A parallel-plate capacitor with plate area A and separation between the plates d, is charged by a constant current i. Consider a plane surface of area A/2 parallel to the plates and drawn symmetrically between the plates. Find the displacement current through this area.
i2
Suppose the charge on the capacitor at time t is Q. The electric field between the plates of the capacitor is E = Qϵ0A. The flux through the area considered is ϕE=Qϵ0A.A2=Q2ϵ0.
The displacement current is
id=ϵ0dϕEdt=ϵ0(12ϵ0)dQdt=i2.