question_answer

A small conducting sphere of radius r is lying concentrically inside a bigger hollow conducting sphere of radius R. The bigger and smaller spheres are charged with Q and q (Q > q)and are insulated from each other. The potential difference between the spheres will be

A)  \[\frac{1}{4\pi {{\varepsilon }_{0}}}\left( \frac{q}{r}-\frac{q}{R} \right)\]

B)  \[\frac{1}{4\pi {{\varepsilon }_{0}}}\left( \frac{q}{R}-\frac{Q}{r} \right)\]

C)  \[\frac{1}{4\pi {{\varepsilon }_{0}}}\left( \frac{q}{r}-\frac{Q}{R} \right)\]

D)  \[\frac{1}{4\pi {{\varepsilon }_{0}}}\left( \frac{Q}{R}-\frac{q}{r} \right)\]

Correct Answer: A

Solution :

The potential \[{{V}_{1}}\]of smaller sphere is given by \[{{V}_{1}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{q}{r}+\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{Q}{R}\] The potential \[{{V}_{2}}\]of bigger sphere is given by  \[{{V}_{2}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{Q}{R}+\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{q}{R}\]         So, the potential difference between the  plates \[V={{V}_{1}}-{{V}_{2}}\] or\[V=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{q}{r}+\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{Q}{R}-\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{Q}{R}-\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{q}{R}\]\[=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{q}{r}-\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{q}{R}=\frac{1}{4\pi {{\varepsilon }_{0}}}\left( \frac{q}{r}-\frac{q}{R} \right)\]