Work, Energy, and Emf — Question 9

Question 9

Constant-voltage and reversible capacitor charging demand different source work

An initially uncharged capacitor C is charged to V0. Method A uses a constant ideal source V0 with a resistor. Method B raises the source voltage quasistatically to track the capacitor voltage, with arbitrarily small losses.

1. Find source work, stored energy and loss for method A.

2. Integrate source work for method B and compare the two methods.

Question 9 - Solution

Constant-voltage and reversible capacitor charging demand different source work

(a) Method and calculation

$$Q_f=CV_0,\quad W_A=V_0Q_f=CV_0^2.$$$$U_f=\tfrac12CV_0^2,\quad W_{{\mathrm{loss}},A}=W_A-U_f=\tfrac12CV_0^2.$$

(b) Interpretation and further calculation

$$W_B=\int_0^{Q_f}\frac qC\,dq=Q_f^2/(2C)=\tfrac12CV_0^2.$$$$W_B/W_A=1/2,\quad W_{{\mathrm{loss}},B}\to0.$$

Physical check

The work depends on the source voltage at each transferred charge, not just the final capacitor state.