The magnetic field of the earth can be represented approximately by a magnetic dipole of magnetic moment M=8.1×1025 gauss-cm3. Consider the motion of energetic electrons in the neighborhood of the earth under the action of this dipole field (Van Allen electron belts). [Note that M points south.]
(a) Show that the equation for a line of magnetic force is r=r0sin2θ, where θ is the usual polar angle (colatitude) measured from the axis of the dipole, and find an expression for the magnitude of B along any line of force as a function of θ.
(b) A positively charged particle circles around a line of force in the equatorial plane with a gyration radius a and a mean radius R (a≪R). Show that the particle's azimuthal position (east longitude) changes approximately linearly in time according to
ϕ(t)=ϕ0−23(Ra)2ωB(t−t0)
where ωB is the frequency of gyration at radius R.
(c) If, in addition to its circular motion of part b, the particle has a small component of velocity parallel to the lines of force, show that it undergoes small oscillations in θ around θ=π/2 with a frequency Ω=(3/2)(a/R)ωB. Find the change in longitude per cycle of oscillation in latitude.
(d) For an electron of 10 MeV kinetic energy at a mean radius R=3×107 m, find ωB and a, and so determine how long it takes to drift once around the earth and how long it takes to execute one cycle of oscillation in latitude. Calculate the same quantities for an electron of 10 keV at the same radius.