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6.013 Electromagnetics and Applications, Fall 2005
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CartesianCoordinates(x, y, z)
af af Vf = -af. , + O i, + i, ax
Oz
ay
aA, aA, + aA, V . A= a+ ax
az
ay
(LAX,
A
AaA\
ay z V2f a' +f Ox jy
+
az
.a,
O aA.
ax
Ox
ay
a'f az
CylindricalCoordinates (r, 4, z)
Of. I •af4 + af.1 r04 az
Vf = arr
1a
1iA,
aA,
rr
rr a
Oz
V *A=- -(rAr)+M +M I aBA,
aA.
rrr) V~ f
la0
af\
-- r-
rr
1 82f
+
I(rAs)
ar
'L
,+
a2f
1
ae+I
r
•O
af. If-14
r sin 0 aO
A 1 (r 1 a(sin OAo) 1 oA* V" -A = (rPA,)+ + r ar r sin 0 ae r sin 0 a4 x
1 r sin
a(sin OAs)
aA]
a80
a 04,
S 1 r sio V'f =
r-r a-"
r+
MA,
arsin
a(rA,))
1 [ra(rAo) dA,1 rOr O-
a+sin 0 O+
I
A
a4J
4)
af.
a.
ar
DA A.
Ora r) 14-2az
SphericalCoordinates (r, 0, Vf=
A, 1xzaz
a•f
Spherical
Cylindrical
Cartesian x
=
r cosc
=
r sin 0 cos 4
y z
=
r sinq
=
r sin 0 sin 4
=
=
=
z
cos
=
i, - sin 0i
os 0
r
sin 0 cos
i, + cos 0 cos
4ie
-sin Ois 1
=
,Y
= sin 0 sin 4i, + cos 0 sin
sin 0i, + cos 0ik
=
iz
cos Oi,- sin Oie
=
Spherical
Cartesian
Cylindrical
=r =
tan- 1y/x
-=
z
=
cos kix, +sin =
i,
sin Oi, +cos
=
i4 cos Oi, -sin 0iO
If,- Ă˝+z
/x 2+y2+z =
ie
Cylindrical
Cartesian
r
cos 0
=
=
i
Spherical
sin 0
Sr
-sin 0ix +cos 4iy
=
0
/ie
+ cos 46 i
-1
cos
=
/x2'+y2+z'
z
cos
2
=
cot- x/y
i,
=
sin 0 cos ,ix +sin 0 sin (i, + cos Oi.
is
=
cos 0 cos oi, +cos 0 sin -sin Oi.
i,
=
-sin 46i, +cos
di,
=
4i, = =
sin Oi,+cos Oi, cos Oi, -sin
Oi,
i4,
Geometric relations between coordinates and unit vectors for Cartesian, cylir drical, and spherical coordinate systems.
VECTOR IDENTITIES (AxB). C= A. (B xC)= (CxA). B Ax(BxC)=B(A
C)-C(A - B)
V* (VxA)=O
Vx(Vf)=o V(fg) = fVg + gVf V(A B) =(A * V)B + (B -V)A +Ax(VxB)+Bx(VxA) V. (fA)= fV. A+(A - V)f V *(A x B)= B (V x A)-A -(V x B)
v x (A x B) = A(V B) - B(V - A) +(B . V)A-(A - V)B Vx(fA)= VfxA+fVxA (V x A) x A = (A V)A - 'V(A . A) Vx (Vx A) = V(V - A) - V A
INTEGRAL THEOREMS Line Integral of a Gradient =f(b) -f(a) Vf dlI Divergence Theorem: sA dS
f V-AdV= Corollaries
t VfdV=f dS VVxAdV=-s AxdS Stokes' Theorem: fA
dl=
(Vx A) dS
Corollary ffdl= -fVfxdS
I
MAXWELL'S EQUATIONS Differential
Integral
Boundary Conditions
Faraday's Law E'*dl=-d B-dS VxE=- aB nx(E2'-E')=0. dtJI at Ampere's Law with Maxwell's Displacement Current Correction H.dI=s J,.dS +
dtiJs
VxH=Jjf+a-
nx (H 2 -HI) =Kf
D dS
Gauss's Law V D=p
pfdV
sD-dS= B dS=0
n *(D 2 -D 1 ) = of
V*B=0
Conservation of Charge JdS+ d
pf
dV = O
V J,+f=0
s dt Usual Linear Constitutive Laws D=eE
n (J2-JI)+
at
at
= 0
B=LH Jf = o(E + vx B) =0E'[Ohm's law for moving media with velocity v] PHYSICAL CONSTANTS Constant
Symbol
Speed of light in vacuum Elementary electron charge Electron rest mass Electron charge to mass ratio
c e m, ee
Value 2.9979 x 108 =3 x 108 1.602 x 10 - '9
9.11 x 10- 3s '
m/sec coul kg
1.76 x 10"
coul/kg
-
Proton rest mass Boltzmann constant Gravitation constant Acceleration of gravity
mn k G g
1.38 x 10-23 6.67 x 10- " 9.807
Permittivity of free space
60
8.854x 10-
Permeability of free space Planck's constant Impedance of free space
Avogadro's number
Al0 h 110=
units
1.67 x 10 27
12=
-7
1036?r
36
kg joule/OK nt-m2 /(kg)2 m/(sec)2 farad/m
4Tr x 10 6.6256 x 10- 3 4
henry/m joule-sec
376.73 - 120ir
ohms
6.023 x 1023
atoms/mole