Q.
A uniformly charged thin spherical shell of radius $R$ carries 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) $\frac{1}{\varepsilon_{0}}\sigma^{2}\ R^{2}$
B) $\frac{1}{\varepsilon_{0}}\sigma^{2}\ R^{\ }$
C) $\frac{1}{\varepsilon_{0}}\ \frac{\sigma^{2}}{R}$
D) $\frac{1}{\varepsilon_{0}}\ \frac{\sigma^{2}}{R^{2}}$
Explanation
(a) Electrical force per unit area
$= \frac{1}{2}\varepsilon_{0}E^{2}$
$= \frac{1}{2}\varepsilon_{0}\left( \frac{\sigma}{\varepsilon_{0}} \right)^{2} = \frac{\sigma^{2}}{2\varepsilon_{0}}$
Projected area =$\pi R^{2}$
∴ Net electrical force
=$\left( \frac{\sigma^{2}}{2\varepsilon_{0}} \right)\left( \pi R^{2} \right)$
In equilibrium, this force should be equal to the applied force.
$\therefore\ \ \ F = \frac{\pi\sigma^{2}R^{2}}{2\varepsilon_{0}}$
Or $F \propto \frac{\sigma^{2}R^{2}}{\varepsilon_{0}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.