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 force $F$ (see figure). $F$ is proportional to

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

Answer 1

Answer: (A)

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