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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.
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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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