• # question_answer 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 [AIPMT (S) 2009] A) ${{V}_{A}}={{V}_{C}}\ne {{V}_{B}}$ B) ${{V}_{C}}={{V}_{B}}\ne {{V}_{A}}$ C) ${{V}_{C}}\ne {{V}_{B}}\ne {{V}_{A}}$ D) ${{V}_{C}}={{V}_{B}}={{V}_{A}}$

 [a] Hence, ${{V}_{A}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\,\frac{\sigma 4\pi {{a}^{2}}}{a}-\frac{1}{4\pi {{\varepsilon }_{0}}}\,\frac{\sigma 4\pi {{b}^{2}}}{b}$ $+\frac{1}{4\,\pi {{\varepsilon }_{0}}},\,\,\frac{\sigma 4\pi {{c}^{2}}}{c}$ $=\frac{\sigma }{{{\varepsilon }_{0}}}(a-b+c)=\frac{\sigma }{{{\varepsilon }_{0}}}(2a)\,$$(\because \,c=a+b)$ ${{V}_{B}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\,-\,\frac{\sigma 4\pi {{\alpha }^{2}}}{a}\,-\frac{1}{4\pi {{\varepsilon }_{0}}}\,\frac{\sigma 4\pi {{b}^{2}}}{b}$ $+\frac{1}{4\,\pi {{\varepsilon }_{0}}},\,\,\frac{\sigma 4\pi {{c}^{2}}}{c}$ $=\frac{\sigma }{{{\varepsilon }_{0}}}\,\left( \frac{{{a}^{2}}}{c}-b+c \right)\,=\frac{\sigma }{{{\varepsilon }_{0}}}(2a)\,\,(\because \,c=a+b)$ And      ${{V}_{C}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\cdot \frac{\sigma \,4\,\pi {{\sigma }^{2}}}{c}-\frac{1}{4\,\pi {{\varepsilon }_{0}}}\frac{\sigma \,4\,\pi \,{{b}^{2}}}{c}$ $+\frac{1}{4\,\pi {{\varepsilon }_{0}}},\,\,\frac{\sigma 4\pi {{c}^{2}}}{c}$ $=\frac{\sigma }{{{\varepsilon }_{0}}}\,\left( \frac{{{a}^{2}}}{c}-\frac{{{b}^{2}}}{c}+c \right)\,=\frac{\sigma }{{{\varepsilon }_{0}}}(2a)\,\,(\because \,c=a+b)$ Hence,  ${{V}_{A}}={{V}_{C}}\ne {{V}_{B}}$