According to Hint 2, the surface charge can be represented as a superposition of almost spherical oblate and prolate spheroids with equal and opposite volume charges. Based on this, one can deduce the functional form of how surface charge density depends on the polar angle. This functional form has two parameters, the values of which can be determined by knowing the electrostatic potential distribution inside the sphere
It appears that the rotating sphere has zero dipole moment; hence, its field at large distances is a quadrupole field, and therefore its quadrupole moment needs to be calculated.
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