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 = ρ_f,  ∇×E = -∂B/∂t,
∇·B = 0,  ∇×H = j_f + ∂D/∂t.

In lih materials with D = εE and B =  μH and  ε, μ = constant we have
∇·E = ρ_f/ε,  ∇×E = -∂B/∂t,
∇·B = 0,  ∇×B = μj_f + 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.

A and Φ are not unique.
A --> A + ∇ψ,  Φ --> Φ - ∂ψ/∂t, with ψ an arbitrary scalar field, is called a gauge transformation.
E and B are invariant under such a transformation.

Energy and momentum in electrodynamics

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.

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/ε₀,  ∇²A - (1/c²)∂²A/∂t² = -μ₀j.
Φ(r,t) = (4πε₀)^-1∫ρ(r', t')|_retdr'/|r - r'|,  A_i(r,t) = (μ₀/(4π))∫j_i(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.