Magnetostatics

The magnetic vector potential

∇·B = 0 --> B = ∇×A.
A is not unique.  A' = A + ∇ψ + C, with ψ an arbitrary scalar field and C an arbitrary constant vector is also a vector potential for the same field.
In magnetostatics we choose ∇·A = 0. 
Then  ∇²A = -μ₀j,  A(r) = (μ₀/(4π)) ∫_V'dV' j(r')/|r - r'|.

The uniqueness theorem:

If the current density j is specified throughout a volume V and A or its normal derivatives are specified at the boundaries of a volume V, then a unique solution exists for A inside V.
Or, if the current density j is specified throughout a volume V and and either A or B are specified at the boundaries of a volume V, then a unique solution exists for B inside V.

Boundary conditions in magnetostatics

(B₂ - B₁)·n₂ = 0,  (B₂ - B₁)·t = μ₀k·n.
A is continuous across the boundary.

Magnetic Materials

The magnetization M = dm/dV is defined as the magnetic dipole moment per unit volume. 

The total current density is due to free and to magnetization current densities.
j = j_f + j_m,  k_m = k_f + k_m,
j_m = ∇×M,  k_m = M×n.
H = B/μ₀ - M.  (This definition is not unique.)
∇×H = j_f,  (Ampere's law for H).

For linear, isotropic, homogeneous (lih) magnetic materials we have
M = Χ_mH,  B = μ₀(H + M) = μ₀(1 + Χ_m)H =  μ₀κ_mH =  μH.
Χ_m < 0 for diamagnetic materials, Χ_m > 0 for paramagnetic materials, permanent magnets are not lih.

Boundary conditions for H

(H₂ - H₁)·t₂ = k_f·n,  ∇·H ≠ 0  in general.

Energy in magnetostatics

The magnetostatic energy stored in a current distribution is given by 
U = (2μ₀)^-1∫_{all space}B·B dV.

In the presence of a magnetic material, the total work done in establishing a free current distribution is
W = ½∫_{all space}B·H dV,
or, in the presence of a lih magnetic material
U = (2μ)^-1∫_{all space}B·B dV.

Electromagnetic radiation

Radiation produced by moving charges

In the Lorentz gauge the potentials A and Φ satisfy the inhomogeneous wave equation.
∇²Φ - (1/c²)∂²Φ/∂t² = -ρ/ε₀,    ∇²A - (1/c²)∂²A/∂t² = -μ₀j.
Solutions are
Φ(r,t) = [1/(4πε₀)]∫_v' dV' ρ(r',t_r)/|r - r'|,
A(r,t) = [μ₀/(4π)]∫_v' dV' j(r',t_r)/|r - r'|.
ρ(r',t_r) = ρ(t_r(r')) is evaluated at the retarded time t_r = t - |r - r'|/c.
For a point charge moving in an arbitrary this yields the Lienard-Wiechert potentials
Φ(r,t) = [1/(4πε₀)][q/((1 - β∙n)|r - r'|)]_ret,
A(r,t) = [μ₀c/(4π)][qβ/([1 - β∙n)|r - r'|)]_ret.
Here r' is the position of the point charge at the retarded time,  n|r - r'| is the vector pointing from the position of the point charge at the retarded time to the observer at r.
The potentials of a point charge depend only on the position and the velocity at the retarded time.  The fields E and B depend also on the acceleration.
We can find E(r,t) and B(r,t) using E = -∂A/∂t -∇Φ,  B = ∇×A.

Assume an observer is located at the origin.
The electric field produced by a point charge q which moves in an arbitrary way at the location of the observer is
E(t) = -(q/(4πε₀))[(r'/r'³) + (r'/c)(d(r'/r'³)/dt) + (1/c²)(d²(r'/r')dt²)].
Here r' is the position of the charge at the retarded time (t - r'/c); r' points from the observer to the charge.  [Note r'/r' is the unit vector.]
E = E₁ + E₂ + E₃.
E₁ = E_c(t - r'/c) = retarded Coulomb field.  E₂ = (r'/c)(dE₁/dt).
E(t) = E₁(t - r'/c) + (r'/c)(dE₁(t - r'/c)/dt) + ...
The retardation is removed to first order.  For the near field it is a better approximation to use the instantaneous Coulomb field than to use the retarded Coulomb field.

E₃ is the radiation field.  For a point charge moving non-relativistically we have
E₃ = -(q/(4πε₀c²r'))a_⊥(t - r'/c).
If the observer is not located at the origin but at position r then the radiation field E(r,t) of a point charge moving non-relativistically is
E(r,t) = -(4πε₀)^-1[(q/(c²r'')]a_⊥(t - r''/c),
where
r'' = r - r'(t - |r - r'|/c),
i.e. the vector from the charge to the observer at the retarded time t -|r - r'|/c, and r' is the position of the charge at the retarded time.
For the radiation field we have  B = r''/(r''c) × E.

The energy flux associated with the fields of a point charge is calculated from the Poynting vector S. 
The total power radiated by a point charge moving non-relativistically is
P =∮_A S∙dA = ⅔e²a²/c³,
with e² = q²/(4πε₀).  This is the Larmor formula.

Dipole radiation field (non-relativistic)

The radiation field of an oscillating electric dipole with p = p₀cos(ωt) is
E_R(r,t) = -(1/(4πε₀c²r''))(d²p_⊥(t - r''/c)/dt²).
An electric dipole radiates energy at a rate P_rad = <(d²p/dt²)²>/(6πε₀c³). 
For an oscillating dipole the average total power radiated is <P> =  ω⁴p₀²/(12πε₀c³).
A magnetic dipole radiates energy at a rate P_rad = <(d²m/dt²)²>/(6πε₀c⁵).