Richard Feynman
In Sect. 17_4 of The Feynman Lectures on Physics, V2, Feynman presents a “paradox.” A superconducting solenoid is mounted on a plastic disk that is free to rotate on a frictionless axle. Around the periphery of the disk are embedded small, charged spheres. A constant current initially circulates in the solenoid. The disk is initially at rest. There is a magnetostatic field and no electric field.
As the solenoid temperature rises toward room temperature, the coil loses its superconductivity. At some temperature the current begins to drop toward zero. dB/dt is nonzero as the current drops, and a circulating E field is induced. Each charged ball experiences an electric force in this field. There is a nonzero torque, and the entire disk assembly begins to rotate.
According to Feynman, here is the paradox: If the initial angular momentum of the entire apparatus is zero, momentum conservation suggests that the final angular momentum should also be zero. But the final angular momentum is clearly not zero. Is momentum conservation violated? Or does the initial solenoid current have angular momentum, even though the solenoid itself is initially perceived to be at rest?
Let us simplify Feynman’s apparatus by dispensing with the charged balls. Again we suppose that the solenoid is initially superconducting and has a constant current.
Now Maxwell was aware of the inertia-like character of electric currents. In Article 547 of A Treatise on Electricity and Magnetism he quotes Faraday: “…the first thought that arises in the mind is that the electricity circulates with something like momentum or inertia in the wire.” Thus even though the solenoid’s wire might initially be at rest, the circulating electricity evidently has inertia!
Maxwell appears not to have appreciated the role of interactive forces in a current’s momentum, perhaps because the particulate nature of electricity had not yet been discovered. In his own words, “momentum’ is an intrinsic property of a physical entity, independent of other entities in the environment.” Thus he believed that the total momentum of a system is simply the sum of the momenta of its constituent parts. In Article 549 of his Treatise he states, “…if the phenomena are due to momentum, the momentum is certainly not that of the electricity in the wire, because the same wire, conveying the same current, exhibits effects which differ according to its form…”
We now know from Maxwell’s own equations that an accelerated charged particle has “electromagnetic mass.” If such a particle is accelerated, a non-conservative electric field is induced right at the charge and in the surrounding space. This electric field results in a “self-force” which is oppositely directed to the acceleration. It acts on the charge, and the particle evidently has more inertia than if it were uncharged. Note that this induced field is maximum in the space to the “side” of the accelerating charge, and is zero “in front of and in back of” the charge.
A straight line of N such charged particles accordingly has a total electromagnetic mass that is practically N times the electromagnetic mass of a single charged particle. But what about the (ideal) case where all N charges are superimposed? In this case a given particle experiences not only its own self-induced electric force. In the acceleration-induced electric field of the other charges it experiences a total force of N times its own self-force. And there are N such charges. So the net electric force is N2 times the force experienced by a single particle. And collectively this N-charge “super particle” has an electromagnetic mass which is proportional to the square of its total charge.
A solenoid is in effect a compromise to N superimposed charged particles. When the solenoid’s current varies, a given constituent particle experiences not only its own self-induced electric force, but to a lesser extent the interactive electric forces of the other particles. Thus (contrary to what Maxwell thought) the net electromagnetic momentum of the solenoid is more than N times the momentum of a single particle, but somewhat less than N2 times the single-charge momentum.
In conclusion, our modified Feynman apparatus will begin to rotate when the current begins to drop (assuming the solenoid is attached to the platform). It all stems from the fact that the electric current in the wire does indeed have inertia (as Faraday surmised). And the effect can be magnified by winding the host wire into a spiral.