Electrodynamics Flashcards
Monopole term
Vmon(r) = 1/(4pi eps0) Q/r
Dipole term
Vdip(r) = 1/(4pi eps0) (p*r^)/r^2
Dipole moment
p = int(r’ rho(r’) dtau’)
Dipole moment of an atom in an electric field
p = alpha E
Atomic polarizability of a spherical atom
alpha = 4pi eps0 a^3 = 3 eps0 v
Continuity equation (charge)
drho/dt = - div J
Total energy stored in EM fields
u = 1/2(eps0 E^2 + 1/mu0 B^2)
Poynting vector
S = 1/mu0 (E x B)
Work done on charges
dW/dt = - d/dt int( u dtau) - (S * da)
Continuity equation (energy)
du/dt = - div S (only valid if no work is done)
Maxwell stress tensor
Tij = Eps0 (EiEj - 1/2 delt_ij(E^2)) + 1/mu0(BiBj - 1/2 delt_ij(B^2))
Force per unit volume
f = div T - eps0 mu0 dS/dt
Momentum density
g = mu0 eps0 S = eps0 (E x B)
Continuity equation (EM momentum)
dg/dt = div T
Angular momentum
l = r x g
General solution to Laplace eq. in spherical coordinates
V = sum(A_l r^l + B_l/(r^(l+1)) P_l(cos theta)
Torque on a dipole in a field
N = p x E
Bound surface charge
sig_b = P * n^
Bound volume charge
rho_b = - div P
Electric displacement
D = eps_0 E + P = eps E
Polarization in a dielectric
P = eps_0 X_e E
Permittivity
eps = eps_0(1 + X_e)
Relative permittivity/dielectric constant
E_r = 1 + X_e = eps/eps_0
Gauge transformations
A’ = A + grad lambda
V’ = V - dlambda/dt
Lorentz gauge
div A = -mu0 eps0 dV/dt
Bound volume current
J_b = curl M
Bound surface current
K_b = M x n^
Lorentz transformation rules for E and B (for an observer moving in the +x direction)
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)
Field tensor
(eq 12.119)
Maxwell equations in tensor form
dF/dx = mu0 Jmu
dG/dx = 0
Dual tensor
eq 12.120
Continuity equation in 4vector form
dJ/dx = 0
Current density four-vector
Jmu = (c rho, Jx, Jy, Jz)
Position four-vector
xmu = (ct, x, y, z)
general wave equation
d^2 f/dz^2 = 1/v^2 d^2 f / dt^2
Polarization vector
n^ = cos (theta) x^ + sin(theta) y^
Wave equations for E and B
del^2 E = mu0 eps0 d^2e/dt^2
del^2 B = mu0 eps0 d^2/B dt^2
General wave solutions for E and B
E(z, t) = E0 e^(i(kz - wt))
B(z, t) = B0 e^(i(kz - wt))
B0 in terms of E0
B0 = 1/c (z^ x E0)
Propagation velocity of EM waves through a linear medium
v = 1/sqrt(eps mu) = c/n
Index of refraction
n = sqrt(eps mu/eps0 mu0)
Electrodynamic boundary conditions
eps1E1 = eps2E2 (perpendicular)
B1 = B2 (perpendicular)
E1 = E2 (parallel)
(1/mu1) B1 = (1/mu2) B2 (parallel
Law of refraction
sin(theta_T)/sin(theta_I) = n1/n2
Radiation pressure
P = I/c
Wave number in a conductor
k~^2 = k + i kappa = mu eps w^2 + i mu sig omega
skin depth
d = 1/kappa
Characteristic time
tau = eps/sig
Fresnel equations
E0R= (alpha - beta)/(alpha + beta) E0I
E0T = 2/(alpha + beta)E0I
alpha (in Fresnel)
alpha = cos(theta_T)/cos(theta_I)
beta(in Fresnel)
beta = mu1n1/mu2n2
