hw8_p5405_f06 - reference frame 2b What equation does Ψ...

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HW 8 Phys5405 f06 due 11/2/06 1) A homopolar generator of electric current consists of a solid, cylindical metal disk of moment of inertia I, inner radius R i , and outer radius R o that is free to rotate without friction about an insulating cylindrical axle of radius R i . The axle is directed along the z axis and there is everywhere a constant magnetic field B = B 0 e z . Via brush contacts a load resistance R at point a is connected to the disk at R i and at point b on the other side of the resistance to the disk at R o . The disk is started with angular velocity ω 0 and thereafter there is no external force. Find a value for the current through R. Does the current flow fron R i to R o or visa versa? Show that the disk will slow down and find an expression for ω ( t ). 2a) Show that the 4-vector potential A μ is not unique. That is the 4-vector potential, A 0 μ = A μ + Ψ ∂r μ , gives the same F μν , here the primed quantity does not refer to a different
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Unformatted text preview: reference frame. 2b) What equation does Ψ have to satisfy in order that you can work with an A μ that satisfies, ∇ · A = 0 or ∇ · A + ∂ Φ /∂t = 0, even if A μ doesn’t have these properties? 3) Show explicitely that Eqs. 5.46 and 5.47 in your notes yield Maxwell’s 4 equations. 4) You are familiar with the completely anti-symmetric tensor in 3D, ± ijk . In 4D one can similarly define a completely anti-symmetric tensor ± ijkn , with ± 0123 ≡ 1 and interchange of any two indices means multiplication by -1. And of course if any two indices are the same the tensor is zero. Evaluate ± μναβ F μν F αβ and note that it is a Lorentz invariant. Can you think of an easy explanation why without doing any explicit calculations. 1...
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