Consider a spatially nonuniform magnetostatic field expressed in terms of a Cartesian coordinate system by

where B₀ and α are positive constants, |αx| ≪ 1 and |αz| ≪ 1.

(a) Show that this magnetic field is consistent with Maxwell equations, so that both gradient and curvature terms are present. Determine the equation of a magnetic flux line.

(b) Write down the Cartesian components of the equation of motion for an electron moving in the region near the origin under the action of this magnetic field.

(c) Consider the following initial conditions for the electron:

Solve the equation of motion using a perturbation technique, retaining only terms up to the first order in the small parameter α. Show that the leading terms in the velocity components, after eliminating the time periodic parts, are given by

(d) Show that the average position of the electron in the (x, z) plane follows the magnetic flux line that passes through its initial position.

(e) Show that the gradient and curvature drift velocities are given, respectively, by

so that the total drift velocity is precisely the nonperiodic part of v_y.

Transcribed Image Text:

B(x, z)= Bo[azx + (1 + ax)2]