A charge Q is divided in two parts such that when these two parts are kept at some sepeation then the force between them is maximum.find charge on each other

let parts are sepereted at a distance of d

so repulsion force F= k q(Q-q)/d²

     here Q and d are constant so for maximum force differentiate the force w.r.t q

 dF/dq = k/d² [Q -2q]

 for maximum force dF/dq=0

                             Q-2q=0

                              Q/q =2

We get this with the help of some calculus :

F = K*[q1*q2]/r², q1+q2=Q

To maximize F(q1,q2) is to maximize q1*q2 so dF/dq=0

rewriting :

F(q1) = K*[q1*(Q-q1)] / r² => F(q1) = K*[q1Q-q1²)] / r²

dF(q1)/dq1 = [K/r²]* [Q-2q1] = 0 so Q- 2q1 = 0 .: q1 = Q/2 and q2=q1.

Remember and keep this result : the maximum A*B you can get is when A=B.

Any field.