Question

A solid sphere with uniform charge distribution, of self energy $U_0$ and radius $R$ is divided into $n$ number of smaller spheres of radius $r$. The self energy of each smaller sphere is

• A. $\frac{U_0}{n^{7/6}}$
• B. $\frac{U_O}{n^{5/3}}$
• C. $U=U_0{n}^{5/6}$
• D. $\frac{U_O}{n^{5/6}}$

Answer 1

The correct option is B $\frac{U_O}{n^{5/3}}$

Let the charge on the solid sphere be$Q$ and radius of the solid sphere be$R$

Self energy of a solid sphere $$U_0=\frac{3k{Q}^2}{5R}[k=\frac{1}{4\pi {\epsilon }_0}]$$
$\Rightarrow Q^2=\frac{5R{U}_0}{3k}$

Let the charge on each small sphere be $q$ and the radius of each smaller sphere be$r$.

Self energy of each smaller sphere $$U=\frac{3k{q}^2}{5r}.....\left(1\right)$$
From conservation of charge, we can write that
$$q=\frac{Q}{n}...\left(2\right)$$
Given that, Solid sphere is made into $n$ smaller spheres

Thus, the volume remains constant,

$\therefore \frac{4}{3}\pi R^3=n\times \frac{4}{3}\pi r^3$

$\Rightarrow r=\left(\frac{R}{n^{1/3}}\right).....\left(3\right)$

From $\left(1\right))$ , $\left(2\right)$ and $\left(3\right)$ we can write that,

$U=\frac{U_O}{n^{5/3}}$

Hence, option (b) is the correct answer.