Applying Gauss' Law: Planar Symmetry — Question 3

Question 3

Desired piecewise fields determine three sheet charges

Three infinite insulating sheets lie at $x=-d$, 0 and $+d$. The desired field is zero outside the stack, $+E_0\hat{\mathbf x}$ in $-d<x<0$ and $+2E_0\hat{\mathbf x}$ in $0<x<d$, with $E_0>0$.

1. Find the three surface densities from the field jumps.

2. Check the total charge per area and verify one of the interior fields by adding individual sheet contributions.

Question 3 - Solution

Desired piecewise fields determine three sheet charges

(a) Method and calculation

$$\sigma_i=\epsilon_0(E_{x,\mathrm{right}}-E_{x,\mathrm{left}}).$$$$\sigma_{-d}=\epsilon_0E_0,\quad\sigma_0=\epsilon_0E_0,\quad\sigma_{+d}=-2\epsilon_0E_0.$$

(b) Interpretation and further calculation

$$\Sigma=\sigma_{-d}+\sigma_0+\sigma_{+d}=0.$$$$-d<x<0:\quad E_x=\frac{\sigma_{-d}-\sigma_0-\sigma_{+d}}{2\epsilon_0}=E_0.$$Zero net sheet charge allows zero exterior field, while distinct interior jumps create different gap fields.

Physical check

The field data also fix the otherwise possible added uniform field through the specified zero exterior values.