Nature of electromagnetic waves

The E and B fields of an electromagnetic wave are at right angles to each other, and both are at right angles to the direction of travel: the wave is transverse. The capacitor hints at this: E between the plates is perpendicular to them, and the B it produces runs in circles parallel to them.

For a plane wave travelling along z with E along x and B along y: Ex = E₀ sin(kz − ωt) and By = B₀ sin(kz − ωt). Both vary sinusoidally with position and time and are in phase.

Here k = 2π/λ is the size of the wave vector (propagation vector) k, whose direction is the direction of travel, and ω is the angular frequency. The wave moves with speed ω/k.

Maxwell's equations give ω = ck with c = 1/√(μ₀ε₀). Written with ν = ω/2π and λ = 2π/k this is νλ = c.

The amplitudes are related by B₀ = E₀/c. Because c is so large, the magnetic field in tesla is a tiny number next to the electric field in V/m.

The wave travels in the direction of E × B. Given two of the three directions, this fixes the third.

Example 8.1: a 25 MHz wave travels along x, and at some point E = 6.3 ĵ V/m. Then B = E/c = 6.3/(3 × 10⁸) = 2.1 × 10⁻⁸ T, and since ĵ × k̂ = î, B = 2.1 × 10⁻⁸ k̂ T.

Example 8.2: By = (2 × 10⁻⁷ T) sin(0.5 × 10³x + 1.5 × 10¹¹t). Then λ = 2π/(0.5 × 10³) m = 1.26 cm and ν = (1.5 × 10¹¹)/2π = 23.9 GHz. E₀ = B₀c = 60 V/m, and E lies along z: Ez = 60 sin(0.5 × 10³x + 1.5 × 10¹¹t) V/m.