Three concentric spherical shells have radii $a$, $b$, and $c$ ($a < b < c$) and have surface charge…

Three concentric spherical shells have radii $a$, $b$, and $c$ ($a < b < c$) and have surface charge densities $\sigma$, $-\sigma$, and $\sigma$ respectively. If $V_A$, $V_B$, and $V_C$ denote the potentials of the three shells, then, for $c=a+b$, we have:
  1. $V_C=V_B=V_A$
  2. $\mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}}$
  3. $V_C=V_B \neq V_A$
  4. $\mathrm{V}_{\mathrm{C}} \neq \mathrm{V}_{\mathrm{B}} \neq \mathrm{V}_{\mathrm{A}}$

Solution

$\begin{aligned} \mathrm{q}_{\mathrm{A}} & =4 \pi \mathrm{a}^2 \sigma, \\ \mathrm{q}_{\mathrm{B}} & =-4 \pi \mathrm{b}^2 \sigma, \\ \mathrm{q}_{\mathrm{C}} & =4 \pi \mathrm{c}^2 \sigma, \mathrm{c}=\mathbf{a}+\mathrm{b} \\ \mathrm{V}_{\mathrm{A}} & =\frac{1}{4 \pi \in_0}\left(\frac{\mathrm{q}_{\mathrm{A}}}{\mathrm{a}}+\frac{\mathrm{q}_{\mathrm{B}}}{\mathrm{b}}+\frac{\mathrm{q}_{\mathrm{C}}}{\mathrm{c}}\right) \\ & =\frac{2 \sigma \mathrm{a}}{\epsilon_0} \\ \mathrm{~V}_{\mathrm{B}} & =\frac{1}{4 \pi \epsilon_0}\left(\frac{\mathrm{q}_{\mathrm{A}}}{\mathrm{a}}+\frac{\mathrm{q}_{\mathrm{B}}}{\mathrm{b}}+\frac{\mathrm{q}_{\mathrm{C}}}{\mathrm{c}}\right) \end{aligned}$ $\begin{aligned} & =\frac{\sigma}{\epsilon_0}\left(\frac{\mathbf{a}^2}{\mathbf{b}}-\mathbf{b}+\mathrm{c}\right) \\ & =\frac{\sigma}{\epsilon_0}\left(\mathbf{a}+\frac{\mathbf{a}^2}{\mathbf{b}}\right) \\ \mathrm{V}_{\mathrm{C}} & =\frac{1}{4 \pi \in_0}\left(\frac{\mathrm{q}_{\mathrm{A}}}{\mathbf{a}}+\frac{\mathrm{q}_{\mathrm{B}}}{\mathrm{b}}+\frac{\mathrm{q}_{\mathrm{C}}}{\mathrm{c}}\right) \\ & =\frac{\sigma}{\epsilon_0}\left(\frac{\mathrm{a}^2-\mathbf{b}}{\mathrm{c}}+\mathrm{c}\right)=\frac{2 \sigma \mathbf{a}}{\epsilon_0} \end{aligned}$ So, $V_C=V_A \neq V_B$

Asked in: NEET 2009 (Mains)

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