Maxwell's equations

Maxwell's equations (SI units) are
E = ρ/ε0×E = -∂B/∂t,
B = 0,  ×B = μ0j + (1/c2)∂E/∂t,
or, in macroscopic form,
D = ρf×E = -∂B/∂t,
B = 0,  ×H = jf + ∂D/∂t.

In lih materials with D = εE and B =  μH and  ε, μ = constant we have
E = ρf/ε,  ×E = -∂B/∂t,
B = 0,  ×B = μjf + εμ∂E/∂t.

Maxwell's equations are linear equations and the principle of superposition holds.

B = 0 --> B = ×A.
×(E + ∂A/∂t) = 0 --> E + ∂A/∂t = -Φ.


Energy and momentum in electrodynamics

Poynting's theoremEj =  -(∂u/∂t) -∇∙S   is a statement of energy conservation.
u = (1/(2μ0))B2 + (ε0/2)E2  is the energy density and
S = (1/μ0)(E×B)  is the energy flux in the electromagnetic field.
We define the momentum density as g = S/c2.


The Lorentz gauge

If in electrodynamics we choose the Lorentz gauge defined through
A = -(1/c2)∂Φ/∂t,
then Φ and each Cartesian component of A satisfy the inhomogeneous wave equation.
2Φ - (1/c2)∂2Φ/∂t2 = -ρ/ε02A - (1/c2)∂2A/∂t2 = -μ0j.
Φ(r,t) = (4πε0)-1∫ρ(r', t')|retdr'/|r - r'|,  Ai(r,t) = (μ0/(4π))∫ji(r', t')|retdr'/|r - r'|,
with ρ(r', t')|ret = ρ(r', t - |r - r'|/c).

Choosing  A = 0  is called choosing the Coulomb gauge.