Force and Hidden Momentum for Classical Microscopic Dipoles

The concept of hidden momentum is reviewed and the first rigorous derivation from Maxwell's equations is provided for the electromagnetic force on electrically small perfect electric conductors of arbitrary shape in bandlimited but otherwise arbitrarily time-varying fields. It is proven for the Amperian magnetic dipoles of these perfect conductors that a "hidden-momentum" electromagnetic force exists that makes the force on these time varying Amperian magnetic dipoles equal to the force on magnetic-charge magnetic dipoles with the same time varying magnetic dipole moment in the same time varying externally applied fields. The exact Mie solution to the perfectly conducting sphere under plane-wave illumination is used to prove that the expressions for the total and hidden-momentum forces on the arbitrarily shaped electrically small perfect conductors correctly predict the forces on perfectly conducting spheres. Remarkably, it is found that the quadrupolar fields at the surface of the sphere are required to obtain the correct total force on the sphere even though the quadrupolar moments are negligible compared to the dipole moments as the electrical size of the sphere approaches zero.