A uniformly charged thin spherical shell of radius $R$ carries uniform surface charge density of $\sigma$…
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
$\frac{1}{\varepsilon_0} \sigma^2 R^2$
$\frac{1}{\varepsilon_0} \sigma^2 R$
$\frac{1}{\varepsilon_0} \frac{\sigma^2}{R}$
$\frac{1}{\varepsilon_0} \frac{\sigma^2}{R^2}$
Solution
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 $=\pi R^2$
$\therefore$ 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.
$\therefore \quad F=\frac{\pi \sigma^2 R^2}{2 \varepsilon_0}$
or $\quad F \propto \frac{\sigma^2 R^2}{\varepsilon_0}$
$\therefore$ The correct option is (a)
