A charge Q is distributed over three concentric spherical shells of radii a, b, c (a`lt` b `lt` c) 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`lt`a, would be :

Options

A. `(Q(a+b+c))/(4pi epsilon_0(a^2+b^2+c^2))`
B. `Q/(4pi epsilon_0(a+b+c))`
C. `Q/(12pi epsilon_0)(ab+bc+ca)/(abc)`
D. `(Q(a^2+b^2+c^2))/(4pi epsilon_0(a^3+b^3_c^3))`

Answer

A

Explanation

`q_1/a^2=q_2/b^2=(Q-q_1-q_2)/c^2`
`q_1=a^2/b^2 q_2`
`q_2=-b^2/c^2 Q-b^2/c^2 q_1-b^2c^2 q_2`
`implies q_2 (1+b^2/c^2)=b^2/c^2 Q-b^2/c^2 q_1 implies b^2/a^2 q_1(1+b^2/c^2)=b^2/c^2Q=b^2/c^2 q_1`
`implies (b^2/a^2+b^4 (a^2c^2))q_1+b^2/c^2 q_1=b^2/c^2 Q implies b^2/c^2 q_1(1+b^2/c^2+a^2/c^2)=b^2/c^2Q`
`q_1=(a^2/c^2 Q)/((c^2+b^2+a^2)/c^2) , q_1=a^2/(a^2+b^2+c^2) Q, q_2= b^2/(a^2+b^2+c^2)Q`
`Q-q_1-q_2= c^2/(a^2+b^2+c^2) Q ,V=(kq_1)/a+(kq_2)/b+(k(Q-q_1-q_2))/c`
`=Q/(4 pi e_0) [a/(a^2+b^2+c^2) +b/(a^2+b^2+c^2) +c/(a^2+b^2+c^2)] =Q/(4 pi e_0) ((a+b+c)/(a^2+b^2+c^2))`