Question

Two parallel uniformly charged infinite plane sheets, ${}^{\prime }{1}^{\prime }$ and ${}^{\prime }2$, have charge densities $+\sigma$ and $-2\sigma$ respectively. Give the magnitude and direction of the net electric field at a point.
(i) In between the two sheets and
(ii) Outside near the sheet ${}^{\prime }{1}^{\prime }$.

Answer 1

(i) Let ${\overrightarrow{E}}_1$ and ${\overrightarrow{E}}_2$ be the electric field intensity at the point $P_1$, between the plates. So, $|E_{P_1}|=|E_1|+|E_2|$
$=\frac{\sigma }{{ϵ}_0}+\frac{2\sigma }{{ϵ}_0}$
$=\frac{3\sigma }{{ϵ}_0}$ (directed towards sheet $2$)
${\overrightarrow{E}}_{P_1}=\frac{3\sigma }{{ϵ}_0}(-\hat{j})=-\frac{3\sigma }{{ϵ}_0}\hat{j}$
(ii) Outside near the sheet ${}^{\prime }{1}^{\prime },|{\overrightarrow{E}}_{P_2}|={\overrightarrow{E}}_2-{\overrightarrow{E}}_1$
$=\frac{2\sigma }{2{ϵ}_0}-\frac{\sigma }{2{ϵ}_0}=\frac{\sigma }{2{ϵ}_0}$ (directed towards sheet $2$)
${\overrightarrow{E}}_{P_2}=\frac{\sigma }{2{ϵ}_0}(-\hat{j})=-\frac{\sigma }{{ϵ}_0}\hat{j}$.