Three concentric metal shells A ,   B and C of respective radii a ,   b and c   ( a  …

1. $\frac{\sigma }{{\epsilon }_0}\left[\frac{b^2-c^2}{c}+a\right]$
2. $\frac{\sigma }{{\epsilon }_0}\left[\frac{a^2-b^2}{a}+c\right]$
3. $\frac{\sigma }{{\epsilon }_0}\left[\frac{a^2-b^2}{b}+c\right]$
4. $\frac{\sigma }{{\epsilon }_0}\left[\frac{b^2-c^2}{b}+a\right]$

Solution

Net potential at B is due to the superposition of potential due to all shells, 

${\vee }_B={\vee }_{B\left(I\right)}+{\vee }_{B\left(II\right)}+{\vee }_{B\left(III\right)}$

$=\frac{K4\pi a^2\sigma }{b}+\frac{K4\pi b^2\left(-\sigma \right)}{b}+\frac{K4\pi c^2\left(\sigma \right)}{c}$

Required potential of b

$=\frac{\sigma a^2}{{\epsilon }_{0}b}-\frac{\sigma b^2}{{\epsilon }_{0}b}+\frac{\sigma c^2}{{\epsilon }_{0}c}$

$=\frac{\sigma }{{\epsilon }_0}\left(\frac{{a^2-b}^2}{b}+c\right)$

Asked in: JEE Main 2018 (08 Apr)

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