Electrostatics 3 Question 4
4. Three concentric metal shells $A, B$ and $C$ of respective radii $a, b$ and $c(a<b<c)$ have surface charge densities $+\sigma,-\sigma$ and $+\sigma$, respectively. The potential of shell $B$ is (2018 Main)
(a) $\frac{\sigma}{\varepsilon_{0}} \frac{b^{2}-c^{2}}{c}+a$
(b) $\frac{\sigma}{\varepsilon_{0}} \frac{a^{2}-b^{2}}{a}+c$
(c) $\frac{\sigma}{\varepsilon_{0}} \frac{a^{2}-b^{2}}{b}+c$
(d) $\frac{\sigma}{\varepsilon_{0}} \frac{b^{2}-c^{2}}{b}+a$
Show Answer
Answer:
Correct Answer: 4. (c)
Solution:
- (c) $q_{I}=\left(4 \pi a^{2}\right)(\sigma), q_{I I}=4 \pi b^{2}(-\sigma), q_{I I I}=4 \pi c^{2}(\sigma)$
$ \begin{aligned} V_{B}=(\text { Potential due to I })+ & \text { (Potential due to II }) \\ & +(\text { Potential due to III }) \end{aligned} $
$ V_{B}=\frac{K 4 \pi a^{2} \sigma}{b}+\frac{K 4 \pi b^{2}}{b}(-\sigma)+\frac{K 4 \pi c^{2}(\sigma)}{c} $
Substituting $K=\frac{1}{4 \pi \varepsilon_{0}}$, we get
$ V_{B}=\frac{\sigma}{\varepsilon_{0}} \Big(\frac{a^{2}-b^{2}}{b}+c\Big) $
