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HW-571-4

# HW-571-4 - Determine E and B(c In frame K E ⊥ B Find the...

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1 Phys 571 , Fall 2010 Assignment #4 Due:9/29/10 1. Prove that for a constant magnetic field B the following two gauges are equivalent, i.e. lead to the same field: A = - By ˆ x and A 0 = 1 2 B × r . Find the gauge transformation that connects A and A 0 . 2. Given the 4-potential A μ = { ϕ, ~ A } . Derive equations describing transformations of electric and magnetic fields ~ E and ~ B under boosts. 3. (a) Write down the transformation laws of the electric and magnetic fields in vector form (i.e. in terms of E , B , E 0 , B 0 and V ). (b) In frame K there is a homogeneous electromagnetic field E , B . Find the relative velocity V of frame K 0 with respect to K in which E 0 k B 0 . Is the solution unique?
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Unformatted text preview: Determine E and B . (c) In frame K E ⊥ B . Find the velocity of K with respect to K such that either B = 0 or E = 0. Is the solution unique? 4. (a) Prove that that the following two equations are equivalent to the two homogeneous Maxwell equations ∇ · B = 0 and c ∇ × E =-∂ t B : (a) ∂ λ F μν + ∂ μ F νλ + ∂ ν F λμ = 0, (b) ε ρλμν ∂ λ F μν = 0. (b) Using the Lorentz transformation equations for E and B prove that E 2-B 2 and E · B are 4-scalars....
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