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. $\frac{1}{\epsilon _{0}}\sigma ^{2} \, R^{2}$
• B. $\frac{1}{\epsilon _{0}}\sigma ^{2} \, R^{ \, }$
• C. $\frac{1}{\epsilon _{0}} \, \frac{\sigma ^{2}}{R}$
• D. $\frac{1}{\epsilon _{0}} \, \frac{\sigma ^{2}}{R^{2}}$

Answer 1

Answer: (A)

Electrical force per unit area
$=\frac{1}{2}\epsilon _{0}E^{2}$
$=\frac{1}{2}\left(\epsilon \right)_{0}\left(\frac{\sigma }{\left(\epsilon \right)_{0}}\right)^{2}=\frac{\left(\sigma \right)^{2}}{2 \left(\epsilon \right)_{0}}$
Projected area = $\pi R^{2}$
∴ Net electrical force
= $\left(\frac{\left(\sigma \right)^{2}}{2 \left(\epsilon \right)_{0}}\right)\left(\pi R^{2}\right)$
In equilibrium, this force should be equal to the applied force.
$\therefore \, \, \, F=\frac{\pi \sigma ^{2} R^{2}}{2 \epsilon _{0}}$
Or $F \propto \frac{\sigma ^{2} R^{2}}{\epsilon _{0}}$