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6.013 Electromagnetics and Applications, Fall 2005

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Markus Zahn, Electromagnetic Field Theory. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License:Creative Commons Attribution-NonCommercial-Share Alike

CartesianCoordinates(x, y, z)af. af afVf = - , + O i, + i,ax ay Oz

aA, aA, aA,V . A= a+ +ax ay az

(LAX, A AaA\ .a, OaA.

ay z az Ox ax ay

V2f a' +f + a'fOx jy az

CylindricalCoordinates (r, 4, z)

Of. I af af.Vf = r • 4 + 1ar r04 az

1 a 1iA, aA,V *A=- -(rAr)+M +Mrr rr a Oz

I aBA, aA. A, DAA. I(rAs) Ar rr) 1xzaz 'L ar a4JV~ f l a0 af\ 1 82f a2f

rr -- r-Ora + r) 14 -2az

SphericalCoordinates (r, 0, 4)a. af. 1 af.

Vf= ar ,+ ae+I If-14r •O 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 a(sin OAs) aA]r sin a80 04,a

S 1 MA, a(rA,)) 1[ra(rAo) dA,1

r sio arsin rOr O-

V'f = a-"r-r r+ a+sin0 O+ I a•f

Cartesian Cylindrical Spherical

x = r cosc = r sin 0 cos 4

y = r sinq = r sin 0 sin 4

z = z = r os 0

= cos i, - sin 0i = sin 0 cos i, + cos 0 cos 4ie-sin Ois

1 = sin 0 sin 4i, + cos 0 sin ,Y = sin 0i, + cos 0ik /ie+ cos 46 i

= iz = cos Oi,- sin Oie

Cylindrical Cartesian Spherical

=r sin 0

= tan- 1y/x

-= z Sr cos 0

= cos kix,+sin i, = sin Oi, +cos ie

= -sin 0ix +cos 4iy = i4

= i = cos Oi, -sin 0iO

Spherical Cartesian Cylindrical

r /x 2+y2+z If,- ý+z

0 -1 z

= cos = cos

/x2'+y2+z' 2

= cot- x/y

i, = sin 0 cos ,ix +sin 0 sin (i, = sin Oi,+cos Oi,+ cos Oi.

is = cos 0 cos oi, +cos 0 sin 4i, = cos Oi, -sin Oi,-sin Oi.

i, = -sin 46i, +cos di, = i4,

Geometric relations between coordinates and unit vectors for Cartesian, cylirdrical, 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

Vf dlI =f(b) -f(a)

Divergence Theorem:

f V-AdV= sA dS

Corollaries

t VfdV=f dS

V VxAdV=-s AxdS

Stokes' Theorem:

fA dl= (Vx A) dS

Corollary

ffdl= -fVfxdS

I

MAXWELL'S EQUATIONS

Integral Differential 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 VxH=Jjf+a- nx (H2 -HI) =Kf

+ D dSdtiJs

Gauss's Law

sD-dS= pfdV

B dS=0

Conservation of Charge

V D=p

V*B=0

n *(D 2 -D 1 ) = of

JdS+ d pfdV = O V J,+f=0 n (J2-JI)+ = 0s dt at at

Usual Linear Constitutive LawsD=eE

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 cElementary electron charge eElectron rest mass m,

eElectron charge to mass ratio e

Proton rest mass mnBoltzmann constant kGravitation constant GAcceleration of gravity g

Permittivity of free space 60

Permeability of free space Al0Planck's constant h

Impedance of free space 110=

Avogadro's number

Value

2.9979 x 108 =3 x 1081.602 x 10 - '99.11 x 10- s3 '

1.76 x 10"

1.67 x 10- 271.38 x 10-236.67 x 10- "9.807

10-

8.854x 10- 12= 3636?r

4Tr x 10- 7

6.6256 x 10-3

4

376.73 - 120ir

6.023 x 1023

units

m/seccoulkg

coul/kg

kgjoule/OKnt-m2/(kg)2

m/(sec)2

farad/m

henry/mjoule-sec

ohms

atoms/mole