n
dt
h
er
a
di
a
t
i
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e
a
c
t
i
on
f
or
c
ei
sc
a
l
l
e
dt
h
eAb
r
a
h
a
m–L
or
e
n
t
zf
o
r
c
e
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tc
a
nb
ema
d
er
e
l
a
t
i
v
i
s
t
i
cb
ec
on
v
e
r
t
i
n
gt
o
p
r
op
e
rt
i
mea
su
s
u
a
l
.
Not
et
h
a
tt
h
i
si
sn
otn
e
c
e
s
s
a
r
i
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t
i
s
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yt
h
ei
n
t
e
g
r
a
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on
s
t
r
a
i
n
ta
b
ov
e
.
An
ot
h
e
rwa
yt
os
a
t
i
s
f
yi
ti
st
or
e
qu
i
r
et
h
a
tt
h
ed
i
ffe
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e
n
c
eb
eor
t
h
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on
a
l
t
ov.E
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e
nt
h
i
si
st
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s
p
e
c
i
f
i
c
,
t
h
ou
g
h
.Th
eon
l
yt
h
i
n
gt
h
a
ti
sr
e
q
u
i
r
e
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st
h
a
tt
h
et
ot
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li
n
t
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g
r
a
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ez
e
r
o,
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n
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ofd
e
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omp
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i
n
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i
t
yt
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a
j
e
c
t
or
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nor
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h
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on
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ls
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t
e
ma
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e
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h
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n
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e
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l
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l
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sofv
a
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on
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ti
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otp
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i
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oma
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es
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a
t
e
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t
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b
ou
tt
h
en
e
c
e
s
s
a
r
y
f
or
mofFr
.
a
d
Th
i
s“
s
u
ffic
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e
n
t
”s
ol
u
t
i
oni
sn
otwi
t
h
ou
tp
r
ob
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e
msofi
t
sown
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l
e
mst
h
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ts
e
e
m
u
n
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i
k
e
l
yt
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oa
wa
yi
fwec
h
oos
es
omeot
h
e
r“
s
u
ffic
i
e
n
t
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r
i
t
e
r
i
on
.Th
i
si
sa
p
p
a
r
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n
t
f
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omt
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e
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v
a
t
i
ont
h
a
tt
h
e
ya
l
ll
e
a
dt
oa
ne
qu
a
t
i
onofmot
i
ont
h
a
ti
st
h
i
r
do
r
d
e
ri
n
t
i
me
.
Now,
i
tma
yn
ots
e
e
mt
oy
ou(
y
e
t
)t
h
a
tt
h
a
ti
sad
i
s
a
s
t
e
r
,
b
u
ti
ti
s
.
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p
p
os
et
h
a
tt
h
ee
x
t
e
r
n
a
l
f
or
c
ei
sz
e
r
oa
ts
omei
n
s
t
a
n
toft
i
met=0
.
Th
e
n
˙ ¨
v ≈τv
or
˙
(
2
1
.
1
7
)
t
/
τ
v(
t
)=
a0e
(
2
1
.
1
8
)
wh
e
r
e
a0i
st
h
ei
n
s
t
a
n
t
a
n
e
ou
sa
c
c
e
l
e
r
a
t
i
onoft
h
ep
a
r
t
i
c
l
ea
tt=0
.
Re
c
a
l
l
i
n
gt
h
a
t
v
˙
·
v =0a
tt
n
dt
,
wes
e
et
h
a
tt
h
i
sc
a
non
l
yb
et
r
u
ei
f
1a
2
a0=0(
orwec
a
nr
e
l
a
xt
h
i
sc
on
d
i
t
i
ona
n
dp
i
c
ku
pa
na
d
d
i
t
i
on
a
lb
ou
n
d
a
r
yc
on
d
i
t
i
ona
n
d
wor
kmu
c
hh
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r
d
e
rt
oa
r
r
i
v
ea
tt
h
es
a
mec
on
c
l
u
s
i
on
)
.Di
r
a
ch
a
das
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mp
l
yl
ov
e
l
yt
i
me
wi
t
ht
h
et
h
i
r
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d
e
re
qu
a
t
i
on
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f
or
ea
t
t
a
c
k
i
n
gi
t
,
t
h
ou
g
h
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e
tu
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t
a
i
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u
t
i
ont
h
a
t
d
oe
s
n
’
th
a
v
et
h
ep
r
ob
l
e
msa
s
s
oc
i
a
t
e
dwi
t
hi
ti
nad
i
ffe
r
e
n
t(
mor
eu
p
f
r
on
t
)wa
y
.
L
e
tu
sn
ot
et
h
a
tt
h
er
a
d
i
a
t
i
onr
e
a
c
t
i
onf
or
c
ei
na
l
mos
ta
l
lc
a
s
e
swi
l
lb
ev
e
r
ys
ma
l
l
c
omp
a
r
e
dt
ot
h
ee
x
t
e
r
n
a
l
f
or
c
e
.
