Two concentric spheres kept in air have radii R and r. They have similar charge and equal surface charge density `sigma`. The electrical potential at their common centre is (where, `epsi_(0) =` permittivity of free space)

Options

A. `(sigma (R + r))/(epsi_(0))`
B. `(sigma (R - r))/(epsi_(0))`
C. `(sigma (R + r))/( 2 epsi_(0))`
D. `(sigma ( R + r))/(4 epsi_(0))`

Answer

A

Explanation

Potential at the centre due to the sphere 1
`V_(1) = (1)/(4pi epsi_(0)) (q)/(R)`
and due to sphere `2 ` is
`V_(2) = (1)/(4pi epsi_(0)) (q)/(r)`
`therefore ` Electric potential at the common centre
`V = V_(1) + V_(2) = (1)/(4pi epsi_(0)) (q)/(R) + (1)/(4 pi epsi_(0)) (q)/(r)`
`= (q)/(4 pi epsi_(0)) [ (1)/(R) + (1)/(r)] = (q)/(4 pi epsi_(0)) [( R + r)/(Rr)]`
We have `(q)/(4 pi R r) = sigma therefore v = (sigma (R + r))/(epsi_(0))`