Three concentric charged metallic spherical shells $A, B$ and $C$ have radii $a, b$ and $c$; charge densities $\sigma,-\sigma$ and $\sigma$ and potentials $V_{A}, V_{B}$ and $V_{C}$ respectively. Then which of the following relations is correct?

Question

Aicharya · Accepted Answer

The three shells are shown in figure.
(i) Potential of A
$\begin{aligned}
& =(\text { Potential of A due to }+\sigma \text { on A }) \\
& +(\text { Potential of A due to }-\sigma \text { on B }) \\
& +(\text { Potential of A due to }+\sigma \text { on C }) \\
& =K\left[\frac{4 \pi \mathrm{a}^2 \sigma}{\mathrm{a}}-\frac{4 \pi \mathrm{b}^2 \sigma}{\mathrm{b}}+\frac{4 \pi \mathrm{c}^2 \sigma}{\mathrm{c}}\right] \\
& =\frac{\sigma}{\epsilon_0}[\mathrm{a}-\mathrm{b}+\mathrm{c}]
\end{aligned}$
Potential of $\mathrm{B}=($ Potential due to $+\sigma$ on $\mathrm{A})+($ Potential due to $-\sigma$ on $\mathrm{B})$

$\begin{aligned}
& +(\text { Potential due to }+\sigma \text { on } \mathrm{C}) \\
& =\mathrm{K}\left[\frac{4 \pi \mathrm{a}^2 \sigma}{\mathrm{a}}-\frac{4 \pi \mathrm{b}^2 \sigma}{\mathrm{b}}+\frac{4 \pi \mathrm{c}^2 \sigma}{\mathrm{c}}\right] \\
& =\frac{\sigma}{\epsilon_0}\left[\frac{\mathrm{a}^2}{\mathrm{~b}}-\mathrm{b}+\mathrm{c}\right]
\end{aligned}$
Potential of $\mathrm{C}=($ Potential due to $+\sigma$ on $\mathrm{A})+($ Potential due to $-\sigma$ on $\mathrm{B})+($ Potential due to $+\sigma$ on $\mathrm{C}$ )
$\begin{aligned}
& =K\left[\frac{4 \pi \mathrm{a}^2 \sigma}{\mathrm{a}}-\frac{4 \pi \mathrm{b}^2 \sigma}{\mathrm{b}}+\frac{4 \pi \mathrm{c}^2 \sigma}{\mathrm{c}}\right] \\
& =\frac{\sigma}{\epsilon_0}\left[\frac{\mathrm{a}^2}{\mathrm{~b}}-\frac{\mathrm{b}^2}{\mathrm{c}}+\mathrm{c}\right]
\end{aligned}$

Three concentric charged metallic spherical shells $A, B$ and $C$ have radii $a, b$ and $c$; charge…

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