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ee
x
t
e
r
n
a
l
f
or
c
e
,
i
na
dd
i
t
i
on
,
wi
l
l
g
e
n
e
r
a
l
l
yb
e“
s
l
owl
y
−2
4
v
a
r
y
i
n
g
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,
a
tl
e
a
s
tonat
i
me
s
c
a
l
ec
omp
a
r
e
dt
oτ
0 s
e
c
on
d
s
.I
fwea
s
s
u
met
h
a
tF
r≈1
(
t
)i
ss
moo
t
h(
c
on
t
i
n
u
ou
s
l
yd
i
ffe
r
e
n
t
i
a
b
l
ei
nt
i
me
)
,
s
l
owl
yv
a
r
y
i
n
g
,
a
n
ds
ma
l
le
n
ou
g
h
e
x
t
t
h
a
tFr
a
nu
s
ewh
a
ta
mou
n
t
st
op
e
r
t
u
r
b
a
t
i
ont
h
e
or
yt
ode
t
e
r
mi
n
eFr
a
d≪ Fe
x
twec
a
d
a
n
dob
t
a
i
nas
e
c
on
dor
d
e
re
qu
a
t
i
onofmot
i
on
.
˙
Un
d
e
rt
h
e
s
ec
i
r
c
u
ms
t
a
n
c
e
s
,
wec
a
na
s
s
u
met
h
a
tFext≈mv,
s
ot
h
a
t
:
˙ ¨
F
e
x
t
( −τ
v
r)
= mv
˙
F
d ext
t
≈ mv −τr d
or
(
2
1
.
1
9
)
F
˙F
mv =
ex
t
F
+τ
r
d
e
x
t
dt
∂
(
2
1
.
2
0
)
=
+τ
t+v( ·∇)Fext
r ∂
Th
i
sl
a
t
t
e
re
qu
a
t
i
onh
a
sn
or
u
n
a
wa
ys
ol
u
t
i
on
sora
c
a
u
s
a
l
b
e
h
a
v
i
ora
sl
on
ga
s
ex
t
Fexti
sd
i
ffe
r
e
n
t
i
a
b
l
ei
ns
p
a
c
ea
n
dt
i
me
.
Wewi
l
ld
e
f
e
rt
h
ed
i
s
c
u
s
s
i
onoft
h
ec
ov
a
r
i
a
n
t
,s
t
r
u
c
t
u
r
ef
r
e
eg
e
n
e
r
a
l
i
z
a
t
i
onoft
h
e
Ab
r
a
h
a
m–L
or
e
n
t
zd
e
r
i
v
a
t
i
onu
n
t
i
ll
a
t
e
r
.Th
i
si
sb
e
c
a
u
s
ei
ti
n
v
ol
v
e
st
h
eu
s
eoft
h
ef
i
e
l
d
s
t
r
e
s
st
e
n
s
or
,
a
sd
oe
sDi
r
a
c
’
sor
i
g
i
n
a
l
p
a
p
e
r—wewi
l
l
di
s
c
u
s
st
h
e
ma
tt
h
es
a
met
i
me
.
Wh
a
ta
r
et
h
e
s
er
u
n
a
wa
ys
ol
u
t
i
on
so
ft
h
ef
i
r
s
t(
Ab
r
a
h
a
mL
or
e
n
t
z
)e
qu
a
t
i
onof
mot
i
on
?Cou
l
dt
h
e
yr
e
t
u
r
nt
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l
a
g
u
eu
swh
e
nt
h
ef
or
c
ei
sn
ots
ma
l
la
n
dt
u
r
n
son
q
u
i
c
k
l
y
?L
e
t
’
ss
e
e
.
.
.
21.
3 I
n
t
e
gr
o
d
i
ffe
r
e
n
t
i
a
l
E
q
u
a
t
i
o
n
sofMo
t
i
on
Wes
e
e
ks
ol
u
t
i
on
st
ot
h
et
h
i
r
dor
d
e
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qu
a
t
i
onofmot
i
o
nt
h
a
te
v
ol
v
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n
t
ot
h
e“
n
a
t
u
r
a
l
”
on
e
swh
e
nt
h
ed
r
i
v
i
n
gf
or
c
ei
st
u
r
n
e
doff.I
not
h
e
rwor
d
s
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a
di
a
t
i
onr
e
a
c
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h
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s
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l
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y
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e
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t
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e
a
r
l
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v
e
nt
h
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sr
e
qu
i
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e
me
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k
e
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os
e
n
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mer
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e
r
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ls
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t
r
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on
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e
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e
d
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l
li
n
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h
et
r
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c
h
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n
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e
t
a
r
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e
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n
t
e
r
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c
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onon
l
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r
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n
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n
da
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t
h
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n
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i
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l
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ea
b
a
n
d–a
i
d
.
L
e
tu
si
n
t
r
od
u
c
ea
n“
i
n
t
e
g
r
a
t
i
n
gf
a
c
t
or
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n
t
ot
h
ee
qu
a
t
i
on
sofmot
i
on
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fwea
s
s
u
me
(
qu
i
t
eg
e
n
e
r
a
l
l
y
)t
h
a
t
˙
t
/
τ
r
v(
t
)=e
t
)
u(
(
2
1
.
