§ 12. Electromagnetic Light Wave (E, B & Poynting Vector) (Grade 11)
Mutually perpendicular fields E ⊥ B, phase wavefront, and light polarization. Light is a transverse electromagnetic wave: within every plane of the light wavefront, vectors E and B oscillate in phase perpendicular to the direction of propagation.
Theory Summary & Scientific Mechanism
According to James Clerk Maxwell’s electrodynamics, a time-varying magnetic field induces a solenoidal electric field (∇ × E = -(∂ B)/(∂ t)), while a time-varying electric displacement current generates a magnetic field (∇ × B = μ₀ε₀ (∂ E)/(∂ t)). Together they sustain a self-propagating transverse electromagnetic wave traveling at the speed of light c = (1)/(√(ε₀μ₀)) ≈ 3× 10⁸ m/s.
In a plane travelling wave, the electric field E(x,t) = E₀sin (kx - ω t) and magnetic field B(x,t) = B₀sin (kx - ω t) oscillate strictly in phase, are mutually orthogonal (E ⊥ B), and form a right-handed triad with the wavevector k, with amplitudes linked by E₀ = c B₀.
Directional energy flux is given by the Poynting vector S = (1)/(μ₀)[E × B], and the instantaneous electric energy density w_E = (1)/(2)ε₀ E² and magnetic energy density w_B = (B²)/(2μ₀) are equal at every point and instant (w_E = w_B).
Everyday Analogy
Flashlight beams, Wi-Fi signals, radio, and X-rays are all electromagnetic waves! They need no air or water to cross empty space: a changing electric field E generates a magnetic field B, which in turn regenerates E.
Interactive Question & Derivation
How are the electric field vector E and magnetic field vector B oriented relative to the light wavefront plane and propagation vector k in a free light wave?
- A: Vectors E and B point along the ray k (longitudinal wave)
- B: Vectors E and B lie in the plane of the light wavefront, are mutually perpendicular (E⊥B), and oscillate in phase, forming a right-handed triad with k
- C: Oscillations of E and B are shifted in phase by π/2
Study Hint: How are vectors E and B oriented relative to each other and to propagation vector c, and do they oscillate in phase in vacuum?
Proof & Explanation: Maxwell's equations show that light waves are strictly transverse: E and B always lie in the wavefront plane (E⊥k, B⊥k), are mutually orthogonal (E⊥B), and reach their maxima and zeros simultaneously (in phase). Their cross product S = (1)/(μ₀)[E×B] gives the direction and density of light energy flux.
Key Equations & 3D Simulation Parameters
w_E = (εε₀ E²)/(2) = w_B = (B²)/(2μ₀), S = (EB)/(μ₀) = c· wₜot
- Wavefront spacing λ: 1.8 … 4.8 m (default: 3.2 m)
- Field amplitude E₀: 0.8 … 2.2 V/m (default: 1.6 V/m)
- Medium refractive index n: 1 … 2.2 (default: 1)
- Polarization ellipticity ε: 0 … 1 (default: 0)