Question

The potential energy of the system of two identically charged spherical shell as shown in the figure is equal to (Assume the charge distribution to be uniform on surface)

• A. $\frac{q^2}{4\pi {ϵ}_0}(\frac{1}{R}+\frac{1}{r})$
• B. $\frac{q^2}{4\pi {ϵ}_0}\frac{1}{r}$
• C. $\frac{q^2}{4\pi {ϵ}_0}\frac{1}{(R+r)}$
• D. $\frac{q^2}{8\pi {ϵ}_0}(\frac{1}{R}+\frac{1}{r})$

Answer 1

The correct option is A $\frac{q^2}{4\pi {ϵ}_0}(\frac{1}{R}+\frac{1}{r})$

Each of the charged sphere will have potential energy due to its own charge or self energy.
$U_1=\frac{1}{8\pi {ϵ}_0}\frac{q^2}{r}$
The mutual potential energy stored in one sphere in the elastic field of the other is
$\frac{q_1{q}_2}{4\pi {ϵ}_0{r}_{12}}=U_2$
$\Rightarrow U_2=\frac{1}{4\pi {ϵ}_0}\frac{q^2}{R}$
Therefore, the P.E. stored in the system of spheres is
$U=2U_1+U_2=\frac{q^2}{4\pi {ϵ}_0}[\frac{1}{R}+\frac{1}{r}]$