Applying Gauss' Law: Planar Symmetry — Question 2

Question 2

Unequal opposing sheets leave a residual exterior field

Two infinite nonconducting sheets lie at $x=0$ and $x=d>0$, with fixed densities $+\sigma$ and $-2\sigma$, respectively, where $\sigma>0$. No external uniform field is added.

1. Use superposition to find $E_x$ in all three regions.

2. Find the difference between the far-right and far-left fields and compare it with total sheet charge per area divided by $\epsilon_0$.

Question 2 - Solution

Unequal opposing sheets leave a residual exterior field

(a) Method and calculation

$$E_x=\frac\sigma{2\epsilon_0}\operatorname{sgn}(x)-\frac{2\sigma}{2\epsilon_0}\operatorname{sgn}(x-d).$$$$E_x=\begin{cases}\sigma/(2\epsilon_0),&x<0,\\3\sigma/(2\epsilon_0),&0<x<d,\\-\sigma/(2\epsilon_0),&x>d.\end{cases}$$

(b) Interpretation and further calculation

$$E_x(+\infty)-E_x(-\infty)=-\sigma/\epsilon_0.$$$$\Sigma=\sigma-2\sigma=-\sigma\Rightarrow\Delta E_x=\Sigma/\epsilon_0.$$The fields add in the gap; the unequal charges prevent cancellation in the exterior.

Physical check

Each field jump is fixed by its own sheet: $+\sigma/\epsilon_0$ at 0 and $-2\sigma/\epsilon_0$ at d.