Two identical very small metal spheres, A and B, that are electrically neutral are fixed at a distance R On insulating handles. A third identical sphere C carrying charge Q is also connected to an insulating handle. C ia first brought into contact with A and B and then moved away from these two spheres. It is found that A and B apply `F_1` force on each other.
Now C with the remaining charge is again brought into contact with A and then moved away from A and B. It is found that A and B apply `F_2` force on each other. Now C is further brought into contact with B and then moved away from A and B. It is found that A and B apply `F_3` force on each other Finally B is moved closer to A at a distance r, such that the force between them again becomes `F_1`.
The magnitude of `F_1//F_2` is

Options

A. `3//4`
B. `4//3`
C. `4//5`
D. `5//4`

Answer

B

Explanation

Charge acquired by A when C touches it for the first time , `q_1=Q//2`
Charge acquired by B and C touches it for the first time, `q_2=Q//4`
Force between A and B :
`F_1=1/(4piepsilon_0) ((Q/2)(Q/2))/R^2 =1/8 1/(4piepsilon_0) Q^2/R^2`…(i)
Charge acquired by A when C touches if for the second time ,
`q_3=1/2 (Q/2+Q/4)=(3Q)/8`
Force between A and B :
`F_2=1/(4piepsilon_0) (("3Q"/8)(Q/4))/R^2 =3/32 1/(4piepsilon_0) Q^2/R^2`...(ii)
`F_1/F_2=1/8xx32/3=4/3`