Question

A thin spherical shell of radius $R$ is cut into two equal halves and then each of its halves is charged to the same uniform surface charge density $\sigma $ . If $F$ is the force exerted on each half to keep them at equilibrium, then $F$ is proportional to

• A. $\frac{1}{ε_{0}} σ^{2} \textit{R}$
• B. $\frac{1}{ε_{0}} \frac{σ^{2}}{\textit{R}} $
• C. $\frac{1}{ε_{0}} \frac{σ^{2}}{\textit{R}^{2}} $
• D. $\frac{1}{ε_{0}} σ^{2} \textit{R}^{2}$

Answer 1

Answer: (D)

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 $=\pi R^{2}$
$\therefore \quad$ 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.
$∴$ $\textit{F} = \frac{\pi σ^{2} \textit{R}^{2}}{2 ε_{0}}$
or $\textit{F} ∝ \frac{σ^{2} \textit{R}^{2}}{ε_{0}}$