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

Options

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

Answer

A

Explanation

Potential at the centre due to sphere with radius
`V_(1) = (q_(1))/(4piepsilon_(0)R)`
Potential at the centre due to sphere with radius r ,
`V_(2) = (q_(2))/(4piepsilon_(0)r)`
Given , surface charge densities for both spheres
`sigma = sigma_(R) = sigma_(r )`
i.e `(q_(1))/(4piepsilon_(0)R^(2)) = (q_(1))/(4piepsilon_(0)r^(2))`
`rArr " " q_(1) = (R^(2))/(r^(2)) q_(2)`
Ttal potential at the centre due to both spheres
`v_= V_(1) +V_(2)`
`= (q_(1))/(4piepsilon_(0)R) +(q_(1))/(4piepsilon_(0)r) = 1/(4piepsilon_(0)) ((q_(1))/R+(q_(2))/r)`
`=1/(4piepsilon_(0)R) ((R^(2))/(r^(2))(q_(2))/R) +(q_(2))/r = (q_(2))/(4piepsilon_(0)) ((R+r)/r^(2))`
`= (q_(2))/(4pir^(2)epsilon_(0)) (R+r) = sigma/(epsilon_(0)) (R+r)` .