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))/(4piepsilon_(0)(a^(2)+b^(2)c^(2)))`
B. `(Q)/(4piepsilon_(0)(a+b+c))`
C. `(Q)/(12piepsilon_(0))(ab+bc+ca)/(abc)`
D. `(Q(a^(2)+b^(2)+c^(2)))/(4piepsilon_(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^(2))/(C^(2))q_(2)`
`rArrq_(2)(1+(b^(2))/(C^(2)))=(b^(2))/(C^(2))Q-(b^(2))/(C^(2))q_(1)`
`rArr(b^(2))/(a^(2))q_(1)(1+(b^(2))/(C^(2)))=(b^(2))/(C^(2))Q-(b^(2))/(C^(2))q_(1)`
`rArr((b^(2))/(a^(2))+(b^(4))/(a^(2)c^(2)))q_(1)+(b^(2))/(C^(2))q_(1)=(b^(2))/(C^(2))Q`
`rArr(b^(2))/(a^(2))q_(1)(1+(b^(2))/(C^(2))+(a^(2))/(C^(2)))=(b^(2))/(C^(2))Q`
`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/(4piepsilon_(0))[a/(a^(2)+b^(2)+c^(2))+b/(a^(2)+b^(2)+c^(2))+c/(a^(2)+b^(2)+c^(2))]=Q/(4piepsilon_(0))((a+b+c)/(a^(2)+b^(2)+c^(2)))`