Propagation of Electromagnetic Waves

Propagation of Plane Electromagnetic Waves in Infinite Media

• Electromagnetic waves can transport energy.
• Plane waves in infinite media are the simplest solutions.
• Consider a material with: Β = μΗ, D = E, J = p = 0.
• Maxwell equations, in vector form:
  • ∇ × E = - (1/c) (dB/dt)
  • ∇ × B = μ₀(J + (1/c)(dE/dt))
  • ∇ ⋅ E = ρ/ε₀
  • ∇ ⋅ B = 0

Solving the Wave Equation in Infinite Media

• The curl of the curl equations, e.g., Faraday's Law, results in:

  • ∇ × (∇ × E) = μ₀ ε₀ (d²E/dt²) / c²

• The divergence of E is zero, so the equation becomes

  • μ₀ ε₀ (d²E/dt²) = ∇²(E) / c²

• Identical manipulations with Ampère's law gives a similar equation for B

  • μ₀ ε₀ (d²B/dt²) = ∇²(B) / c²

• The wave equation has complex traveling wave solutions:

  • Uk(x, t) = ei(kx - wt) where w = vk and k is a real vector

• Derivatives of the complex function:

  • ∇Uk = ikUk
  • ∇²Uk = - k²Uk
  • dUk/dt = -iwUk
  • d²Uk/dt² = -w²Uk

• Solutions represent waves traveling in the direction of k.

• Surfaces of constant phase are planes perpendicular to k.

• Velocity of the wave is v = ω/k.

Electromagnetic Fields in Terms of Plane Waves

• Electric and magnetic fields can be represented as:
  • E(x, t) = E₀e^(i(kx - wt))
  • B(x, t) = B₀e^(i(kx - wt))
• The true fields are the real parts of these complex expressions.
• Maxwell's equations require:
  • k ⋅ B₀ = 0
  • k ⋅ E₀ = 0
• This means that B₀ and E₀ are perpendicular to k (and thus to the direction of wave propagation).

Conditions Imposed by Maxwell's Equations

• Further conditions on the amplitudes of the fields from other Maxwell equations.

• The electric field amplitude is related to the magnetic field:

  • E₀ = (k x B₀) / (iωε₀) or Eo = n x Bo/sqrt(μ₀ε₀)

• The magnetic field is also expressed in terms of the electric field:

  • B₀ = (k × E₀) / (iωμ₀) or Bo = √(μ₀ε₀) n x Eo

• Electromagnetic waves are transverse waves (E, B, and k are mutually orthogonal).

• Time-averaged energy density () and Poynting vector (S) are calculated using amplitudes.