Classical Electrodynamics · Problem 12.12

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bmt-equation-spin-precession

Problem

In Section 11.11 the BMT equation of motion for the spin of a particle of charge ee and a magnetic moment with an arbitrary gg factor was obtained. (a) Verify that (11.171) is the correct equation for the time derivative of the longitudinal component of the rest-frame spin vector s\mathbf{s}. (b) Let n^\hat{\mathbf{n}} be a unit 3-vector perpendicular to β^\hat{\boldsymbol{\beta}} and coplanar with β^\hat{\boldsymbol{\beta}} and s\mathbf{s} (n^\hat{\mathbf{n}} is generally time dependent). Let θ\theta be the angle between β^\hat{\boldsymbol{\beta}} and s\mathbf{s}. Show that the time rate of change of θ\theta can be written as dθdt=emc[(g2−1)n^⋅(β^×B)+(gβ2−1β)n^⋅E]\frac{d\theta}{dt} = \frac{e}{mc}\left[\left(\frac{g}{2}-1\right)\hat{\mathbf{n}}\cdot(\hat{\boldsymbol{\beta}} \times \mathbf{B}) + \left(\frac{g\beta}{2} - \frac{1}{\beta}\right)\hat{\mathbf{n}}\cdot\mathbf{E}\right] where E\mathbf{E} and B\mathbf{B} are the fields in the laboratory and cβ=cββ^c\boldsymbol{\beta} = c\beta\hat{\boldsymbol{\beta}} is the particle's instantaneous velocity in the laboratory. (c) For a particle moving undeflected through an E×BE \times B velocity selector and with (n^×β^)⋅B=B(\hat{\mathbf{n}} \times \hat{\boldsymbol{\beta}})\cdot\mathbf{B} = B, find dθ/dtd\theta/dt in terms of the gyration frequency eB/γmceB/\gamma mc. (d) By defining the two 4-vectors, Lα=(γβ,γβ^)L^\alpha = (\gamma\beta, \gamma\hat{\boldsymbol{\beta}}) and Nα=(0,n^)N^\alpha = (0, \hat{\mathbf{n}}), show that dθ/dτd\theta/d\tau can be written in the quasi-covariant form dθdτ=emc[g2 Lα−1v Uα]FαβNβ\frac{d\theta}{d\tau} = \frac{e}{mc}\left[\frac{g}{2}\,L_\alpha - \frac{1}{v}\,U_\alpha\right]F^{\alpha\beta}N_\beta where UαU^\alpha is the particle's 4-velocity.

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