Question

A conducting sphere of radius $a$ has a charge $Q$ on it. It is enclosed by a neutral concentric spherical shell having an inner radius $2a$ and outer radius $3a$. Find the electrostatic energy of the system.

• A. $\frac{5}{12}\frac{Q^2}{4\pi {ϵ}_{0}a}$
• B. $\frac{11}{12}\frac{Q^2}{4\pi {ϵ}_{0}a}$
• C. $\frac{1}{2}\frac{Q^2}{4\pi {ϵ}_{0}a}$
• D. $\frac{1}{12}\frac{Q^2}{4\pi {ϵ}_{0}a}$

Answer 1

The correct option is A $\frac{5}{12}\frac{Q^2}{4\pi {ϵ}_{0}a}$

Radius $=a$

charge $=Q$

radius $=2a$ (inner)

outer radius $=3a$

Electric field due to each sphere

$E_1=E_3=0$ (inside the inner conductor from $O$ to a and from $2a$ to $3a$ in conducting shell)

$E_2$ and $E_4=\frac{KQ}{r^2}$

Now,

$dU=0.5e\{E_2{r}^2.dr\left(ato2a\right)\}+E_4{2r}^{2}dr\left\{3ato\alpha \right\}$

$=\frac{5}{12}\frac{Q^2}{4\pi {ϵ}_{0}a}$