PHYS3211: Electromagnetic theory

Equation sheet

Electrostatics

Coulomb’s law: ~F12 =q1q24⇡✏0

(~r � ~r0)

|~r � ~r0|3Electric fields: ~E (~r) =

1

4⇡✏0

Zdv ⇢v (~r)

(~r � ~r0)

|~r � ~r0|3

Electric potential: V (~rb)� V (~ra) = �Z ~rB

~rA

~E (~r) · d~ ~E (~r) = �~rV (~r)

~D = ✏ ~E

Gauss’s law:for closed surface:

I

S

~D · d~S = Qencl~r · ~D = ⇢v

for closed loop:

I

C

~E · d~= 0 ~r⇥ ~E = 0

Specialized results: Field of an infinite sheet in the z = 0 plane: ~D =⇢S2z

Field of an infinite line with line density ⇢L: ~D = ⇢L2⇡⇢ ⇢

Ohm’s law: ~J = � ~E

Boundary conditions: n ·⇣~D1 � ~D2

⌘= ⇢S; E1t = E2t.

Magnetostatics

Magnetic force: ~Fm = q~u⇥ ~B = I

Z

L

d~⇥ ~B

Biot-Savart:

~H (~r) =1

4⇡

Z

wire

Id~⇥ (~r � ~r0)

|~r � ~r0|3=

1

4⇡

Z

S

dS ~K ⇥ (~r � ~r0)

|~r � ~r0|3=

1

4⇡

Z

v

dv ~J ⇥ (~r � ~r0)

|~r � ~r0|3

Gauss’s law:for closed surface:

I

S

~B · d~S = 0 ~r · ~B = 0

for closed loop:

I

C

~H · d~= Iencl ~r⇥ ~H = ~J

Specialized results: Field of an infinitely long wire with current I: ~B =µ0I

2⇡⇢�

Field of a loop in xy plane (z = 0) for a point on the z axis: ~B = zIµ0a

2

2 (z2 + a2)3/2

Field of an infinite sheet of current (+y direction) in the z = 0 plane:

~H =K

2x, (for z > 0); ~H = �K

2 x, (for z < 0)

Boundary conditions: n ·⇣~B1 � ~B2

⌘= 0; H2t �H1t = Js

Magnetic moment: ~m = NIA

1

From: http://physics.nist.gov/constants

Fundamental Physical Constants — Frequently used constantsRelative std.

Quantity Symbol Value Unit uncert. ur

speed of light in vacuum c, c0 299 792 458 m s�1 (exact)magnetic constant µ0 4π⇥ 10�7 N A�2

= 12.566 370 614...⇥ 10�7 N A�2 (exact)electric constant 1/µ0c2 ✏0 8.854 187 817...⇥ 10�12 F m�1 (exact)Newtonian constantof gravitation G 6.6742(10)⇥ 10�11 m3 kg�1 s�2 1.5⇥ 10�4

Planck constant h 6.626 0693(11)⇥ 10�34 J s 1.7⇥ 10�7

h/2π h 1.054 571 68(18)⇥ 10�34 J s 1.7⇥ 10�7

elementary charge e 1.602 176 53(14)⇥ 10�19 C 8.5⇥ 10�8

magnetic flux quantum h/2e �0 2.067 833 72(18)⇥ 10�15 Wb 8.5⇥ 10�8

conductance quantum 2e2/h G0 7.748 091 733(26)⇥ 10�5 S 3.3⇥ 10�9

electron mass me 9.109 3826(16)⇥ 10�31 kg 1.7⇥ 10�7

proton mass mp 1.672 621 71(29)⇥ 10�27 kg 1.7⇥ 10�7

proton-electron mass ratio mp/me 1836.152 672 61(85) 4.6⇥ 10�10

fine-structure constant e2/4π✏0hc ↵ 7.297 352 568(24)⇥ 10�3 3.3⇥ 10�9

inverse fine-structure constant ↵�1 137.035 999 11(46) 3.3⇥ 10�9

Rydberg constant ↵2mec/2h R1 10 973 731.568 525(73) m�1 6.6⇥ 10�12

Avogadro constant NA, L 6.022 1415(10)⇥ 1023 mol�1 1.7⇥ 10�7

Faraday constant NAe F 96 485.3383(83) C mol�1 8.6⇥ 10�8

molar gas constant R 8.314 472(15) J mol�1 K�1 1.7⇥ 10�6

Boltzmann constant R/NA k 1.380 6505(24)⇥ 10�23 J K�1 1.8⇥ 10�6

Stefan-Boltzmann constant(π2/60)k4/h3c2 � 5.670 400(40)⇥ 10�8 Wm�2 K�4 7.0⇥ 10�6

Non-SI units accepted for use with the SI

electron volt: (e/C) J eV 1.602 176 53(14)⇥ 10�19 J 8.5⇥ 10�8

(unified) atomic mass unit1 u = mu = 1

12m(12C) u 1.660 538 86(28)⇥ 10�27 kg 1.7⇥ 10�7

= 10�3 kg mol�1/NA

Page 1 Source: Peter J. Mohr and Barry N. Taylor, CODATA Recommended Values of the Fundamental PhysicalConstants: 2002, to be published in an archival journal in 2004.

