Charge Q is distributed between two concentric hollow sphere placed in vacuum in such a way that their surface densities of charges are equal. If the radii of the two spheres be r and R (R gt r), calculate the potential at their centre.

Explanation

Let the charge on the smaller and the bigger spheres be `Q_(1) and Q_(2)`, respectively.
`:. Q = Q_(1)+Q_(2)`
As the surface densities of charge of the two spheres are equal,
`:. (Q_1)/(4 pi r^(2)) = (Q_2)/(4 pi R^(2)) or, (Q_1)/(Q_2) = (r^2)/(R^2)`
or, (Q_(1)+Q_(2)) = (r^2)/(R^2)`
or, (Q_(1)+Q_(2))/(Q_2) = (r^(2)+R^(2))/(R^2) or, (Q_2)/Q = (R^2)/(r^2 + R^2) or, Q_(2) = (Q cdot R^(2))/(r^(2)+R^(2))`
Similarly, `Q_(1) = (Q cdot r^(2))/(r^(2) + R^(2))`
`:.` Potential at the centre,
`V = (Q_1)/(r ) + (Q_2)/R = (Q cdot r^(2))/(r(r^(2)+R^(2)))+(Q cdot R^(2))/(R(r^(2)+R^(2))) = (Q (r+R))/(r^(2)+R^(2))`.