Question

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

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

Answer 1

Answer: (A)

Electrical force per unit area $=\frac{1}{2}{\epsilon }_0{E}^2$
$=\frac{1}{2}{\epsilon }_0{\left(\frac{\sigma }{{\epsilon }_0}\right)}^2=\frac{{\sigma }^2}{2{\epsilon }_0}$
Projected area $=\pi R^2$
$\therefore$ Net electrical force $=\left(\frac{{\sigma }^2}{2{\epsilon }_0}\right)(\pi R^2)$
In equilibrium, this force should be equal to the applied
force.
$\therefore F=\frac{\pi {\sigma }^2{R}^2}{2{\epsilon }_0}$ or $F\propto \frac{{\sigma }^2{R}^2}{{\epsilon }_0}$