EM wave in dielectric from Maxwell's equations

Maxwell's equations

Four basic laws of electromagnetism: Gauss's law for E, Gauss's law for B, Faraday's law and the Ampere-Maxwell law.

For a non-conducting dielectric (ρ = 0, J = 0, permittivity ε = ε₀ε_r, permeability μ = μ₀μ_r):

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

Wave equation for E

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

2. Use ∇×(∇×E) = ∇(∇·E) − ∇²E = −∇²E, and put (4) on the right side:

   −∇²E = −με ∂²E/∂t²

3. So ∇²E = με ∂²E/∂t².

Wave equation for B

Take the curl of (4) and use (3) and ∇·B = 0 in the same way:

∇²B = με ∂²B/∂t²

Plane-wave form

For a wave travelling along x, with E along y and B along z:

∂²E_y/∂x² = με ∂²E_y/∂t², ∂²B_z/∂x² = με ∂²B_z/∂t²

Comparing with the standard wave equation ∂²y/∂x² = (1/v²) ∂²y/∂t²:

v = 1/√(με)

Solutions: E_y = E₀ sin(kx − ωt), B_z = B₀ sin(kx − ωt), with E₀/B₀ = v.

Speed less than c

c = 1/√(μ₀ε₀)
v = 1/√(μ₀μ_r ε₀ε_r) = c/√(μ_r ε_r) = c/n,   n = √(μ_r ε_r)

For a dielectric ε_r > 1 and μ_r ≈ 1, so n > 1. Therefore v < c: the wave is slower in the medium (e.g. glass n ≈ 1.5, so v ≈ 2×10⁸ m/s).