Energy Stored in an Electric Field — Question 3

Question 3

At fixed voltage the source energy is essential to the force calculation

A vacuum plate capacitor of area A is held at fixed voltage V while its gap increases slowly from d to 2d. Let $U_i=\epsilon_0AV^2/(2d)$. Neglect fringing.

1. Find the change in stored energy and the work done by the voltage source.

2. Find the required external mechanical work and explain why differentiating U at fixed V with a minus sign alone gives the wrong force.

Question 3 - Solution

At fixed voltage the source energy is essential to the force calculation

(a) Method and calculation

$$U_f=U_i/2\Rightarrow\Delta U=-U_i/2.$$$$W_{\mathrm{source}}=V\Delta Q=V^2\Delta C=-U_i.$$

(b) Interpretation and further calculation

$$\Delta U=W_{\mathrm{source}}+W_{\mathrm{ext}}\Rightarrow W_{\mathrm{ext}}=+U_i/2.$$$$F_d=+\frac12V^2\frac{dC}{dd}=-\frac{\epsilon_0AV^2}{2d^2}.$$

Physical check

The source absorbs more energy than the field loses; the difference comes from the external agent separating the attracting plates.