Electrostatics 7 Question 8
8. 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
(2010)
(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) $\frac{1}{\varepsilon _0} \frac{\sigma^{2}}{R^{2}}$
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Solution:
- Electrical force per unit area $=\frac{1}{2} \varepsilon _0 E^{2}$
$$ =\frac{1}{2} \varepsilon _0 \quad \frac{\sigma^{2}}{\varepsilon _0}=\frac{\sigma^{2}}{2 \varepsilon _0} $$
Projected area $=\pi R^{2}$
$\therefore$ Net electrical force $=\frac{\sigma^{2}}{2 \varepsilon _0}\left(\pi R^{2}\right)$
In equilibrium, this force should be equal to the applied force.
$$ \therefore \quad F=\frac{\pi \sigma^{2} R^{2}}{2 \varepsilon _0} \text { or } F \propto \frac{\sigma^{2} R^{2}}{\varepsilon _0} $$
$\therefore$ The correct option is (a).
