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A charge Q is distributed over two conce...

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.

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Let q and q' be the charges on the inner and outer sphere.
As surface charge densities are equal
`:. (q)/( 4pi r^(2)) = (q')/( 4piR^(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 by
`V_( c) = (q)/( 4pi epsilon_(0)r)+ ( q')/ ( 4pi epsilon_(0)R)`
`= (Q(r+ R))/( 4pi epsilon_(0)(R^(2) + r^(2)))`
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