Problem 1, hint 5

First, check if your results have correct dimensionality (units). Few examples of mistakes leading to dimensional mismatch: forgotten square root or forgotten moment of inertia when deriving circular frequency from the equation of motion.

Second, check all the numerical factors. Few examples of possible mistakes: forgotten factor of 2 because there are two balls, and external field exerts torque on the both balls; mistake by a factor of 2 when deriving the equation of motion from energy conservation law, and failing to fix the induced dipole moments to a constant when using the virtual displacement method; doing too many simplifications when considering small angle approximation (neither radial nor tangential components of the electric field in the expression of the electric field of a dipole can be neglected); wrong expression for the polarizability of a sphere; mistake in the expression for the moment of inertia of the “dumbell”.

Third, in order to check your expression for the polarizability of a sphere, you may use the fact that if a metallic sphere is placed into a homogeneous electric field \vec E, such a charge distribution is induced on the surface of the sphere that outside the sphere, the field due to these charges corresponds to the field of an ideal electric dipole \vec p. The electric potential created by this dipole field compensates the potential due to the external field \vec E: the sum of these two contributions to the net potential is constant over the entire surface of the sphere. Check if your polarizability satisfies this condition.

NB! Those contestants who have submitted almost correct solutions with minor mistakes will have one final opportunity to submit the corrected solution by 24th December, 23:59 GMT.