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 two spheres of radii r and R respectively, then the surface charge density is given by
`sigma = (q_(1))/(4pi r^(2)) = (q_(2))/(4pi R^(2))` or `(q_(1))/(q_(2)) = (r^(2))/(R^(2))`
`(q_(1))/(q_(2)) + 1 = (r^(2))/(R^(2)) + 1 rArr (q_(1) + q_(2))/(q_(2)) = (r^(2) + R^(2))/(R^(2))`
But `q_(1) + q_(2)` = total charge Q.
`:. (Q)/(q_(2)) = (r^(2) + R^(2))/(R^(2))` or `(QR)/(q_(2)) = (r^(2) + R^(2))/(R)`
or `(q_(2))/(R ) = (QR)/(r^(2) + R^(2))`
Similarly, `(q_(1))/(r ) = (Qr)/(r^(2) + R^(2))`
Potential at the common centre `= (1)/(4pi epsilon_(0))(q_(1))/(r ) + (1)/(4pi epsilon_(0))(q_(2))/(R)`
`= (1)/(4pi epsilon_(0))((q_(1))/(r) + (q_(2))/(R)) = (1)/(4pi epsilon_(0))((Qr)/(r^(2) + R^(2)) + (QR)/(r^(2) + R^(2)))`