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) $\frac{\sigma}{\varepsilon _0} \frac{b^{2}-c^{2}}{b}+a$
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Answer:
Correct Answer: 4. (d)
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
$$ V _B=\frac{\sigma}{\varepsilon _0} \frac{a^{2}-b^{2}}{b}+c $$
