Question

A large conducting plane has a surface charge density $1·0\times {10}^{-4}\mathrm{C}{\mathrm{m}}^{-2}$. Find the electrostatic energy stored in a cubical volume of edge 1⋅0 cm in front of the plane.

Answer 1

Given,
Surface charge density of the plane, $\sigma =1.0\times {10}^{-4}\mathrm{C}/{\mathrm{m}}^2$
Volume of the cube, $V=a^3={10}^{-6}{\mathrm{m}}^3$

Electric field near the charged conducting plane is given as,
$E=\frac{\sigma }{{\in }_0}...\left(\mathrm{i}\right)$

Energy density of electric filed,
$$\begin{gathered} u=\frac{1}{2}{\in }_0{E}^2 \\ \Rightarrow u=\frac{1}{2}{\in }_0{\left(\frac{\sigma }{{\in }_0}\right)}^2 \\ \Rightarrow u=\frac{1}{2}\frac{{\sigma }^2}{{\in }_0} \\ \Rightarrow u=\frac{1}{2}\times \frac{(1.0\times {10}^{-4}{)}^2}{8.85\times {10}^{-12}} \\ \Rightarrow u=0.056\times {10}^{4}j/m^3 \\ volume={10}^{-6}{m}^3 \\ \Rightarrow U=u\times V \\ \Rightarrow U=0.056\times {10}^4\times {10}^{-6} \\ \Rightarrow U=5.6\times {10}^{-4}J \end{gathered}$$