Question

Three concentric spherical metallic shells A, B and C of radii a, b and c(c > b > a) have charge densities $\sigma ,-\sigma$ and $\sigma$ respectively.The potential of shell B is :

• A. $(a+b+c)\frac{\sigma }{{\epsilon }_0}$
• B. $(\frac{a^2}{b}-b+c)\frac{\sigma }{{\epsilon }_0}$
• C. $(\frac{a^2}{c}-\frac{b^2}{c}+c)\frac{\sigma }{{\epsilon }_0}$
• D. $\frac{\sigma c}{{\epsilon }_0}$

Answer 1

The correct option is B $(\frac{a^2}{b}-b+c)\frac{\sigma }{{\epsilon }_0}$

The net potential at any point ($r=b)$ on shell$=$potential due to shell A $+$ potential due to shell B $+$ potential due to shell C.

or $V=V_{A,out}+V_{B,surface}+V_{C,in}=V_{A,out}+V_{B,surface}+V_{C,surface}$ (for shell $V_{in}=V_{surafce}$)

or $V=\frac{1}{4\pi {ϵ}_0}[\frac{q_a}{b}+\frac{q_b}{b}+\frac{q_c}{c}]$

$=\frac{1}{4\pi {ϵ}_0}[\frac{\sigma 4\pi a^2}{b}+\frac{(-\sigma )4\pi b^2}{b}+\frac{\sigma 4\pi c^2}{c}]$

$=\frac{\sigma }{{ϵ}_0}[\frac{a^2}{b}-b+c]$