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hw7_p5405_f04 - HW 7 Phys5405 f04 due Nov 4 2004 1 Using...

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HW 7 Phys5405 f04 due Nov. 4, 2004 1) Using only F μν , and ijkl , show E · B and B 2 - E 2 /c 2 are Lorentz invariants. The second tensor is the completely antisymmetric tensor of rank 4 with the same properties as that of rank 3 ijk that we have studied. 2a) Show that the 4-vector potential A μ is not unique, by showing the 4-vector potential, A μ = A μ + Ψ ∂r μ , gives the same F μν . 2b) What equation does Ψ have to satisfy in order that you can work with an A μ that sat- isfies, ∇ · A = 0 or ∇ · A + Φ /∂t = 0, even if the given A μ doesn’t have these properties. 3)This problem is like JDJ 12-6. You can forget about part a because in the last homework you showed you can find a frame where E and B are parallel. Thus assume you transferred to this frame just show that part b) holds in this frame. If you want to get JDJ’s solution you should work in cgs units as he does. There F = q ( E + v × B /c ). What are the appropriate integration constants corressponding to JDJ’s solution?
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