Q40 / 57 · JEE Advanced 2010 (Paper 2) · Physics

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 Figure

  • Correct answer
    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
    $\frac{1}{\varepsilon_0} \frac{\sigma^2}{R^2}$

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).

Hence, the correct answer is option: 'A'.

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