3

Cartesian coordinates

Coordinate variables: (x, y, z) Scalar functions: f(x, y, z), g(x, y, z) Basis vectors: x, y, z or a

x

,ay

,az

Vector Vector Calculus

~A = xAx

+ yAy

+ zAz

~r = x@

@x+ y

@

@y+ z

@

@z

~Af(x, y, z) = xAx

f(x, y, z) + yAy

f(x, y, z) + zAz

f(x, y, z) Gradient : ~rf(x, y, z) = x@f

@x+ y

@f

@y+ z

@f

@z

dot product: ~A · ~B = Ax

Bx

+ Ay

By

+ Az

Bz

Divergence : ~r · ~A =@

@xA

x

+@

@yA

y

+@

@zA

z

cross product : ~A£ ~B =

ØØØØØØØØØØx y z

Ax

Ay

Az

Bx

By

Bz

ØØØØØØØØØØCurl: ~r£ ~A =

ØØØØØØØØØØx y z

@

@x

@

@y

@

@z

Ax

Ay

Az

ØØØØØØØØØØr2 V =

@2V

@x2

+@2V

@y2

+@2V

@z2

Cylindrical coordinates

Coordinate variables: (Ω, ¡, z) Scalar functions: f(Ω,¡, z) Basis vectors: Ω, ¡, z or a

Ω

,a¡

,az

Vector Vector Calculus

~A = ΩAΩ

+ ¡A¡

+ zAz

~r = Ω@

@Ω+ ¡

1Ω

@

@¡+ z

@

@z

~Af = ΩAΩ

f + ¡A¡

f + zAz

f Gradient : ~rf(Ω,¡, z) = Ω@f

@Ω+ ¡

1Ω

@f

@¡+ z

@f

@z

dot product: ~A · ~B = AΩ

BΩ

+ A¡

B¡

+ Az

Bz

Divergence : ~r · ~A =1Ω

@

@Ω(ΩA

Ω

) +1Ω

@

@¡A

¡

+@

@zA

z

cross product : ~A£ ~B =

ØØØØØØØØØØΩ ¡ z

AΩ

A¡

Az

BΩ

B¡

Bz

ØØØØØØØØØØCurl: ~r£ ~A =

1Ω

ØØØØØØØØØØΩ Ω¡ z

@

@Ω

@

@¡

@

@z

AΩ

ΩA¡

Az

ØØØØØØØØØØr2 V =

1Ω

@

@Ω

µΩ@V

@Ω

∂+

1Ω2

@2V

@¡2

+@2V

@z2

,

Spherical coordinates

Coordinate variables: (r, µ,¡) Scalar functions: f(r, µ,¡) Basis vectors: r, µ, ¡ or a

r

,aµ

,a¡

Vector Vector Calculus

~A = rAΩ

+ µAµ

+ ¡A¡

~r = r@

@r+ µ

1r

@

@µ+ ¡

1r sin µ

@

@¡

~Af = rAr

f + µAµ

f + ¡A¡

f Gradient : ~rf(r, µ, ¡) = r@f

@r+ µ

1r

@f

@µ+ ¡

1r sin µ

@f

@¡

dot product: ~A · ~B = Ar

Br

+ Aµ

Bµ

+ A¡

B¡

Divergence : ~r · ~A =1r2

@

@r

°r2A

r

¢+

1r sin µ

@

@µ(A

µ

sin µ) +1

r sin µ

@

@¡A

¡

cross product : ~A£ ~B =

ØØØØØØØØØØr µ ¡

Ar

Aµ

A¡

Br

Bµ

B¡

ØØØØØØØØØØCurl: ~r£ ~A =

1r2 sin µ

ØØØØØØØØØØr rµ r sin µ¡

@

@r

@

@µ

@

@¡

Ar

rAµ

r sin µA¡

ØØØØØØØØØØr2 V =

1r2

@

@r

µr2

@V

@r

∂+

1r2 sin µ

@

@µ

µsin µ

@V

@µ

∂+

1r2 sin2 µ

@2V

@¡2

.