Question

A uniformly charged thin spherical shell of radius $R$ carries uniform surface charge density of σ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$. $F$ is proportional to :

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

Answer 1

The correct option is A $\frac{1}{{ϵ}_0}{\sigma }^2{R}^2$

As all the four options have same quantites (i.e $\sigma ,R$ and ${ϵ}_0$), all we need to do is to just check which option has same dimension as that of force.
We know that $\sigma \times$Area = $Q$
$⟹\sigma \times R^2=Q$
Also, $F=\frac{Q^2}{4\pi {ϵ}_0}$
$⟹F=\frac{\sigma R^2}{{ϵ}_0}$ (dimensionally)