Calculating the Field from the Potential — Question 7

Question 7

A piecewise continuous potential determines the sign of a surface charge

Near the plane x=0, $V=V_0+a x$ for x<0 and $V=V_0+b x$ for x>0. The medium is vacuum and the only singular source is a surface charge at the plane.

1. Find the one-sided fields and the surface charge density using $E_x(0^+)-E_x(0^-)=\sigma/\epsilon_0$.

2. For a=300 V/m and b=-100 V/m find $\sigma/\epsilon_0$ and the electric force on a test charge away from the sheet on each side.

Question 7 - Solution

A piecewise continuous potential determines the sign of a surface charge

(a) Method and calculation

$$E_x(0^-)=-a,\quad E_x(0^+)=-b.$$$$\sigma=\epsilon_0(a-b).$$

(b) Interpretation and further calculation

$$\sigma/\epsilon_0=400\,\mathrm{V/m}>0.$$$$\mathbf F=q\mathbf E=\begin{cases}-300q\hat{\mathbf x}\,\mathrm{V/m}&x<0,\\+100q\hat{\mathbf x}\,\mathrm{V/m}&x>0.\end{cases}$$

Physical check

Potential remains continuous; the slope jump carries the surface-charge information. The ideal sheet value itself requires a separate force convention.