Question

A number of spherical conductors of different radii have same potential. Then the surface charge density on them:

• A. is proportional to their radii
• B. is inversely proportional to their radii
• C. are equal
• D. is proportional to square of their radii

Answer 1

Answer: (B)

is inversely proportional to their radii

We know voltage on a spherical conductor is given by $V=\frac{Kq}{R}$
where $q$ is the charge on that sphere and $K=\frac{1}{4\pi {ϵ}_o}$.
$\Rightarrow q=\frac{VR}{K}$

Let $R$ be the radius of that sphere.
Now surface charge density is $\sigma =\frac{q}{A}=\frac{q}{4\pi R^2}$
$\therefore \sigma =\frac{VR}{K4\pi R^2}$ $=\frac{V}{4\pi RK}$
$⟹$ $\sigma \propto \frac{1}{R}$

Hence option B is correct.