Applying Gauss' Law: Planar Symmetry — Question 8

Question 8

Forces between charged sheets use only the field of the other sheet

Two infinite thin insulating sheets with fixed densities $\sigma_1$ and $\sigma_2$ lie at $x=0$ and $x=d>0$. Ignore their mechanical supports. To compute force on a sheet, exclude its own field.

1. Derive the signed force per area on each sheet and check their sum.

2. For $\sigma_1=\sigma$ and $\sigma_2=-2\sigma$, find the attraction pressure. Explain why multiplying the second sheet density by the total gap field gives an incorrect result.

Question 8 - Solution

Forces between charged sheets use only the field of the other sheet

(a) Method and calculation

$$E_{1\to2}=\sigma_1/(2\epsilon_0),\quad E_{2\to1}=-\sigma_2/(2\epsilon_0).$$$$f_{2x}=\frac{\sigma_1\sigma_2}{2\epsilon_0},\quad f_{1x}=-\frac{\sigma_1\sigma_2}{2\epsilon_0},\quad f_{1x}+f_{2x}=0.$$

(b) Interpretation and further calculation

$$\sigma_1=\sigma,\ \sigma_2=-2\sigma:\quad f_{2x}=-\sigma^2/\epsilon_0,\quad P=\sigma^2/\epsilon_0.$$$$\sigma_2E_{\mathrm{gap}}=(-2\sigma)\frac{3\sigma}{2\epsilon_0}=-3\sigma^2/\epsilon_0\ne f_{2x}.$$The gap field includes the second sheet self-field, which must not be used as an external force on that same sheet.

Physical check

The force magnitude is independent of d in the ideal infinite-plane approximation; opposite signs attract.