Maxwell's Equations & EM Waves
Four equations, and light falls out.
Lesson at a glanceShow the key points
- Maxwell's equations: Gauss (E), Gauss (B), Faraday, Ampère–Maxwell.
- Plane wave: E_y = E₀ sin ω(t − x/c), B_z = B₀ sin ω(t − x/c) (40.4).
- E₀ = cB₀ (40.5); c = 1/√(μ₀ε₀) (40.6): light is electromagnetic.
Four equations sum up all of electricity and magnetism. In empty space they have a remarkable solution: a wave of electric and magnetic fields that travels at exactly the speed of light.
Maxwell’s equations
| Law | Equation | Says |
|---|---|---|
| Gauss (electricity) | ∮E⃗·dS⃗ = q/ε₀ | charges are sources of E |
| Gauss (magnetism) | ∮B⃗·dS⃗ = 0 | no magnetic poles |
| Faraday | ∮E⃗·dl⃗ = −dΦ_B/dt | changing B makes E |
| Ampère–Maxwell | ∮B⃗·dl⃗ = μ₀i + μ₀ε₀ dΦ_E/dt | currents and changing E make B |
In vacuum there are no charges or currents, leaving
A plane wave
Try a wave travelling along x, with E along y and B along z:
Faraday’s law on a rectangle in the x–y plane (sides at x₁ and x₂, height l): the circulation of E is E₀l[sin ω(t − x₂/c) − sin ω(t − x₁/c)], and −dΦB/dt works out to cB₀l times the same bracket. They agree if
Ampère–Maxwell law on a rectangle in the x–z plane does the same job for B and gives B₀ = μ₀ε₀cE₀. Together with E₀ = cB₀:
With μ₀ and ε₀ measured in purely electrical and magnetic experiments, this gave c = 2.99793 × 10⁸ m/s: the measured speed of light. That was the proof that light is an electromagnetic wave. (Today c and μ₀ are fixed by definition and ε₀ is defined through this equation.)
A plane wave travelling along x has its electric field along y with a maximum of 450 N/C. Find the maximum magnetic field and its direction.
A wave along x has wavelength 3.0 cm and its electric field along y with amplitude 60 V/m. Write E and B as functions of x and t.
- 1Maxwell's four equations describe all of electromagnetism.
- 2In vacuum they allow plane waves with E ⊥ B ⊥ direction of travel, in phase.
- 3E₀ = cB₀ (40.5); c = 1/√(μ₀ε₀) (40.6): light is an electromagnetic wave.
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