Solved Electrically! The mistake was in ignoring the relativistic Fitzgerald-Lorentz contraction, even though the velocities involved are millimeters per second ! In the frame in which the wire is at rest, the positive and negative charge densities exactly balance, otherwise there will be extra electrostatic fields that the electrons will quickly move to neutralize. However, this necessarily means that the densities cannot balance exactly in the frame in which the drifting electrons are at rest. In the electrons’ rest frame, the positive charges, which had density λ coulombs per meter in their rest frame, are moving at speed v , so relativistic contraction will increase their density to: On the other hand, in this electron frame the electrons are at rest, so their density is actually less than it was in the frame of the wire, it has decreased by λv 2 /2 c 2 . The net effect is that the wire, electrically neutral in the lab frame, has a positive charge density λv 2 / c 2 in the frame of the moving electrons (and the outside charge). Recall Gauss’s Law relating
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This note was uploaded on 11/09/2011 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.