homework_2

# homework_2 - ELECTRODYNAMICS PROBLEM SET 2 due February 9...

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ELECTRODYNAMICS PROBLEM SET 2 due February 9, before class Problem 1.: Field invariants a) Write the Lorentz scalars F μν F μν and ± μνλρ F μν F λρ in terms of ~ E and ~ B . b)Show that if ~ E and ~ B are seen as perpendicular in one reference frame, they are perpendicular in any other reference frame. c) Show that if | ~ E | = | ~ B | in one reference frame, | ~ E 0 | = | ~ B 0 | in any other reference frame. d) Show that if | ~ E | > | ~ B | in one reference frame, | ~ E 0 | > | ~ B 0 | in any other reference frame. Problem 2.: Relativistic invariance of Maxwell’s equations in your face An inﬁnitely long straight wire of negligible cross sectional area is at rest ans has an uniform charge density q 0 in the in the inertial frame K 0 . The frame K 0 (and the wire) move with velocity v parallel to the direction of the wire with respect to the lab frame K . a) Write down the electric and magnetic ﬁelds in cylindrical coordinates in the rest frame of the wire. Using the Lorentz transformation properties of the ﬁelds, ﬁnd the components of the electric and magnetic ﬁelds in the lab. b) What are the charges and current densities associated with the wire in its rest frame? In the lab frame?
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