A charge Q is distributed over two concentric hollow spheres of radii r and `R (gt r)` such that the surface charge 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 respectievely , then in accordance with conservation of charge .
`q_(1) + q_(2) = Q…………(1)`
And according to give problem `sigma_(1) = sigma_(2)`
`i.e., (q_(1))/(4pi r^(2)) = (q_(2))/(4pi R^(2)) or (q_(1))/(q_(2)) or (r^(2))/(R^(2)) ..........(2)`
So from Eqs (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 surface , so potential at the common.
Centre ,
`V = V_(1) + V_(2) = (1)/(4pi epsi_(0)) [ (q_(1))/(r) + (q_(2))/(R)]`
Substituting the value of `q_(1)` and `q_(2)` form Eq. (2)
` = (1)/(4 pi epsi_(0)) [ (Qr)/((R^(2) + r^(2))) + (QR)/((R^(2) + r^(2)))]`
` = (1)/(4pi epsi_(0)) (Q(R+r))/((R^(2) + r^(2))`