2
1
)
wh
e
r
e
u(
t
)i
st
ob
ed
e
t
e
r
mi
n
e
d
,
t
h
e
nt
h
ee
qu
a
t
i
on
sofmot
i
ons
i
mp
l
i
f
yt
o
˙ 1 −t/τ
mu=− τ
t
)
.
re F(
(
2
1
.
2
2
)
Wec
a
nf
or
ma
l
l
yi
n
t
e
g
r
a
t
et
h
i
ss
e
c
on
de
qu
a
t
i
on
,
ob
t
a
i
n
i
n
g
˙
mv(
t
)=
C
t
/
τ
e
r
τ
r
′
−t/
τ
rF
′
′
e
(
t
)
d
t
t
(
2
1
.
2
3
)
Th
ec
on
s
t
a
n
tofi
n
t
e
g
r
a
t
i
oni
sd
e
t
e
r
mi
n
e
db
you
rr
e
qu
i
r
e
me
n
tt
h
a
tn
or
u
n
a
wa
y
s
ol
u
t
i
on
se
x
i
s
t
!
Not
ewe
l
l
t
h
a
ti
ti
sac
on
s
t
r
a
i
n
tt
h
a
tl
i
v
e
si
nt
h
ef
u
t
u
r
eoft
h
ep
a
r
t
i
c
l
e
.
I
n
or
d
e
rt
ou
s
et
h
i
st
of
i
n
d
v(
t
)
,
wemu
s
tk
n
owt
h
ef
or
c
eF(
t
)f
ors
omet
i
me(
ofor
d
e
rτ
)i
n
r
t
h
ef
u
t
u
r
e
!
Af
t
e
rt
h
i
s
,
t
h
ei
n
t
e
g
r
a
n
di
s“
c
u
toff”b
yt
h
ed
e
c
a
y
i
n
ge
x
p
on
e
n
t
i
a
l
.
Th
i
ss
u
g
g
e
s
t
st
h
a
twec
a
ne
x
t
e
n
dt
h
ei
n
t
e
g
r
a
lt
oC=∞ wi
t
h
ou
td
i
ffic
u
l
t
y
.I
nt
h
e
l
i
mi
tτ
,
wer
e
c
ov
e
rNe
wt
on
’
sl
a
w,
a
swes
h
ou
l
d
.
Tos
e
et
h
i
s
,
l
e
t
r→ 0
1
′
t−t
)
r(
s=τ
s
ot
h
a
t
˙
∞
mv(
t
)=
0
(
2
1
.
2
4
)
−
s
e F(
t+τ
s
)
d
s
.
r
(
2
1
.
2
5
)
Th
ef
or
c
ei
sa
s
s
u
me
dt
ob
es
l
owl
yv
a
r
y
i
n
gwi
t
hr
e
s
p
e
c
tt
oτ(
orn
on
eoft
h
i
sma
k
e
ss
e
n
s
e
,
j
u
s
ta
swa
st
h
ec
a
s
ea
b
ov
e
)s
ot
h
a
taTa
y
l
ors
e
r
i
e
se
x
p
a
n
s
i
onc
on
v
e
r
g
e
s
:
∞
F(
t
+τs
)=
F
n
2
(
τ
r
s
)ad (
t
)
n
n
!
n
=
0
dt
wh
i
c
h
,
u
p
ons
u
b
s
t
i
t
u
t
i
ona
n
di
n
t
e
g
r
a
t
i
onov
e
rs
,
y
i
e
l
d
s
˙
mv =
∞
n
n
τ
dF
n
d
t
.
(
2
1
.
2
6
)
(
2
1
.
2
7
)
˙
F
i
g
u
r
e2
1
.
1
:F(
t
)
,
v(
t
)a
n
dv(
t
)onat
i
me
s
c
a
l
eofτ
.Not
et
h
a
tt
h
ep
a
r
t
i
c
l
e
r
“
p
r
e
a
c
c
e
l
e
r
a
t
e
s
”b
e
f
or
e“
t
h
ef
or
c
eg
e
t
st
h
e
r
e
”
,
wh
a
t
e
v
e
rt
h
a
tme
a
n
s
.
I
nt
h
el
i
mi
tτ→ 0o
n
l
yt
h
el
owe
s
tor
de
rt
e
r
ms
u
r
v
i
v
e
s
.Th
i
si
sNe
wt
on
’
sl
a
wwi
t
h
ou
t
r
a
d
i
a
t
i
onr
e
a
c
t
i
on
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eh
i
g
h
e
ror
d
e
rt
e
r
msa
r
es
u
c
c
e
s
s
i
v
er
a
d
i
a
t
i
v
ec
or
r
e
c
t
i
on
sa
n
d
ma
t
t
e
ron
l
yt
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h
ee
x
t
e
n
tt
h
a
tt
h
ef
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