Question

Two uniformly charged large plane sheets $S_{1}$ and $S_{2}$ having charge densities $\sigma_{1}$ and $\sigma_{2}\left(\sigma_{1}>\sigma_{2}\right)$ are placed at a distance $d$ parallel to each other. A charge $q_{0}$ is moved along a line of length $a(a < d)$ at an angle $45^{\circ}$ with the normal to $S _{1}$. Calculate the work done by the electric field, in joule $\left( q _{0}=2 C\right.$, $\sigma_{1}=3 \varepsilon_{0} C / m ^{2}, \sigma_{2}=\varepsilon_{0} C / m ^{2}, a =\sqrt{2}$ metre).

Answer 1

$\overrightarrow{ E }=\frac{\sigma_{1}}{2 \varepsilon_{0}}-\frac{\sigma_{2}}{2 \varepsilon_{0}}=\frac{\sigma_{1}-\sigma_{2}}{2 \varepsilon_{0}}$
$W = q _{0} \overrightarrow{ E } \cdot \overrightarrow{ dr }= q _{0} E a \cos 45$
$W = q _{0}\left(\frac{\sigma_{1}-\sigma_{2}}{2 \varepsilon_{0}}\right) \frac{ a }{\sqrt{2}}$
$ W =\frac{ q \left(\sigma_{1}-\sigma_{2}\right) a }{2 \sqrt{2} \varepsilon_{0}}$