Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two.

Options

A. `(1)/(4 pi epsilon_(0))((q)/(R ))^(2)`
B. `(sqrt(3))/(4 pi epsilon_(0))((q)/(R ))^(2)`
C. `(sqrt(3))/(16 pi epsilon_(0))((q)/(R ))^(2)`
D. `(sqrt(5))/(16 pi epsilon_(0))((q)/(R ))^(2)`

Answer

C

Explanation

For external points a charged sphere behaves as if the whole of its vharge is concentrated at its center.

Force on A due to B,
`F_(AB)= (1)/(4 pi epsilon_(0))(q^2)/((2R)^(2))= (1)/(4 pi epsilon_(0))(q^2)/((4R)^(2)) along vec(BA)`.
and force on A due to C
`F_(AC)= (1)/(4 pi epsilon_(0))(q^2)/((2R)^(2))= (1)/(4 pi epsilon_(0))(q^2)/((4R)^(2)) along vec(CA)`
Now as angle between BA and CA is `60^(@)` and
`|F_(AB)|=|F_(AC)|=F`
`:. F_(A)=sqrt(f^(2)+F^(2)+2FF cos 60)=sqrt(3) F`
`(1)/(4 pi epsilon_(0))(sqrt(3))/(4)((q)/(R ))^(2)`.