Problem 1 A thin straight conducting cable, oriented at rest along the z-axis in an inertial reference frame S, carries

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Problem 1 A Thin Straight Conducting Cable Oriented At Rest Along The Z Axis In An Inertial Reference Frame S Carries 1 (207.5 KiB) Viewed 39 times

Problem 1 A thin straight conducting cable, oriented at rest along the z-axis in an inertial reference frame S, carries a constant current I. The cable is charge neutral. a) Show, by the use of Ampere's law, that the current produces a magnetic field B = B(r)ep, where (r, ) are polar coordinates in the (x, y)-plane, and ep is a unit vector in the direction of increasing 0. Determine the function B(r). Consider next the same situation in a reference frame S' that moves with velocity v along the z-axis. b) Use the fact that charge and current densities transform under Lorentz transformation as components of a current four-vector to show that in S' the conducting cable will be charged. Determine the charge per unit length, \', and the current I' in this reference frame. c) Use Gauss' and Ampere's laws to determine the electric and magnetic fields, E' and B', as functions of the polar coordinates (r',¢') in refer- ence frame S'. d) Show that if the fields in S' are derived from the fields in S by use of the relativistic transformation formulas for E and B, this gives the same results as found in c).