What is displacement current? Using a charging capacitor, explain why it is needed and show that it equals the conduction current. Write the Ampere-Maxwell law.

1. 1 · Define. Displacement current is the current due to a changing electric field: i_d = ε₀ dΦ_E/dt.
2. 2 · State the puzzle. For a loop round the wire, a flat surface gives a current i, but a surface passing between the plates gives zero, so Ampere's law contradicts itself.
3. 3 · Draw the set-up. [Charging capacitor with wires; a circular loop round one wire; one flat surface and one pot-shaped surface passing between the plates]
4. 4 · Show that i_d = i. E = Q/(ε₀A), Φ_E = Q/ε₀, so ε₀ dΦ_E/dt = dQ/dt = i.
5. 5 · Write the law. ∮ B·dl = μ₀(i_c + i_d) = μ₀i_c + μ₀ε₀ dΦ_E/dt.
6. 6 · Close in one line. So, a changing electric field produces a magnetic field, and the current is continuous everywhere.

Scoring tip: Write the flux step Φ_E = Q/ε₀ clearly; it links the field to the current.