Additional details to the steps described by Hint 4.
First Step: To determine the volume charge density inside the rotating ball induced by the Lorentz force, note that the electrostatic force must compensate for the Lorentz force. Apply Gauss’s law to the electrostatic field to find the charge density.
Second Step: Using the volume charge density found in the first step, you can determine the electric field due to the surface charge needed to ensure that the total force acting on charges inside the rotating sphere is zero. From this electric field, you can find the electrostatic potential (and therefore the potential difference between the ball’s center and the point (0,0,R)) through integration.
Fourth Step: Find the potential at the center of the ball due to the surface charge by integrating over all surface charges. This is relatively straightforward since all surface charges are equidistant from the center. Then find the potential at point (0,0,R) through a similar integration over all surface charges, though this requires more complex integration over angles in the spherical coordinate system.
Sixth Step: The calculation of the dipole moment induced on a conducting sphere by a homogeneous electric field has been addressed in several Physics Cup problems. Consult the published solutions of these problems for reference.
Please submit the solution to this problem via e-mail to physcs.cup@gmail.com.