This hint is just a proof that generalized momentum is conserved for a set of mutually interacting isolated set of charges in homogeneous magnetic field (more general statements for inhomogeneous magnetic fields are not considered here).

\sum_im_i\frac{\mathrm d\mathbf v_i}{\mathrm dt}=\sum q_i\mathbf v_i \times \mathbf B=\frac{\mathrm d}{\mathrm dt}\sum_i q_i\mathbf r_i \times \mathbf B.

After integration, we obtain

\mathbf P\equiv \sum_im_i\mathbf v_i+ q_i \mathbf B\times\mathbf r_i =\mathbf p_m+\mathbf B\times \mathbf d=\mathrm{const},

where \mathbf p_m=\sum_im_i\mathbf v_i is the total mechanical momentum, and \mathbf d=\sum_iq_i \mathbf r_i is the total dipole momentum.

Regarding the polarization of the “dumbbell”: keep in mind that in the co-moving frame, (where there is no Lorentz force due to the speed being zero, but where there is an electric field which appears due to the Lorentz transform for electromagnetic fields), the two spheres are connected with a bar and hence, are **equipotential**!

NB! This will be the last hint! The next update of the intermediate results will be on 7th February. If your solution will have still some mistake by then, you’ll be having one more chance as the final deadline for submissions will be Saturday, 13th February, 23:59:59 GMT.

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