The problem with Ampere's law
Ampere's circuital law says ∮B·dl = μ₀i, where i is the current through any surface bounded by the loop. Consider a capacitor being charged. Take a loop around the wire. A flat surface bounded by the loop is crossed by the conduction current i, but a balloon-shaped surface bounded by the same loop that passes between the plates is crossed by no current. The law gives two different answers, so something is missing.
Maxwell's displacement current
Between the plates the electric field is changing as charge builds up. Maxwell proposed that a changing electric flux acts like a current, called the displacement current: i_d = ε₀ dΦ_E/dt The Ampere–Maxwell law becomes: ∮B·dl = μ₀(i_c + ε₀ dΦ_E/dt) Between the plates of a charging capacitor, the displacement current is exactly equal to the conduction current in the leads, so current is continuous. For a capacitor, i_d = C dV/dt.
Consequences: electromagnetic waves
A changing magnetic field produces an electric field (Faraday), and a changing electric field produces a magnetic field (Maxwell). Oscillating charges therefore send out electromagnetic waves in which E and B oscillate together. Properties:
- E and B are perpendicular to each other and to the direction of travel (transverse waves); the wave travels along E × B.
- E and B oscillate in phase, and E₀/B₀ = c.
- In vacuum, speed c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s; in a medium, v = 1/√(με).
- They need no material medium and carry energy and momentum.
Hertz first produced and detected such waves in the laboratory.
Key points
- i_d = ε₀ dΦ_E/dt; it exists wherever the electric field changes.
- In a charging capacitor, i_d (between plates) = i_c (in wires).
- E ⊥ B ⊥ direction of propagation.
- E₀ = cB₀.
Exam tips
- For "find B₀", divide E₀ by 3 × 10⁸.
- Use the right-hand rule E × B to find a missing direction.