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

Let q and q' be the charges on the inner and outer sphere.
As surface charge densities are equal
`therefore (q)/(4 pi r^(2)) = (q')/(4 pi R^(2))`
or `qR^(2) = q' r^(2)`
Also, q + q' + Q . This gives q = Q = q'
Solving the two equations, we get q = `(Qr^(2))/(R^(2) + r^(2)) , q' = (QR^(2))/(R^(2) + r^(2))`
Now potential at the centre is given gy
`V_(c ) = (q)/(4 pi epsilon_(0) r) + (q')/(4 pi epsilon_(0)R) = (Q(r + R))/(4 pi epsilon_(0)(R^(2) + r^(2))) `
This is equal to potential at any point inside the smaller sphere.