question_answer

Two concepts spheres of radii R and r have similar charges with equal surface charge densities \[\left( \sigma  \right).\] what is the electric potential at their common center?

A) \[\sigma /{{\varepsilon }_{0}}\]

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

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

D) None of these

Correct Answer: C

Solution :

[c] Charge on the outer sphere \[={{q}_{1}}=4\pi {{R}^{2}}\sigma \] Charge on the inner sphere \[={{q}_{2}}=4\pi {{r}^{2}}\sigma \] \[v=\frac{1}{4\pi {{\in }_{0}}}\frac{{{q}_{1}}}{R}+\frac{1}{4\pi {{\in }_{0}}}\frac{{{q}_{2}}}{r}\] \[=\frac{1}{4\pi {{\in }_{0}}}\left[ \frac{4\pi {{R}^{2}}\sigma }{R}+\frac{4\pi {{r}^{2}}\sigma }{r} \right]\]  \[=\frac{4\pi \sigma }{4\pi {{\in }_{0}}}\left( R+r \right)=\frac{\sigma }{{{\in }_{0}}}\left( R+r \right)\]