Three concentric spherical shells have radii a, b, and ( a < b < ) and have surface charge densities σ,-σ and σ respectively. If V_A, V_B and V_C…

$$\begin{aligned} \mathrm{q}_{\mathrm{A}} & =4 \pi \mathrm{a}^2 \sigma, \\ \mathrm{q}_{\mathrm{B}} & =-4 \pi \mathrm{b}^2 \sigma, \\ \mathrm{q}_{\mathrm{C}} & =4 \pi \mathrm{c}^2 \sigma, \mathrm{c}=\mathbf{a}+\mathrm{b} \\ \mathrm{V}_{\mathrm{A}} & =\frac{1}{4 \pi \in_0}\left(\frac{\mathrm{q}_{\mathrm{A}}}{\mathrm{a}}+\frac{\mathrm{q}_{\mathrm{B}}}{\mathrm{b}}+\frac{\mathrm{q}_{\mathrm{C}}}{\mathrm{c}}\right) \\ & =\frac{2 \sigma \mathrm{a}}{\epsilon_0} \\ \mathrm{~V}_{\mathrm{B}} & =\frac{1}{4 \pi \epsilon_0}\left(\frac{\mathrm{q}_{\mathrm{A}}}{\mathrm{a}}+\frac{\mathrm{q}_{\mathrm{B}}}{\mathrm{b}}+\frac{\mathrm{q}_{\mathrm{C}}}{\mathrm{c}}\right) \end{aligned}$$qA​qB​qC​VA​ VB​​=4πa2σ,=−4πb2σ,=4πc2σ,c=a+b=4π∈0​1​(aqA​​+bqB​​+cqC​​)=ϵ0​2σa​=4πϵ0​1​(aqA​​+bqB​​+cqC​​)​ $$\begin{aligned} & =\frac{\sigma}{\epsilon_0}\left(\frac{\mathbf{a}^2}{\mathbf{b}}-\mathbf{b}+\mathrm{c}\right) \\ & =\frac{\sigma}{\epsilon_0}\left(\mathbf{a}+\frac{\mathbf{a}^2}{\mathbf{b}}\right) \\ \mathrm{V}_{\mathrm{C}} & =\frac{1}{4 \pi \in_0}\left(\frac{\mathrm{q}_{\mathrm{A}}}{\mathbf{a}}+\frac{\mathrm{q}_{\mathrm{B}}}{\mathrm{b}}+\frac{\mathrm{q}_{\mathrm{C}}}{\mathrm{c}}\right) \\ & =\frac{\sigma}{\epsilon_0}\left(\frac{\mathrm{a}^2-\mathbf{b}}{\mathrm{c}}+\mathrm{c}\right)=\frac{2 \sigma \mathbf{a}}{\epsilon_0} \end{aligned}$$VC​​=ϵ0​σ​(ba2​−b+c)=ϵ0​σ​(a+ba2​)=4π∈0​1​(aqA​​+bqB​​+cqC​​)=ϵ0​σ​(ca2−b​+c)=ϵ0​2σa​​ So, $V_C=V_A \neq V_B$VC​=VA​=VB​