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.

Options

A. `Q((r + R)/(r^(2) + R^(2)))`
B. `(Q)/(2)((r + R)/(r^(2) + R^(2)))`
C. `2Q((r + R)/(r^(2) + R^(2)))`
D. Zero

Answer

1

Explanation

Let charge on inner shell is `q` , charge on outer shell is `Q - q`
`(q)/(4 pi r^(2)) = (Q - q)/(4 pi R^(2)) rArr q((R^(2))/(r ^(2)) + 1) = Q rArr q = (Qr^(2))/(R^(2) + r^(2))`
`Q - q = (QR^(2))/(R^(2) + r^(2))`
`V_(o) = (1)/(4 pi in_(0)) [ (q)/(r ) + (Q - q)/( R)]`
` = (1)/(4 pi in_(0)) [ (Qr)/(R^(2) + r^(2)) + (QR)/(R^(2) + r^(2))]`
`= (1)/(4 pi in_(0)) .(Q(r + R))/((R^(2) + r^(2)))`