Question

Question 5 [12 10 22 marks] (a) In a given inertial reference frame, S', consider a region of space where there is a uniform and constant electric field, E', and zero magnetic field, i.e. B' = 0. The frame S' moves with respect to an observer, in another frame S, with velocity v. Write an expression for the electric field, E, observed in S? Clearly explain any notation (i) and new quantities introduced Write an expression for the magnetic field, B, observed in S? (ii) Explain your answers to parts (i) and (ii) in the context of transformations of the charge (i and current densities between S' and S If the electric field, E', is in the +z-direction, then the potentials producing these fields are given by cp'(t',x', y', z')- Eoz' and A'(t,x, y, z) 0 Using these, find the scalar potential, D(t, x, y,z), in S if ü = vî? Make sure you express (iv) your answer in terms of coordinates in S What is the vector potential, A(t, x,y,z) if v = vt? (v) Consider the covariant expression, KH = qFHVuv, where q is the charge on a particle, u, is its (b) dp is the Minkowski force, dt four-velocity and F is the electromagnetic field tensor. Also, KH = where pH (E/c,p) is the four-momentum, E and p are the relativistic energy and momentum respectively and ris the proper time. dt 1 = y where y = dt (i) Show that u is the speed of the particle and t is the time /c measured in an observer's frame. (ii) Show that 1 dE dp K YC dt' dt What equation is recovered by setting u = 0 in the expression K" = qFHVu,? (iii) (iv) What equation is reproduced by setting u= 1 in the expression KH = qFHVu,? In your answers to (iii) and (iv) above explain your reasoning and show any working required to reach your conclusion

Answer 1

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