Capacitor with a Dielectric — Question 2

Question 2

A voltage source supplies twice the added stored energy during dielectric insertion

A capacitor remains attached to an ideal voltage source V while a linear lossless dielectric of relative permittivity 4 fills the gap. Initially C=C0 and $U_0=C_0V^2/2$. Insertion is quasistatic.

1. Find charge supplied and change in stored energy.

2. Find source work and external mechanical work.

Question 2 - Solution

A voltage source supplies twice the added stored energy during dielectric insertion

(a) Method and calculation

$$C_f=4C_0,\quad\Delta Q=(C_f-C_0)V=3C_0V.$$$$\Delta U=\tfrac12(C_f-C_0)V^2=3U_0.$$

(b) Interpretation and further calculation

$$W_{\mathrm{source}}=V\Delta Q=3C_0V^2=6U_0.$$$$W_{\mathrm{ext}}=\Delta U-W_{\mathrm{source}}=-3U_0.$$

Physical check

At fixed voltage, rising stored energy does not imply a resisting insertion force; the source must be included in the energy balance.