Question 209 of 311

Three concentric metal shells $A,B$ and $C$ of respective radii $a,b$ and $c(a\lt b\lt c)$ have surface charge densities <c) charge="" densities="" have="" math="" surface=""> $+\sigma ,-\sigma$ and $+\sigma$ respectively. The potential of shell $B$ is:</c)>

A$\dfrac{\sigma }{{\epsilon }_{0}}\left[\dfrac{{b}^{2}-{c}^{2}}{c}+a\right]$B$\dfrac{\sigma }{{\epsilon }_{0}}\left[\dfrac{{a}^{2}-{b}^{2}}{a}+c\right]$C$\dfrac{\sigma }{{\epsilon }_{0}}\left[\dfrac{{a}^{2}-{b}^{2}}{b}+c\right]$D$\dfrac{\sigma }{{\epsilon }_{0}}\left[\dfrac{{b}^{2}-{c}^{2}}{b}+a\right]$

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Answer: (C) $\dfrac{\sigma }{{\epsilon }_{0}}\left[\dfrac{{a}^{2}-{b}^{2}}{b}+c\right]$

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

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

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

Required potential of b

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

$=\dfrac{\sigma }{{\epsilon }_{0}}\left(\dfrac{{{a}^{2}-b}^{2}}{b}+c\right)$