How do two charged insulating spheres separated by a finite distance between their centres interact?

Question

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I was solving problems of electrostatics from "Pathfinder for Olympiad and JEE Advanced Physics" by Arvind Tiwari and Sachin Singh when I encountered this problem. I can understand that the conducting spheres will have the charge on their surface rearranged as the electrons are free to move in them. Thus they will have an orientation where the unlike charges will come to closer hemispheres and show maximum attraction while like charges will move to the alternate hemisphere, showing minimum repulsion.
I also thought that this rearrangement will not be possible in insulating spheres as how on the earth the charges gonna dislocate! So I was sure that the answer would be (b). But to my surprise, its (d). Please guide me where I went wrong.

Andrew Steane · Answer

None of conclusions (a) or (b) or (c) or (d) can be certainly drawn. Here is why.
Let's consider first conducting then insulating spheres.

Conducting spheres with like charges:

$(+ ;; ; )$  $(;; ;+);;;;$   hence $F_c > F_m$.

Conducting spheres with unlike charges:

$(;; ; +)$  $(- ;; ;);;;;$   hence $F_c < F_m$.
Now let's turn to insulating spheres. For an insulator the charge might be anywhere and it will stay wherever it happens to be.

Insulating spheres with like charges:

could be $(+ ;; ; )$  $(;; ;+);;;;$ or $(;; ; +)$  $(+ ;; ;);;;;$
or $(;+;)$  $(;+ ;);;;;$ so $F_c$ could be greater than or less than or equal to $F_m$.

Insulating spheres with unlike charges:

could be $(+ ;; ; )$  $(;; ;-);;;;$ or $(;; ; +)$  $(- ;; ;);;;;$ or $(;+;)$  $(;- ;);;;;$ so $F_c$ could be greater than or less than or equal to $F_m$.
Thus conclusion (d) is possible for all materials, but not guaranteed for all materials. And in case (b) the statement should be that the spheres could be conducting, but we cannot know that for sure. This is assuming the spheres simply stay in place at their respective locations, without rotating.
If, on the other hand, the spheres are each free to rotate, then they will rotate to the cases
$(+ ;; ; )$  $(;; ;+);;;;$
and
$(;; ; +)$ $(- ;; ; )$
so then we will find conclusion (d) for any materials. Perhaps that is what the original setter of the question had in mind.

How do two charged insulating spheres separated by a finite distance between their centres interact?

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