question_answer

A charge Q is distributed over three concentric spherical shells of radii a, b, c \[\left( a<b<c \right)\] such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre, where \[r<a\], would be- [JEE Main 10-Jan-2019 Morning]

A) \[\frac{Q({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}{{{4}_{\pi {{\varepsilon }_{0}}}}({{a}^{3}}+{{b}^{3}}+{{c}^{3}})}\]

B) \[\frac{Q}{{{4}_{\pi {{\varepsilon }_{0}}}}(a+b+c)}\]

C) \[\frac{Q}{{{12}_{\pi \varepsilon {{\,}_{0}}}}}\,\,\frac{ab+bc+ca}{abc}\]

D) \[\frac{Q(a+b+c)}{{{4}_{\pi \varepsilon {{\,}_{0}}}}({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\]

Correct Answer: D

Solution :

[d]

\[x+y+z=Q\]

\[\frac{x}{4\pi {{a}^{2}}}=\frac{y}{4\pi {{b}^{2}}}=\frac{z}{4\pi {{c}^{2}}}\]

(surface charge density for all are same)

\[x:y:z={{a}^{2}}:{{b}^{2}}:{{c}^{2}}\]

\[x=\frac{{{a}^{2}}Q}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}};\,\,\,y\,=\,\frac{{{b}^{2}}Q}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}};\]

\[z=\frac{{{c}^{2}}Q}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]

V at \[r<a\] \[=\,\frac{kx}{a}+\frac{ky}{b}+\frac{kz}{c}\]

\[=\frac{1}{4\pi {{\in }_{0}}}\left[ \frac{Qa}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\,+\,\frac{Qb}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\,+\,\,\frac{Qc}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}} \right]\]\[=\,\,\frac{Q}{4\pi {{\in }_{0}}}\,\,\left[ \frac{a+b+c}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}} \right]\]