Question

A uniformly charged thin spherical shell of radius $R$ carries a uniform surface charge density of $\sigma$ per unit area. It is made of two hemispherical shells, held together by pressing them with a force $F$ (see the figure). $F$ is proportional to

• A. $\frac{1}{{ϵ}_0}{\sigma }^{2}R$
• B. $\frac{1}{{ϵ}_0}\frac{{\sigma }^2}{R^2}$
• C. $\frac{1}{{ϵ}_0}\frac{{\sigma }^2}{R}$
• D. $\frac{1}{{ϵ}_0}{\sigma }^2{R}^2$

Answer 1

The correct option is D $\frac{1}{{ϵ}_0}{\sigma }^2{R}^2$

Electrical force per unit area
$=\frac{1}{2}{ϵ}_0{E}^2$
$=\frac{1}{2}{ϵ}_0{\left(\frac{\sigma }{{ϵ}_0}\right)}^2=\frac{{\sigma }^2}{2{ϵ}_0}$
$\text{Projected area}=\pi R^2$
$\therefore \text{Net electrical force}=\left(\frac{{\sigma }^2}{2{ϵ}_0}\right)\left(\pi R^2\right)$
In equilibrium, this force should be equal to the applied force.
$\therefore F=\frac{\pi {\sigma }^2{R}^2}{2{ϵ}_0}$ or
$F\propto \frac{{\sigma }^2{R}^2}{{ϵ}_0}$
Hence, option $\left(A\right)$ is correct.

Alternate solution:
Only the option $\left(A\right)$ has the dimensions of force. So, we can choose it by eliminating all others.