Question

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^2R^2 $
• B. $\frac {1}{\varepsilon_0}\sigma^2R $
• C. $\frac {1}{\varepsilon_0}\frac {\sigma^2}{R} $
• D. $\frac {1}{\varepsilon_0}\frac {\sigma^2}{R^2} $

Answer 1

Answer: (A)

Electrical force per unit area $= \frac {1}{2}\varepsilon_0E^2 $
$= \frac {1}{2}\varepsilon_0\left(\frac {\sigma}{\varepsilon_0}\right)^2= \frac {\sigma^2}{2 \varepsilon_0} $
Projected area $= \pi R^2 $
$\therefore $ Net electrical force $=\left(\frac {\sigma^2}{2 \varepsilon_0}\right)(\pi R^2) $
In equilibrium, this force should be equal to the applied
force.
$\therefore F=\frac {\pi \sigma^2R^2}{2 \varepsilon_0}$ or $F \propto \frac {\sigma^2R^2}{\varepsilon_0}$