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. $\frac{{\sigma }^2{R}^2}{{\epsilon }_0}$
• B. $\frac{{\sigma }^{2}R}{{\epsilon }_0}$
• C. $\frac{{\sigma }^2}{{\epsilon }_{0}R}$
• D. $\frac{{\sigma }^2}{{\epsilon }_0{R}^2}$

Answer 1

The correct option is A $\frac{{\sigma }^2{R}^2}{{\epsilon }_0}$

Here $q_1,q_2=\sigma .2\pi R^2$

These charges are located at centre of mass of hemisphere, at a distance of $\frac{R}{2}$ from centre.
$\therefore d=\frac{R}{2}\times 2=R$

$F=\frac{q_1{q}_2}{4\pi {\epsilon }_0{d}^2}=\frac{{\sigma }^{2}4{\pi }^2{R}^4}{4\pi {\epsilon }_0{R}^2}=\frac{\pi {\sigma }^2{R}^2}{{\epsilon }_0}$

$F\propto \frac{{\sigma }^2{R}^2}{{\epsilon }_0}$