## 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.

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 theoremE·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/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.