Electrodynamics

Question 1

Monopole term

Answer 1

Vmon(r) = 1/(4pi eps0) Q/r

Question 2

Dipole term

Answer 2

Vdip(r) = 1/(4pi eps0) (p*r^)/r^2

Question 3

Dipole moment

Answer 3

p = int(r’ rho(r’) dtau’)

Question 4

Dipole moment of an atom in an electric field

Answer 4

p = alpha E

Question 5

Atomic polarizability of a spherical atom

Answer 5

alpha = 4pi eps0 a^3 = 3 eps0 v

Question 6

Continuity equation (charge)

Answer 6

drho/dt = - div J

Question 7

Total energy stored in EM fields

Answer 7

u = 1/2(eps0 E^2 + 1/mu0 B^2)

Question 8

Poynting vector

Answer 8

S = 1/mu0 (E x B)

Question 9

Work done on charges

Answer 9

dW/dt = - d/dt int( u dtau) - (S * da)

Question 10

Continuity equation (energy)

Answer 10

du/dt = - div S (only valid if no work is done)

Question 11

Maxwell stress tensor

Answer 11

Tij = Eps0 (EiEj - 1/2 delt_ij(E^2)) + 1/mu0(BiBj - 1/2 delt_ij(B^2))

Question 12

Force per unit volume

Answer 12

f = div T - eps0 mu0 dS/dt

Question 13

Momentum density

Answer 13

g = mu0 eps0 S = eps0 (E x B)

Question 14

Continuity equation (EM momentum)

Answer 14

dg/dt = div T

Question 15

Angular momentum

Answer 15

l = r x g

Question 16

General solution to Laplace eq. in spherical coordinates

Answer 16

V = sum(A_l r^l + B_l/(r^(l+1)) P_l(cos theta)

Question 17

Torque on a dipole in a field

Answer 17

N = p x E

Question 18

Bound surface charge

Answer 18

sig_b = P * n^

Question 19

Bound volume charge

Answer 19

rho_b = - div P

Question 20

Electric displacement

Answer 20

D = eps_0 E + P = eps E

Question 21

Polarization in a dielectric

Answer 21

P = eps_0 X_e E

Question 22

Permittivity

Answer 22

eps = eps_0(1 + X_e)

Question 23

Relative permittivity/dielectric constant

Answer 23

E_r = 1 + X_e = eps/eps_0

Question 24

Gauge transformations

Answer 24

A’ = A + grad lambda
V’ = V - dlambda/dt

Question 25

Lorentz gauge

Answer 25

div A = -mu0 eps0 dV/dt

Question 26

Bound volume current

Answer 26

J_b = curl M

Question 27

Bound surface current

Answer 27

K_b = M x n^

Question 28

Lorentz transformation rules for E and B (for an observer moving in the +x direction)

Answer 28

Ex -> Ex, Ey -> gamma(Ey - v Bz), Ez --> gamma(Ez + vBy) Bx -> Bx, By -> gamma(By + v/c^2 Ez), Bz -> gamma(Bz - v/c^2 Ey)

Question 29

Field tensor

Answer 29

(eq 12.119)

Question 30

Maxwell equations in tensor form

Answer 30

dF/dx = mu0 Jmu dG/dx = 0

Question 31

Dual tensor

Answer 31

eq 12.120

Question 32

Continuity equation in 4vector form

Answer 32

dJ/dx = 0

Question 33

Current density four-vector

Answer 33

Jmu = (c rho, Jx, Jy, Jz)

Question 34

Position four-vector

Answer 34

xmu = (ct, x, y, z)

Question 35

general wave equation

Answer 35

d^2 f/dz^2 = 1/v^2 d^2 f / dt^2

Question 36

Polarization vector

Answer 36

n^ = cos (theta) x^ + sin(theta) y^

Question 37

Wave equations for E and B

Answer 37

del^2 E = mu0 eps0 d^2e/dt^2 del^2 B = mu0 eps0 d^2/B dt^2

Question 38

General wave solutions for E and B

Answer 38

E(z, t) = E0 e^(i(kz - wt)) B(z, t) = B0 e^(i(kz - wt))

Question 39

B0 in terms of E0

Answer 39

B0 = 1/c (z^ x E0)

Question 40

Propagation velocity of EM waves through a linear medium

Answer 40

v = 1/sqrt(eps mu) = c/n

Question 41

Index of refraction

Answer 41

n = sqrt(eps mu/eps0 mu0)

Question 42

Electrodynamic boundary conditions

Answer 42

eps1E1 = eps2E2 (perpendicular) B1 = B2 (perpendicular) E1 = E2 (parallel) (1/mu1) B1 = (1/mu2) B2 (parallel

Question 43

Law of refraction

Answer 43

sin(theta_T)/sin(theta_I) = n1/n2

Question 44

Radiation pressure

Answer 44

P = I/c

Question 45

Wave number in a conductor

Answer 45

k~^2 = k + i kappa = mu eps w^2 + i mu sig omega

Question 46

skin depth

Answer 46

d = 1/kappa

Question 47

Characteristic time

Answer 47

tau = eps/sig

Question 48

Fresnel equations

Answer 48

E0R= (alpha - beta)/(alpha + beta) E0I E0T = 2/(alpha + beta)E0I

Question 49

alpha (in Fresnel)

Answer 49

alpha = cos(theta_T)/cos(theta_I)

Question 50

beta(in Fresnel)

Answer 50

beta = mu1n1/mu2n2