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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Answer:
Correct Answer: 19. $\begin{aligned} (a)V_A=\frac{\sigma}{\varepsilon_0}(a-b+c), V_B=\frac{\sigma}{\varepsilon_0}\left(\frac{a^2}{b}-b+c\right), \ & V_C=\frac{\sigma}{\varepsilon_0}\left(\frac{a^2}{c}-\frac{b^2}{c}+c\right) \ & \text { (b) } a+b=c \ & \end{aligned}$
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} $
