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Suppose Joe is in the lab frame. He sets up an electromagnet consisting of
a wire with current flowing in it. According to Joe, at some typical point:
* The wire has stationary positive metal ions with a density rho_p = rho_0.
* The wire contains electrons moving with rapidity u.
* The wire is macroscopically electrically neutral.
Therefore the electron density rho_e must be equal to rho_0
(ignoring terms of order u**2).
Now suppose a test charge whizzes by at a rapidity v. Joe sees the test
charge deflected by the magnetic field of his magnet, and not by any
electric field. (Remember, Joe has no electric field.)