Gauss' Law — Question 7

Question 7

A thin pillbox measures the jump in normal field across a sheet

Use $\epsilon_0=8.854\times10^{-12}\,\mathrm{C^2/(N\,m^2)}$. A thin charge sheet lies in $z=0$. Just above it $\mathbf E_+=(4,0,30)$ N/C; just below it $\mathbf E_-=(4,0,-10)$ N/C. Use a pillbox with vanishing height and face area A.

1. Derive and evaluate the surface charge density.

2. Explain the role of the equal tangential components. Would replacing the upper x component by 7 N/C be consistent with a static electric field across this ideal sheet?

Question 7 - Solution

A thin pillbox measures the jump in normal field across a sheet

(a) Method and calculation

$$\Phi=A(E_{z,+}-E_{z,-}),\quad Q_{\mathrm{enc}}=\sigma A.$$$$\sigma=\epsilon_0(30-(-10))=40\epsilon_0=354.16\,\mathrm{pC/m^2}.$$

(b) Interpretation and further calculation

$$\oint\mathbf E\cdot d\boldsymbol\ell=0\Rightarrow(E_{x,+}-E_{x,-})\ell=0\quad\text{as loop height}\to0.$$$$E_{x,+}=E_{x,-};\quad7-4=3\ne0.$$The proposed tangential jump would violate electrostatic circulation, even though the normal-flux calculation alone would still give the same sheet charge.

Physical check

The pillbox side contribution vanishes with its height if the fields remain finite near the sheet.