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Three concentric metallic shells $A, B$ and $C$ of radii $a, b$ and $c (a < b < c)$ have surface charge densities $\sigma ,\, - \sigma $ and $\sigma $ respectively. then ${V_A}$ and ${V_B}$
$\frac{\sigma }{{{\varepsilon _0}}}(a - b +c),\,\frac{\sigma }{{{\varepsilon _0}}}\left( {\frac{{{a^2}}}{b} - b + c} \right)$
$(a - b - c),\,\frac{{{a^2}}}{c}$
$\frac{{{\varepsilon _0}}}{\sigma }(a - b - c),\,\frac{{{\varepsilon _0}}}{\sigma }\left( {\frac{{{a^2}}}{c} - b + c} \right)$
$\frac{\sigma }{{{\varepsilon _0}}}\left( {\frac{{{a^2}}}{c} - \frac{{{b^2}}}{c} + c} \right)$ ,$\frac{\sigma }{{{\varepsilon _0}}}(a - b + c)$
Solution
${\sigma _A} = \sigma = \frac{{{q_a}}}{{4\pi {a^2}}}\,\,\, \Rightarrow \,\,{q_a} = \sigma \times 4\pi {a^2}$,
${\sigma _B} = – \sigma = \frac{{{q_b}}}{{4\pi {b^2}}}\,\, \Rightarrow \,\,{q_b} = – \sigma \times 4\pi {b^2}$
${\sigma _C} = \sigma = \frac{{{q_c}}}{{4\pi {c^2}}}\, \Rightarrow \,{q_c} = \sigma \times 4\pi {c^2}$
${V_A} = {({V_A})_{{\rm{surface}}}} + {({V_B})_{{\rm{in}}}} + {({V_C})_{{\rm{in}}}}$$ = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{{q_a}}}{a} + \frac{{{q_b}}}{b} + \frac{{{q_c}}}{c}} \right]$
$ = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{\sigma \times 4\pi {a^2}}}{a} + \frac{{( – \sigma ) \times 4\pi {b^2}}}{b} + \frac{{\sigma \times 4\pi {c^2}}}{c}} \right]$${V_A} = \frac{\sigma }{{{\varepsilon _0}}}\left[ {a – b +c]} \right]$
${V_B} = {({V_A})_{{\rm{out}}}} + {({V_B})_{{\rm{surface}}}} + {({V_C})_{{\rm{in}}}} = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{{q_a}}}{b} + \frac{{{q_b}}}{b} + \frac{{{q_c}}}{c}} \right]$
$ = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{\sigma \times 4\pi {a^2}}}{b} – \frac{{\sigma \times 4\pi {b^2}}}{b} + \frac{{\sigma \times 4\pi {c^2}}}{c}} \right]$$ = \frac{\sigma }{{{\varepsilon _0}}}\left[ {\frac{{{a^2}}}{b} – b + c} \right]$
