Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 3

The Electromagnetic Field Equations

Physics — Landau & Lifshitz · drift-guiding-centre-quasihomogeneous-magnetic-field

Problem

Determine the velocity of drift of the guiding centre of the orbit of a nonrelativistic charged particle in a quasihomogeneous magnetic field (H. Alfv\'{e}n, 1940). Hint: Write the equation of the trajectory in the form r=R(t)+ζ(t)\mathbf{r} = \mathbf{R}(t) + \boldsymbol{\zeta}(t), where R(t)\mathbf{R}(t) is the radius vector of the guiding centre and ζ(t)\boldsymbol{\zeta}(t) describes the rotational motion about it. Expand H(r)=H(R)+(ζ⋅∇)H(R)\mathbf{H}(\mathbf{r}) = \mathbf{H}(\mathbf{R}) + (\boldsymbol{\zeta}\cdot\nabla)\mathbf{H}(\mathbf{R}) and average the force (e/c)r˙×H(r)(e/c)\dot{\mathbf{r}}\times\mathbf{H}(\mathbf{r}) over a period of the circular motion. Show that the total transverse force (including the centrifugal force mv∥2/ρmv_{\|}^2/\rho when the particle also has a longitudinal velocity v∥v_{\|}) is f⊥=vmρ(v∥2+v⊥22),\mathbf{f}_{\perp} = \mathbf{v}\frac{m}{\rho}\left(v_{\|}^2 + \frac{v_{\perp}^2}{2}\right), and hence find the drift velocity of the guiding centre: vd=1ωρ(v∥2+v⊥22)v×n.v_d = \frac{1}{\omega\rho}\left(v_{\|}^2 + \frac{v_{\perp}^2}{2}\right)\mathbf{v}\times\mathbf{n}.

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