• Tags: displacement-current, Ampere, capacitor
• Difficulty: Moderate

Ampere's circuital law (original): line integral of B.dl = mu_0I_enclosed works for steady currents but fails for time-varying fields. Consider a charging capacitor: a wire carries current I to the plates, but between the plates there is no conduction current. Yet the magnetic field must be continuous. Maxwell's resolution: a changing electric field between the plates acts as a "displacement current" I_d = epsilon_0 * d(Phi_E)/dt. Between parallel plates: E = Q/(epsilon_0A) = sigma/epsilon_0, so Phi_E = EA = Q/epsilon_0. Thus I_d = epsilon_0 * d(Q/epsilon_0)/dt = dQ/dt = I_c (the conduction current in the wire). The modified Ampere-Maxwell law: line integral of B.dl = mu_0(I_c + I_d). Displacement current exists wherever the electric field changes with time, even in empty space. It is not a real flow of charges but produces the same magnetic effects.