Electromagnetic Waves — Electromagnetism

Electromagnetism / Electromagnetic Waves

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Electromagnetic waves — the rules

Geometry. E, B and the direction of travel are mutually perpendicular — the wave is transverse. The wave moves along E × B: point your fingers from E to B, the thumb rides the wave.

The lock-step ratio. E and B oscillate in phase, and at every instant E/B = c. So B₀ = E₀/c — tiny in teslas, but never negligible in energy.

Speed. c = 1/√μ₀ε₀ ≈ 3×10⁸ m/s in vacuum. In a medium v = 1/√με, giving refractive index n = √μ_rε_r.

Energy. The electric and magnetic halves carry equal shares: ⟨u_E⟩ = ⟨u_B⟩. Total average density ⟨u⟩ = ½ε₀E₀², and intensity I = ⟨u⟩c.

Momentum. Absorbed energy U delivers momentum p = U/c. Radiation pressure: I/c if absorbed, 2I/c if perfectly reflected — the bounce doubles the kick.

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EM Lab · all topics

§01

Displacement current — Maxwell's missing term

Charge a capacitor and follow the current: it flows down the wire, stops dead at the plate… yet a compass needle near the gap still deflects. Ampère's law seemed to break — until Maxwell pointed at the gap. The E-field there is growing, and a changing electric flux acts exactly like a current: i_d = ε₀ dΦ_E/dt. It picks up precisely where the conduction current stops, keeping the total continuous — and it makes a real, measurable magnetic field.

THE CURRENT THAT JUMPS THE GAP · plate charge 0%

charging current i_c{{ d1Ilab }}

∮ B·dl = μ₀ ( i_c + ε₀ dΦ_Edt )

in the wire: i_c

in the gap: i_d

They are always equal. The total current i_c + i_d is continuous around the whole circuit — Ampère's law works again, no matter which surface you stretch over the loop.

Why this one term changed everything

Faraday already knew a changing B makes E. Maxwell's term is the mirror image — a changing E makes B. Put the two together and neither field needs wires or charges any more: they can bootstrap each other through empty space. That bootstrap is an electromagnetic wave.

THE FIELD i_d MAKES · Amperian loop of radius r around the axis (plate radius R = 5 cm)

loop radius r{{ d1rlab }}

B on the loop · {{ d1where }}

inside: B = μ₀ i_d r / 2πR² · outside: B = μ₀ i_d / 2πr — the gap behaves exactly like a fat wire carrying i_d.

§02

The wave itself — two fields, one ride

Here is the shape Maxwell's equations demand: E (violet) oscillating in one plane, B (teal) in the perpendicular plane, both transverse to the travel direction, both in phase — they peak together, vanish together. Their sizes are locked at every instant: E/B = c. Nothing material moves; the pattern of fields does, at 3×10⁸ m/s, medium or no medium.

E ⊥ B ⊥ v · IN PHASE · TRAVELLING →

wavelength (visual){{ w2lamlab }}

amplitude E₀{{ w2E0lab }}

c = 1√μ₀ε₀ , B₀ = E₀c

peak E₀

peak B₀ = E₀/c

avg energy density ⟨u⟩

intensity I = ⟨u⟩c

B looks tiny — it isn't. B₀ is a factor c smaller in SI units, yet u_B = B²/2μ₀ carries exactly half the energy. Never drop the magnetic half.

Which way does it go?

Along E × B. If E points along y and B along z, the wave runs along y × z = x. Exam favourite: give E's direction and the travel direction, ask for B — cross-product backwards.