Three concentric metallic shells A, B and C o | vedclass

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Question

${\sigma _A} = \sigma = \frac{{{q_a}}}{{4\pi {a^2}}}\,\,\, \Rightarrow \,\,{q_a} = \sigma 	imes 4\pi {a^2}$,

${\sigma _B} = - \sigma = \frac{{{q_b

Vedclass · Accepted Answer

$\frac{\sigma }{{{\varepsilon _0}}}(a - b +c),\,\frac{\sigma }{{{\varepsilon _0}}}\left( {\frac{{{a^2}}}{b} - b + c} \right)$

Vedclass · Answer

${\sigma _A} = \sigma = \frac{{{q_a}}}{{4\pi {a^2}}}\,\,\, \Rightarrow \,\,{q_a} = \sigma 	imes 4\pi {a^2}$,

${\sigma _B} = - \sigma = \frac{{{q_b}}}{{4\pi {b^2}}}\,\, \Rightarrow \,\,{q_b} = - \sigma 	imes 4\pi {b^2}$

${\sigma _C} = \sigma = \frac{{{q_c}}}{{4\pi {c^2}}}\, \Rightarrow \,{q_c} = \sigma 	imes 4\pi {c^2}$

${V_A} = {({V_A})_{{m{surface}}}} + {({V_B})_{{m{in}}}} + {({V_C})_{{m{in}}}}$$ = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{{q_a}}}{a} + \frac{{{q_b}}}{b} + \frac{{{q_c}}}{c}} ight]$

$ = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{\sigma 	imes 4\pi {a^2}}}{a} + \frac{{( - \sigma ) 	imes 4\pi {b^2}}}{b} + \frac{{\sigma 	imes 4\pi {c^2}}}{c}} ight]$${V_A} = \frac{\sigma }{{{\varepsilon _0}}}\left[ {a - b +c]} ight]$

${V_B} = {({V_A})_{{m{out}}}} + {({V_B})_{{m{surface}}}} + {({V_C})_{{m{in}}}} = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{{q_a}}}{b} + \frac{{{q_b}}}{b} + \frac{{{q_c}}}{c}} ight]$

$ = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{\sigma 	imes 4\pi {a^2}}}{b} - \frac{{\sigma 	imes 4\pi {b^2}}}{b} + \frac{{\sigma 	imes 4\pi {c^2}}}{c}} ight]$$ = \frac{\sigma }{{{\varepsilon _0}}}\left[ {\frac{{{a^2}}}{b} - b + c} ight]$

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 ${V_B}$

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