A charge Q is distributed over two concentric conducting thin spherical shells radii r and R `(R gt 1)` . If the surface charge densities on the two shells are equal, the electric potential at the common centre is:

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.