In the following, we will study the potentials and A in the time-dependent Maxwell equations. Assume , r = 1. a) Startin

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In The Following We Will Study The Potentials And A In The Time Dependent Maxwell Equations Assume R 1 A Startin 1 (54.6 KiB) Viewed 49 times

In the following, we will study the potentials and A in the time-dependent Maxwell equations. Assume , r = 1. a) Starting from the Maxwell equations, show that there are potentials o, A s.t. B = x A and E-V-0₂Ã. b) Plug and A into Maxwell's equations and show that they can be written as □A-VK = -403, +K=-p/€0, where = ²-μocoo and K = A + μOOOO. c) In Lorenz gauge, we set K = 0. However, this does not fix the gauge completely. Which property must the gauge function satisfy to remain in Lorenz gauge. d) Starting from arbitrary potentials A(r, t), o(r, t), perform a gauge transformation (spec- ifying the gauge function) to obtain = 0. Show that the electric field did not change. Under which condition is this compatible with Coulomb gauge, V. A=0? Hints: a) You need the homogeneous Maxwell equations, B = 0 and V × E = -0,B. Use that for any vector field a with x a=0 there is a scalar field b such that a-b. b) You need the two inhomogeneous Maxwell equations, V-E p/co and x B= Moj+Hoco, Ē. c) Gauge transformations are performed by choosing a function A(F, t) and setting A = A + VA and o' = 6-8, A.