Question

A metal sphere of radius R is charged to a potential V. (a) Find the electrostatic energy stored in the electric field within a concentric sphere of radius 2 R. (b) Show that the electrostatic field energy stored outside the sphere of radius 2 R equals that stored within it.

Answer 1

Potential of the inner metallic sphere is given by
$V=\frac{1}{4\mathrm{\pi }{\in }_0}\frac{q}{R}$

Capacitance of the capacitor formed by two concentric spheres of radii R and 2R is given by
$$\begin{gathered} C=4\pi {\in }_0\frac{R\times 2R}{2R-R} \\ \Rightarrow C=4\pi {\in }_0\times 2R \end{gathered}$$

Potential of the outer sphere is given by
$\mathit{}{V}_2=\frac{1}{4\mathrm{\pi }{\in }_0}\frac{q}{2R}=\frac{V}{2}$

Electrostatic energy stored outside the sphere is given by
$$\begin{gathered} E=\frac{1}{2}C{\left(V-V_2\right)}^2 \\ \Rightarrow E=\frac{1}{2}\times 4\mathrm{\pi }{\in }_0\times 2R\times \frac{V^2}{4} \\ \Rightarrow E=\mathrm{\pi }{\in }_0\times R{V}^2 \end{gathered}$$