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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.
Solution
Let the alternating emf charging the plates of capacitor be E = Eo sin ωt
Charge on the capacitor, q = EC = CEo 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.
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