Problem 1, hint 3

So, the net torque acting on the “dumbbell” can be caused only by the external field \vec E. At the first approximation, the dipole \vec p induced by the external field is parallel to the external field, and hence, it does not produce any net torque. Therefore, we need to consider smaller corrections: the field produced by one of the dipoles – one of the metal balls – will create an additional electric field \vec E' at the position of the other ball, and the corresponding additional dipole moment \vec p' is induced; that additional dipole moment is no longer parallel to the external field so that the torque due to the external field is no longer zero.

NB! There will be a new problem on 15th December. However, depending on the number of correct solutions, I might extend the submission deadline by one more week, and give one more hint. This will happen if the number of correct solutions remains smaller than 15% from the number of participants.