Question: Q. 8. Three concentric metallic shells $A, B$ and $C$ of rádii $a, b$ and $c(a<b<c)$ have surface charge densities $+\sigma,-\sigma$ and $+\sigma$ respectively as shown in the figure.
If shells $A$ and $C$ are at the same potential, then obtain the relation between the radii $a, b$ and $c$.
U] [Foreign Set-III, 2014]
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Solution:
Ans. $q_{\mathrm{A}}=\sigma .4 \pi a^{2}, q_{\mathrm{B}}=-\sigma .4 \pi b^{2}, q_{c}=\sigma .4 \pi c^{2}$,
$$ \begin{aligned} V_{A} & =k\left[\frac{q_{A}}{a}+\frac{q_{B}}{b}+\frac{q_{C}}{c}\right] \ V_{C} & =k\left[\frac{q_{A}+q_{B}+q_{C}}{c}\right] \ \because, \quad V_{A} & =V_{C}, \text { we have } \ k\left[\frac{q_{A}}{a}+\frac{q_{B}}{b}+\frac{q_{C}}{c}\right] & =k\left[\frac{q_{A}+q_{B}+q_{C}}{c}\right] \ \therefore \quad \frac{q_{A}}{a}+\frac{q_{B}}{b} & =\frac{q_{A}+q_{B}}{c} \end{aligned} $$
Putting the values of $q_{A}$ and $q_{B}$,
We get
$$ a+b=c $$
