A particle of charge e and mass m moves relativistically in a helical path in a uniform magnetic field B. The pitch angle of the helix is a (a = 0 corresponds to circular motion).
(a) By arguments similar to those of Section 14.4, show that an observer far from the helix would detect radiation with a fundamental frequency w0 = wB/cos2a and that the spectrum would extend up to frequencies in the order of wc = 3/2 ?33 wB cos a where wB = eB/?3mc. (Take care in determining the radius of curvature I? of the helical path.)
(b) From part a and the results of Section 14.6, show that the power received by the observer per unit solid angle and per unit circular frequency interval is
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where wB and a)c are defined above, ? = (w/2wc) (l + ?32I^2)3/2, and I^ is the angle of observation measured relative to the particle’s velocity vector, as inFigure.
(a) By arguments similar to those of Section 14.4, show that an observer far from the helix would detect radiation with a fundamental frequency w0 = wB/cos2a and that the spectrum would extend up to frequencies in the order of wc = 3/2 ?33 wB cos a where wB = eB/?3mc. (Take care in determining the radius of curvature I? of the helical path.)
(b) From part a and the results of Section 14.6, show that the power received by the observer per unit solid angle and per unit circular frequency interval is
where wB and a)c are defined above, ? = (w/2wc) (l + ?32I^2)3/2, and I^ is the angle of observation measured relative to the particle’s velocity vector, as inFigure.
