Electrostatics 3 Question 19
19. Three concentric spherical metallic shells, $A, B$ and $C$ of radii $a, b$ and $c(a<b<c)$ have surface charge densities $\sigma,-\sigma$ and $\sigma$ respectively.
(1990, 7M)
(a) Find the potential of the three shells $A, B$ and $C$.
(b) If the shells $A$ and $C$ are at the same potential, obtain the relation between the radii $a, b$ and $c$.
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Solution:
- (a) Potential at any shell will be due to all three charges.
$$ \begin{aligned} V _A & =\frac{1}{4 \pi \varepsilon _0} \frac{q _A}{a}+\frac{q _B}{b}+\frac{q _C}{c} \\ & =\frac{1}{4 \pi \varepsilon _0} \frac{\left(4 \pi a^{2}\right)(\sigma)}{a}+\frac{\left(4 \pi b^{2}\right)(-\sigma)}{b}+\frac{\left(4 \pi c^{2}\right)(\sigma)}{c} \\ & =\frac{\sigma}{\varepsilon _0}(a-b+c) \\ V _B & =\frac{1}{4 \pi \varepsilon _0} \frac{q _A}{b}+\frac{q _B}{b}+\frac{q _C}{c} \\ & =\frac{1}{4 \pi \varepsilon _0} \frac{\left(4 \pi a^{2}\right)(\sigma)}{b}+\frac{\left(4 \pi b^{2}\right)(-\sigma)}{b}+\frac{\left(4 \pi c^{2}\right)(\sigma)}{c} \\ & =\frac{\sigma}{\varepsilon _0} \frac{a^{2}}{b}-b+c \end{aligned} $$
Similarly, $V _C=\frac{1}{4 \pi \varepsilon _0} \frac{q _A}{c}+\frac{q _B}{c}+\frac{q _C}{c}$
$$ \begin{aligned} & =\frac{1}{4 \pi \varepsilon _0} \frac{\left(4 \pi a^{2}\right)(\sigma)}{c}+\frac{\left(4 \pi b^{2}\right)(-\sigma)}{c}+\frac{\left(4 \pi c^{2}\right)(\sigma)}{c} \\ & =\frac{\sigma}{\varepsilon _0} \frac{a^{2}}{c}-\frac{b^{2}}{c}+c \end{aligned} $$
(b) Given $V _A=V _C$
$$ \begin{aligned} & \therefore \quad \frac{\sigma}{\varepsilon _0}(a-b+c)=\frac{\sigma}{\varepsilon _0} \frac{a^{2}}{c}-\frac{b^{2}}{c}+c \\ & \therefore \quad a-b+c=\frac{a^{2}}{c}-\frac{b^{2}}{c}+c \\ & \text { or } \quad a+b=c \end{aligned} $$
