Electric Current — Question 9

Question 9

A sinusoidal transport current stores charge in a supplied capacitor

A current $I(t)=I_a\cos\omega t$ enters plate A of an ideal capacitor C, with $q_A(0)=q_0$. Take $I_a,\omega,C>0$.

1. Find the plate charge and voltage $V_A-V_B$.

2. Find the condition for the plate charge to reverse sign and the net transfer in one period.

Question 9 - Solution

A sinusoidal transport current stores charge in a supplied capacitor

(a) Method and calculation

$$q_A(t)=q_0+\int_0^t I_a\cos\omega t\prime\,dt\prime=q_0+(I_a/\omega)\sin\omega t.$$$$V_A-V_B=q_0/C+[I_a/(\omega C)]\sin\omega t.$$

(b) Interpretation and further calculation

$$q_{\min}=q_0-I_a/\omega,\quad q_{\max}=q_0+I_a/\omega.$$$$\text{Sign reversal: }|q_0|<I_a/\omega,\qquad\Delta Q_{\mathrm{period}}=0.$$

Physical check

Equality gives a momentary zero without a sign change; the imposed current requires no resistance model.