A charge 'Q' is distributed over two concentric hollow spheres of radii 'r' and 'R' (gtr) such that the surface densities are equal. Find the potential at the common centre.

Explanation

if `q_(1)` and `q_(2)` are the charges on spheres of radii `'r'` and `R` respectively ,then in accordance with conservation of charge
`q_(1)+q_(2)=Q….(1)`
And according to given problem `sigma_(1)=sigma_(2)`
i.e,`(q_(1))/(4pir^(2))=(q_(2))/(4piR^(2))` or `(q_(1))/(q_(2))=(r^(2))/(R^(2)).....(2)`
So from Eq. (1) and (2)
`q_(1)=(Qr^(2))/((r^(2)+R^(2)))` and `q_(2)=(QR^(2))/((r^(2)+R^(2)))....(3)`
Now as potential inside a conducting sphere is equal to that at its suface so potential at the common centre,
`V=V_(1)+v_(2)=(1)/(4piepsilon_(0))[(q_(1))/(r)+q_(2)/(R)]`
Subsituting the value of `q_(1)` and `q_(2)` from Eq(3).
`V=(1)/(4piepsilon_(0))[(Qr)/((R^(2)+r^(2)))+(Qr)/((R^(2)+r^(2)))]`
`=(1)/(4piepsilon_(0))(Q(R+r))/((R^(2)+r^(2)))`