Considering the case of a parallel plate capacitor being charged, show how one is required to generalize Ampere's circuital law to include the term due to displacement current.
Solution
Take an Amperian loop around the wire feeding a charging capacitor. For a flat surface pierced by the wire, Ampere's law gives ∮B⋅dl=μ0ic. But for a bowl-shaped surface that passes between the plates, no conduction current crosses it, which would give ∮B⋅dl=0 --- a contradiction, since B cannot depend on the surface chosen. Maxwell resolved this by noting the changing electric flux between the plates carries a displacement current id=ε0dΦE/dt equal to ic. Adding it gives the generalized (Ampere--Maxwell) law:
