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.
Poynting's theorem, E·j = -(∂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.
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/ε0, ∇2A - (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.