A charge of magnitude q is divided into two parts such that force between resulting two charges is maximum when separated through some distance r.The division of charges would be

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.

PS

About distance. F(r) max occurs when r -> 0 so 1/r ² -> infinite