4ot-u'71 oNS
Spring 200/1
Prof. A. Ni,kneiad
Uniuers'ityof Cali,forni,a,B erkeley
EECS117
Exam II (closed book)
T\resday,March 30, 2004
Guidelines: Closedbook and notes. You mav use a calculator. Do not unstaplethe exam.
Warning: Illustrations not to scale.
L. (25 poi,nts)A conductorwith o : 107S/mand dielectric constant € : €0is placed
into a uniform electric field Eo : *50V/m.
Es
+
55
A./
+
-+
\3
5TD
-+
5,t v6p
e,l
11
Pt p
+
(a) Find the chargedensity everywhere.Showthe location of the charge
distribution schematicallyon the figure above.
//b Q€
rtr
E:O t
77fE CbYDuc-7oKt
Ss a fr- D =
6uRft.<-{/
3=o
€o Eo
sl x r;'' c/^. = 44 ", E
?-
(b) Estimate the time scaleto reach the steady-statechargedistribution.
S€uqe,rz
fl,vtv
7-=
w
Ttrt
ffitul1
CoUSf-/4\N--7j
t6^
)N
g
ls
1Yl;t1
EXPrtn'tNA7fu
(
So;{ f}orvs
Z-
(") Is there a force on the conductor? If so, draw the direction of the force and
calculatethe magnitude.
?er<o
TS
fuXC€
N6T
1Eo
-t Eo
t
1= St'A
(d) Now a dielectricobject is placedin the field as shownbelow. Half of the material
has dielectricconstante1: 2 while the other half has €z: 4. Is there a forceon
the obiect? If so. calculatethe masnitude and indicate the direction of the force.
Es
-4
-*
+
--+
+
+
-+
v
\
-)>
/
V
/Lt
, +o
1,tt)T
43 4
7a
-)>
ryJt)rtD
'flfv
CDF+€UV
PoU#t?flcto
S 4tut€
(N
7)+E
Q v t r t o ^ / s ' S r=
1-ago(t*
1w o
Z)
= -!aue,
S4- +{o6a(t-€)=
A%E'
€
(
t
S > t - ' - E ' E , ? , ) { - " € > ( ( - 2o
s, -9:s -!rt / -41,€o
Et =
f z=
?3
+loEo'-A
_J
L(
4
€d
Eu" -A
2
1 o Eutft
ZF=O
)y
5at-r (1oN9
3 . (30 poi,nts)A transmissionline is formed by running a single wire of radius o parallel
to earth. The cross-sectionof the transmissionline is shown below. We wish to find
the transmissionline parameters.You may assumethat the region abovethe wire is
air and thus e x e6,p x lJ"s.
\
O
(a) Find the potential in the charge-freeregion f.ory ) 0. You may assumethat the
wire radius is negligible. H'int: Use the method of images.
'7V
llt
g tv
oTL( e (e
SWC€ Ex
bLzrritoEr
9P
- 8e
I =--L
bJ 6
1,
tW€
r/
2h
s
Z(co
tat€T
9-/L r
Lr-- ffi
3x"Lt
/
/8. t<-(tv
V
6^/9s/ L^P / 0e
+
=
O\.t
lcatv
4 tr
a%ot2
lrtzt'+G€>
b^ ("n*+ b
{,P-^(r- o\iI
(b) Find the capacitanceper unit length of the line.
\0
(
tr-
|
o
NarE f (r')
?rtL6
x' + dz
^ l ^
.h l-
= -y
a{t$
-.a-,^
- \ lU - | I
lc.
I o= 0 ( r - ), 1 - a-.)
tu
24Co
)
a
\LA-N
7r €o
-
Vo
K
u('4-'t
f
4au, {l ous
(c) Find the ind'uctanceper unit length of the line'
L'Cl -- Vu € o
Surtc-u
L ( =aTTA
r'^(1 -tl
(d)CalculatetheshuntconductivelossG'perunitlength'
RC=
6
Ca
-
<o
ertr
4.1
_
l
I
a.
that current
(e) Estimate the seriesresistanceper unit length R'. You may assume
flowsuniformlyinalayerofthicknessdalorrgtheouteredgesofthecondtrctors.
Assumeearth ltas a conductivity of'o2' Hin't:'
I]_ffi:^
5a'vt-?c,c't
7
\
&rL(28 (T,
Pl'{J
Ll=
-L
Rlu,,ge e
Tn*>=
€cz'uD
+
Q-, o,
(/+
Jnu,
3.*o=
l^\^J
I
*
f
-
2na,b
'1/f
fi.D^lr="
(f) Calculatethe lirre characteristicimpedance'
-rL'* il
Ca
Gl
)u.'r
f=-v/
E5 ( 1 = o )
T(*)=
ii
xn+ A'
K{o
Se u >( x )
9.n
! 'l
=) L
lc (x"+au)
,j
-d
= y _
a
,K'+ d'
=
t"ts-++
Ce++
6 "t lTf av L eP(k
F
f (* L \ '
tLX
I \ c*'+'1"))
=
,
= J We*
2rJ
)
Ru* =
drr-
)4r4 5
h.-
I
t
Q'*
?'crdt
f
2r'L t
JaLu Tl o pJ
stores9ne-r'Yequal to i.CV2, and the
4. (10 points)Using the fact that a capacitor
D, caiculatethe capacitanceof the
electrostaticfield has energydensity of |E'
stored in the cap to the energy
following structure by equiing the totai energ'y
fields'
containedin the eleclrostaticfield. Neglectfringing
r,_w1
i
-!
I
T
t
€2
e1
,:
II
I
I
I
w
Dt=
tt €1
Dt=
t7 9z-
t 1,,E - D o t v +
t fr. E'D
(,-',)
I
+\
L
c(=
ct Vo'
Wr t1
L
,
f
r
\
/
(Ls-utt) 5z
/{.
vu'
Z