question_answer

Two concentric spheres of radii R and r have positive charges \[{{q}_{1}}\]and \[{{q}_{2}}\] with equal surface charge densities. What is the electric potential at their common center?

A)  \[\frac{\sigma }{{{\varepsilon }_{0}}}(R+r)\]

B)  \[\frac{\sigma }{{{\varepsilon }_{0}}}(R-r)\]

C)  \[\frac{\sigma }{{{\varepsilon }_{0}}}\left( \frac{1}{R}+\frac{1}{r} \right)\]

D)  \[\frac{\sigma }{{{\varepsilon }_{0}}}\left( \frac{R}{r} \right)\]

Correct Answer: A

Solution :

: The situation is as shown in the figure. According to the problem         \[{{\sigma }_{1}}={{\sigma }_{2}}=\sigma \] \[\therefore \] \[\frac{{{q}_{1}}}{4\pi {{R}^{2}}}=\sigma \] or  \[{{q}_{1}}=\sigma 4\pi {{R}^{2}}\] and \[\frac{{{q}_{2}}}{4\pi {{r}^{2}}}=\sigma \]  or  \[{{q}_{2}}=\sigma 4\pi {{r}^{2}}\] Electric potential at the common centre O is \[V={{V}_{1}}+{{V}_{2}}\] \[=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{1}}}{R}+\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{2}}}{r}=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{{{q}_{1}}}{R}+\frac{{{q}_{2}}}{r} \right]\] \[=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{\sigma 4\pi {{R}^{2}}}{R}+\frac{\sigma 4\pi {{R}^{2}}}{r} \right]\] \[=\frac{4\pi \sigma }{4\pi \in }[(R+r)]=\frac{\sigma }{{{\in }_{0}}}(R+r)\]