#### Two identical conducting spheres A and B, carry equal charge. They are separated by a distance much larger than their diameters, and the force between them is F. A third identical conducting sphere, C is uncharged. Sphere C is first touched to A, then to B, and then removed. As a result, the force between A and B would be equal to : Option 1) F Option 2) 3F/4 Option 3) 3F/8 Option 4) F/2

As we have learned

Coulombic force -

- wherein

K - proportionality Constant

Q1 and Q2 are two Point charge

Initially force = F

let charge on both sphere = Q

when in contact , charge will transfer till potential become equal , hence Q/2 on each sphere

When in contact with another sphere then change will transfer till charge on both is 34/4

new force

=3F/8

Option 1)

F

This is incorrect

Option 2)

3F/4

This is incorrect

Option 3)

3F/8

This is correct

Option 4)

F/2

This is incorrect

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(D) is the correct option as

Lets us consider the first case when A comes in contact with C Charge transfer occurs till an equilibrium is attained as shown in the following steps.Let A have a charge Q as well as B have a charge Q and Qa,Qb,Qc be the new charges acquired after C comes in contact with A and B and r be radius of A,Band c

Va=Vc

(1/4 pi e0) Qa/r  = (1/4 pi e0) Qc/r

By solving the above equation we would the the following result

Qc=Q*r/(r+r)

Qc=Q/2

Similarly in the second case  Qc=Q/2 in the end both A and B have charges of Q/2 each by Coloumb's inverse square law

F=1/(4 pi e0) Q^2/r^2------->(1)

New force F"=1/(4 pi e0)Q^2/4r^2----------->(2)

Dividing (1) and (2) F/F"=1/4

Therfore new force is F/4 as seen in D option

#### Samanyu

let Q be the initial charge on the spheres.

•  Force  F=kQ^2/r^2

when sphere C is attached to sphere A then net charge on the both spheres(A & C) is   =(Q+0)/2 =Q/2

Now when sphere C containing Q/2 charge is attached with sphere B (containing Q charge)

the net charge on the spheres B & C =(Q/2+Q)/2 =3Q/4

then force between A & B   F1=(kQ/2.3Q/4)/r^2

=(kQ^2/r^2).(3/8)

=3F/8