Two concentric spheres of raddi R and r have positive charges `q_(1) and q_(2)` with equal surface charge densities. What is the electric potential at their common centre?

Options

A. `(sigma)/(in_(0)) (R + r)`
B. `(sigma)/(in_(0)) (R-r)`
C. `(sigma)/(in_(0)) ((1)/(R ) + (1)/(r ))`
D. `(sigma)/(in_(0)) ((R )/(r ))`

Answer

A

Explanation

The situation is as shown in the figure. According to the problem
`sigma_(1) = sigma_(2) = sigma`
`:. (q_(1))/(4pi R^(2)) = sigma or q_(1) = sigma 4pi R^(2)`
and `(q_(2))/(4pi r^(2)) = sigma or q_(2) = sigma 4pi r^(2)`
Electric potential at the common centre O is
`V= V_(1) + V_(2)`
`= (1)/(4pi in_(0)) (q_(1))/(R ) + (1)/(4pi in_(0)) (q_(2))/(r ) = (1)/(4pi in_(0)) [(q_(1))/(R ) + (q_(2))/(r )]`
`= (1)/(4pi in_(0)) [(sigma 4pi R^(2))/(R ) + (sigma 4pi R^(2))/(r )]`
`=(4pi sigma)/(4pi in) [(R+ r)] =(sigma)/(in_(0)) (R + r)`