Capacitance — Question 7

Question 7

A mechanical change alters voltage differently under two constraints

A linear capacitor is initially C0 at voltage V0. A slow geometry change makes its capacitance $C_f=3C_0/2$. Compare an isolated capacitor with one kept at V0 by an ideal source.

1. Find final voltage and charge in the isolated case.

2. Find final voltage and charge in the connected case, including the charge supplied to the positive plate.

Question 7 - Solution

A mechanical change alters voltage differently under two constraints

(a) Method and calculation

$$Q_i=C_0V_0,\quad Q_f=Q_i\quad\text{isolated}.$$$$V_f=Q_i/C_f=\tfrac23V_0.$$

(b) Interpretation and further calculation

$$V_f=V_0,\quad Q_f=C_fV_0=\tfrac32C_0V_0.$$$$\Delta Q=Q_f-Q_i=\tfrac12C_0V_0.$$

Physical check

Charge conservation applies to disconnected terminals; a voltage source can exchange charge with connected terminals.