Question

Two conducting spheres $A$ and $B$ of radius $a$ and $b$ respectively are at the same potential. The ratio of the surface charge densities of $A$ and $B$ is

• A. $ \frac{b}{a} $
• B. $ \frac{a}{b} $
• C. $ \frac{{{a}^{2}}}{{{b}^{2}}} $
• D. $ \frac{{{b}^{2}}}{{{a}^{2}}} $
• E. $ \frac{(a+b)}{(ab)} $

Answer 1

Answer: (A)

Given electric potential of spheres are same ie,
$ {{V}_{A}}={{V}_{B}} $ $ \frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{{{Q}_{1}}}{a}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{{{Q}_{2}}}{b} $
$ \frac{{{Q}_{1}}}{{{Q}_{2}}}=\frac{a}{b} $ ..(i)
Surface charge density $ \sigma =\frac{Q}{4\pi {{r}^{2}}} $
$ \Rightarrow $ $ \frac{{{\sigma }_{1}}}{{{\sigma }_{2}}}=\frac{{{Q}_{1}}}{{{Q}_{2}}}\times \frac{{{b}^{2}}}{{{a}^{2}}} $
$ =\frac{a}{b}\times \frac{{{b}^{2}}}{{{a}^{2}}} $ $ =\frac{b}{a} $