Question

A capacitor, made of two parallel plates each of plate area A and separation d, is being charged by an external ac source. Show that the displacement current inside the capacitor is the same as the current charging the capacitor. 

Answer 1

Let the alternating emf charging the plates of capacitor be E = E_o sin ωt

Charge on the capacitor, q = EC = CE_o sin ωt

Instantaneous Current is I.

`I = (dq)/dt =d/(dt)(CE_0 sin ωt) = ω CE_0 cosωt =I_0cosω t (\text { where} I_0 = ωCE_0)`

Displacement current, `I_D = epsi_0 (dφ_E)/dt = epsi_0A(d(E))/dt = epsi_0Ad/(dt)(q/(epsi_0A)) = epsi_0A d/dt ((CE_0 sinωt)/(epsi_0A))`

`=d/(dt)(CE_0 sin ωt) =ωCE_0cosωt = I_0cosωt `

Thus, the displacement current inside the capacitor is the same as the current charging the capacitor.