EM wave equations in vacuum

Maxwell's equations in free space (ρ = 0, J = 0)

1. ∇·E = 0
2. ∇·B = 0
3. ∇×E = −∂B/∂t
4. ∇×B = μ₀ε₀ ∂E/∂t

Wave equations

1. Take the curl of (3): ∇×(∇×E) = −∂(∇×B)/∂t.

2. Left side: ∇×(∇×E) = ∇(∇·E) − ∇²E = −∇²E (using ∇·E = 0).

3. Right side: use (4): −∂/∂t (μ₀ε₀ ∂E/∂t) = −μ₀ε₀ ∂²E/∂t².

4. So:

   ∇²E = μ₀ε₀ ∂²E/∂t²

Taking the curl of (4) and using (3) gives in the same way:

∇²B = μ₀ε₀ ∂²B/∂t²

These have the form ∇²ψ = (1/c²)∂²ψ/∂t², so the speed is c = 1/√(μ₀ε₀) = 3×10⁸ m/s.

Plane wave solutions

Take a wave travelling along +x. The fields depend on x and t only.

∂²E/∂x² = (1/c²) ∂²E/∂t²
E = E₀ sin(kx − ωt)      (along y)
B = B₀ sin(kx − ωt)      (along z)
k = 2π/λ,  ω = 2πf,  ω/k = c

From ∇·E = 0 and ∇·B = 0, E_x = B_x = 0, so the wave is transverse. From Faraday's law, ∂E_y/∂x = −∂B_z/∂t, which gives kE₀ = ωB₀, so E₀/B₀ = c. E, B and the direction of propagation are mutually perpendicular, with E and B in phase.