question_answer

Three concentric metal shells A, B and C of respective radii a, b and c (a < b < c) have surface charge densities\[\text{+ }\sigma \], \[\text{- }\sigma \] and \[\text{+ }\sigma \] respectively. The potential of shell B is:            [JEE Main Online 08-04-2018]

A)  \[\frac{\sigma }{{{\in }_{\text{O}}}}\left[ \frac{{{b}^{2}}-{{c}^{2}}}{b}+a \right]\]          

B)  \[\frac{\sigma }{{{\in }_{\text{O}}}}\left[ \frac{{{b}^{2}}-{{c}^{2}}}{c}+a \right]\]

C)  \[\frac{\sigma }{{{\in }_{\text{O}}}}\left[ \frac{{{a}^{2}}-{{b}^{2}}}{a}+c \right]\]          

D)  \[\frac{\sigma }{{{\in }_{\text{O}}}}\left[ \frac{{{a}^{2}}-{{b}^{2}}}{b}+c \right]\]

Correct Answer: D

Solution :

\[{{V}_{B}}=\frac{{{a}^{2}}\sigma }{{{\in }_{0}}b}-\frac{{{b}^{2}}\sigma }{{{\in }_{0}}b}+\frac{{{c}^{2}}\sigma }{{{\in }_{0}}c}=\frac{\sigma }{{{\in }_{0}}}\left[ \frac{{{a}^{2}}-{{b}^{2}}}{b}+c \right]\]