Problem 1, hint 2

First, let us consider what is going on. The external field \vec E is inducing dipole moments on the both balls. Since these dipoles are parallel to \vec E, there is no torque applied to them by the external field. You might think that the interaction of these two dipoles can cause the “dumbell” to rotate, but that is not true, recall the first hint. Let us elaborate here. The “dumbbell” cannot be rotated by the internal forces of a closed system, if you calculate the torque acting on a dipole as \vec p\times \vec E_p caused by the field \vec E_p of the other dipole, you are missing one contribution to the total torque. This means that obtaining the equation of motion from the dipole-dipole interaction energy is incorrect, too, as that is basically the same as calculating the dipole-dipole interaction torque with the \vec p\times \vec E formula. However, with these (incomplete) calculations, you may arrive at a correct answer (there is actually a simple reason why those incomplete calculations yield a correct answer). (I have been strict in that all those solutions, even with correct answers, which consider – as described above – either dipole-dipole torque, or dipole-dipole interaction energy only, have been judged as incorrect ones.)

Now, what is causing the “dumbell” to rotate? You just need to consider the paragraph above as an introduction to the problem, and look beyond what is already described here (more recommendations about this will be available on the next Sunday)! Good luck!