Question

A charge $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R(R>r)$ such that their surface densities are equal. The charge on smaller and bigger shells are :

• A. $\frac{Q{r}^2}{r^2+R^2}and\frac{Q{R}^2}{r^2+R^2}$, respectively
• B. $Q(1+\frac{r^2}{R^2})andQ(1+\frac{R^2}{r^2})$, respectively
• C. $Q(1-\frac{r^2}{R^2})andQ(1-\frac{R^2}{r^2})$, respectively
• D. $\frac{Q{R}^2}{r^2+R^2}and\frac{Q{r}^2}{r^2+R^2}$, respectively

Answer 1

The correct option is A $\frac{Q{r}^2}{r^2+R^2}and\frac{Q{R}^2}{r^2+R^2}$, respectively

If $q_{1}and{q}_2$ are the charges on the spheres of radii $r$ and $R$, respectively, then by conservation of charges
$q_1+q_2=Q$ (i)
But it is given that ${\sigma }_1={\sigma }_2$
$\frac{q_1}{4\pi r^2}=\frac{q_2}{4\pi R^2}$ (ii)
or $\frac{q_1}{q_2}=\frac{r^2}{R^2}$
From Eqs (i) and (ii) we get
$q_1=\frac{Q{r}^2}{r^2+R^2}and{q}_2=\frac{Q{R}^2}{r^2+R^2}$