Cylindrical
Cartesian
Spherical
x
=
r cosc/
S
r sin 6 cos 4
y
z
=
r sin4
S
r sin 6 sin4
=
z
=
ix
=
Cos Oi-sin 4k 4
=
=
i
sin Ocos ki,+cos 6 cos
-sin Ois
6 sin
+cos 6 io
0 i,+ Cos 01i
=sin
iY
rcos6
= sin 0 sin 6i, + Cos
il41+C541
S
1
=
=~V
ie
cos Oi,- sin Ois
Spherical
Cartesian
Cylindrical
4ie
sin 6
Sr
Ix+ y7
tan~1y/x
z
=
rcos6
=
sin Oi,+cos 6ie
i4
=
cos (k ,, +sin
=
-sin ki,,+Cos 4i,
=
=
i,
=
i,
.
i4
=
Spherical
Cartesian
r
0
I4x +y
=
=
i,
=
Cos
iO
Cylindrical
If,- +z"
+z
_1 z
cot-'
cos i, -sin
=
cos
-1
z
x/y
sinG cos kix+sin
+cos
6 sin ci,
=
sin
i, +Cos Oi,
i.
=
cos 6 cos oi,+cos 6 sin oi,
-sin Oi.
=
Cos Oi,-sin Oi,
i
=
-sin 46i, +Cos 0i,
=
i4
,
i,
Geometric relations between coordinates and unit vectors for Cartesian, cylir
drical, and spherical coordinate systems.
CartesianCoordinates(x, y, z)
Vf = Ofi.+Ofi,+ Ofi.
ax +-A=,++-i
ay
Oz
V- aA,, aA, aA, 2
ax
ay
az
aA -(aA)
_8A\
_)+i
a.x)
ay
=i. ay_
,(-az
(LAI
ay aza )+
) (aA
VxA
V2f+!L+f+a2f
Z
Ox2 j
CylindricalCoordinates (r, 4, z)
Of.
r+
Vf=
Of.i,+ Of iz
1
+A MA.
V -A= Ia(rA,.)+ -.
r Or
r ao
0/f
1 a2f a2f
ar Of\
+
42 O
r Or On) r
SphericalCoordinates (r, 0,
Vf=i,.+
Or r
=
V A
4,)
afi+
aO r sin 0 a4
(r2A,)+
r2 ar
+r s n
r sin
0 a4
aA.
a4
1 [a(rA#) OA,.
rL Or
aeJ
I rIAM, a(rA)
r sin{ 0r]
r
oA*
1
(sin OA.)
1
r sin 0
aI
a(sin OAs)
VxA=i1
'r sineL
80
V f =
OA r a4
Or +
+
V'f=
+ixaz
-
VxA=i
az
sin0
rf ar' Or. r- -s i n aG
)
+Of I
a2 f
ai r sin 04, 0­
MAXWELL'S EQUATIONS
Boundary Conditions
Differential
Integral
Faraday's Law
d C
B
E'-dl=-B-dS
VxE=nx(E'-E')=0.
dtJI
at
Ampere's Law with Maxwell's Displacement Current Correction
H-dl=
Jf,-dS
VXH=Jf+aD
nX(H 2 -H 1 )=Kf
D'-dS=t pfdV
V - D=p
n - (D 2 -D 1 )= o-
B-dS=0
V-B=0
+
~it-s
D-dS
Gauss's Law
Conservation of Charge
J,- dS+
sVd
pfdV=O
V-J,+
at
=0
n - (J2-J)+"
a
0
Usual Linear Constitutive Laws
D=eE
B= H
Jf= o-(E+vX B)=
a-E' [Ohm's law for moving media with velocity v]
PHYSICAL CONSTANTS
Constant
Symbol
Speed of light in vacuum
Elementary electron charge
Electron rest mass
c
e
Electron charge to mass ratio
Value
units
8
M,
2.9979 x 10 = 3 x 108
1.602 x 10~'9
9.11 x 10 3
m/sec
coul
kg
e
1.76 x 10"
coul/kg
I,
k
G
kg
joule/*K
nt-m2/(kg) 2
m/(sec) 2
M,
g
1.67 x 10-27
1.38 x 10-23
6.67 x 10-"
9.807
Permittivity of free space
60
8.854 x 10~2~36r
Permeability of free space
Planck's constant
A0
4r X
Impedance of free space
Avogadro's number
i1o
10
10
*
Proton rest mass
Boltzmann constant
Gravitation constant
Acceleration of gravity
farad/m
h
6.6256 x 10-34
henry/m
joule-sec
4
376.73- 120ir
ohms
Ar
6.023 x 1023
atoms/mole
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the
I