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 $(q_0=2C$, ${\sigma }_1=3{\epsilon }_{0}C/m^2,{\sigma }_2={\epsilon }_{0}C/m^2,a=\sqrt{2}$ metre).

Answer 1

Answer: 2.00

$\overrightarrow{E}=\frac{{\sigma }_1}{2{\epsilon }_0}-\frac{{\sigma }_2}{2{\epsilon }_0}=\frac{{\sigma }_1-{\sigma }_2}{2{\epsilon }_0}$
$W=q_0\overrightarrow{E}\cdot \overrightarrow{dr}=q_{0}Ea\cos 45$
$W=q_0\left(\frac{{\sigma }_1-{\sigma }_2}{2{\epsilon }_0}\right)\frac{a}{\sqrt{2}}$
$W=\frac{q\left({\sigma }_1-{\sigma }_2\right)a}{2\sqrt{2}{\epsilon }_0}$