The x-component of the generalized momentum of a charged particle in magnetic field is conserved as long as the magnetic field does not depend on the x-coordinate. The generalized momentum is expressed in terms of the vector potential which is outside of the IPhO syllabus (although it is a very useful concept); meanwhile, for a homogeneous magnetic field, the conserved generalized momentum can be deduced directly from the Newton’s 2nd law, expressed in terms of the Lorentz force. Finally, due to the Newton’s third law, the total generalized momentum of an isolated set of particles is conserved regardless of their mutual interaction.
The generalized momentum of the “dumbbell” can be expressed in terms of its dipole momentum. So, you need to express the dipole momentum induced on the “dumbbell” due to the Lorentz force as a function of its orientation. There are two effects: the balls will obtain equal and opposite charges, and each of the balls will become polarized. Depending on the orientation of the rod, you may be able to neglect one of these contributions.
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