Question

Of the charge Q initially on a tiny sphere, a portion q is to be transferred to a second, nearby sphere. Both spheres can be treated as particles. For what value of q/Q>0.5 will the electrostatic force between the two parts have 1/2 of the maximum possible value?

Answer 1

After q is transferred to nearby sphere, charge on tiny sphere=Q-q

Electrostatic force between the two charges=k*(Q-q)*q/r²

K is constant and r is distance between them which again is constant for the given scenario

Force=(Qq-q2)*k/r²

For maximum force, dF/dq=0

Or, Q-2q=0

q=0.5Q

Maximum force=0.5Q*0.5Q*k/r²=0.25Q²*k/r²

Required force=0.5*maximum force=0.5*0.25*Q²k/r²=0.125Q²k/r²

(Q-q)*q*k/r²=0.125Q²*k/r²

(Q-q)*q=0.125Q²

Qq-q2=0.125Q2

Dividing the equation by Q² , we get

(q/Q)-(q/Q)² =0.125

Let q/Q=a

a-a² =0.125

a²-a+0.125=0

a=0.853 or 0.146

required q/Q>0.5

Therefore, 0.853 is the answer