Three identical spheres, each having a charge q and radius R. are kept in such a way that each touches the other two. The magnitude of the electric force on any sphere due to the other two is

Options

A. `(1)/(4pi in_(0)) ((q)/(R ))^(2)`
B. `(1)/(4pi in_(0)) (3)/(4) ((q)/(R ))^(2)`
C. `(1)/(4pi in_(0)) (sqrt(3))/(4) ((q)/(R ))^(2)`
D. `(1)/(4pi in_(0)) (3)/(2) ((q)/(R ))^(2)`

Answer

C

Explanation

Force one sphere `C` due to `A, F_(AC)` = Force on sphere
`C` due to `B, F_(BC) =(1)/(4pi in_(0)) (q^(2))/((2R)^(2))`

Now `F_(AC)` and `F_(BC)` are inclined at `60^(@)` to each other, therefore resultant force on `C` is
`F_(R) = sqrt(F_(AC)^(2) + F_(BC)^(2) + 2 F_(AC) F_(BC) cos 60^(@))`
`= sqrt(F^(2) + F^(2) + 2F^(2) cos 60^(@))`
`= sqrt(3) F` [Taking `F_(AC) = F_(BC) = F`]
`= (1)/(4pi in_(0)) (sqrt(3))/(4) ((q)/(R))^(2)`