Here is a step-by-step sketch how to solve this problem. Some details will be provided in the fifth (final) hint.
First, determine the volume charge density inside the rotating ball induced by the Lorentz force. There is also surface charge that will be addressed later.
Second, the electrons inside the metal ball need to be at equilibrium, and are subject to: (a) Lorentz force; (b) force due to the electric field of the volume charge; (c) force due to the electric field of the surface charge. The electric field of the surface charge must compensate for contributions (a) and (b). You need to show that hence, the electrostatic potential of field (c) must be a quadratic polynomial of the coordinates, and calculate the potential difference between the ball’s center and the point (0,0,R).
Third, knowing that inside homogeneously charged ellipsoids, the potential is a quadratic polynomial of the coordinates, we can construct a surface charge with the same property by superposing two ellipsoids with equal and opposite charge densities, with slightly different shapes (take the limit as the shape difference approaches zero and charge volume density approaches infinity). The only thing we need from here is how the surface charge depends on the polar angle: \sigma=\sigma_1f(\theta), where the constant \sigma_1 will be found later. Find f(\theta) by calculating the radial shape difference between a sphere and a slightly oblate or prolate ellipsoid. Remember that any ellipsoid can be represented as an affine transformation of a sphere.
Fourth, determine \sigma_1 using the previously calculated potential difference between the ball’s center and point latex[/latex].
Fifth, notice that the total charge of the rotating ball is zero, and, as it turns out, its dipole moment is also zero. Hence, it creates a quadrupole field at large distances; calculate the potential (and subsequently the electric field) by determining its quadrupole moment Q_{xx}.
Sixth, note that this quadrupole field will polarize the other ball; calculate the resulting dipole moment of the other ball and hence, determine the force exerted on it.
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