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Re: [Phys-L] feeler-dealer, third law, et cetera



In the paper "Newton's Third Law and Electrodynamics", J. M. Keller, Am. J.
Phys. 10, 302 (1942), Keller treats in detail the case I described, of two
charged particles whose magnetic forces on each other don't sum to zero. To
analyze this situation in detail, Keller makes the following clever move:

"To make the problem definite, we will consider two particles of charge e,
moving along the x and y axes with velocities v_1 and v_2, respectively. To
maintain this uniform motion, we will supply whatever (nonelectric) forces
are required. These forces will of course just balance the electromagnetic
forces on the two charges: and the sum of these forces should be equal to
the time-rate of change of the momentum in the field, since the momenta of
the charges do not change."

After setting up the calculation in a general manner, including
retardation, Keller says this: "The writer was unable to evaluate these
integrals exactly. However, the calculations can be carried out for low
velocities." He goes on to show in the low-velocity case that the time rate
of change of the field momentum at this instant, produced by the (uniformly
moving) charges, is equal to the (nonelectric) forces that keep the
particles moving uniformly.

I'll mention that Ruth and I were unaware of this asymmetry in the magnetic
forces when we assigned an exercise at Carnegie Mellon to calculate the
electric and magnetic forces on one of the two charges in this situation
and a student went ahead and calculated the forces on the other charge as
well and reported that the magnetic forces weren't equal and opposite. Of
course our immediate initial reaction was that the student had made a
mistake.....

Bruce