Question

A shell of radius $R$ having uniform surface change density $\sigma {\text{Cm}}^{-2}$ is divided into two parts as shown in the figure. Find the minimum force $F$ required to keep the two parts together.

• A. $\frac{1}{4}{\sigma }^2\frac{\pi R^2}{{\epsilon }_0}$
• B. $\frac{1}{2}{\sigma }^2\frac{\pi R^2}{{\epsilon }_0}$
• C. $\frac{1}{8}{\sigma }^2\frac{\pi R^2}{{\epsilon }_0}$
• D. zero

Answer 1

The correct option is B $\frac{1}{2}{\sigma }^2\frac{\pi R^2}{{\epsilon }_0}$


Net electrostatic pressure acting at each point on the surface of charge density $\sigma$ is,
$P_e=\frac{1}{2}\frac{{\sigma }^2}{{\epsilon }_0}$

There will be a force due to electrostatic pressure on each half surface which should be balanced by $F$ to keep the parts together.
Thus, $\frac{F}{A}=P_e$
where $A=\pi R^2$ is the projected area where $P_e$ is acting.

$\Rightarrow F=P_e\times A=\frac{1}{2}\frac{{\sigma }^2}{{\epsilon }_0}\pi R^2$

Hence, option (b) is the correct answer.