The Earth's magnetic field can be represented, in a first approximation, by a magnetic dipole placed in the Earth's center, at least up to distances of a few Earth radii (R_E)·

(a) Using the fact that, at one of the magnetic poles, the field has a magnitude of approximately 0.5 gauss near the surface, calculate the dipole magnetic moment.

(b) Consider the motion of an electron of energy E₀ at a radial distance r₀, where r₀ > R_E. Calculate its cyclotron frequency and gyroradius.

(c) Assuming that the electron is confined to move in the equatorial plane, calculate its gradient and curvature drift velocities, and determine the time it takes to drift once around the Earth, at the radial distance r₀.

(d) Calculate the period of the bounce motion of the electron, as it gets reflected back and forth between the magnetic mirrors near the poles. What is the altitude of the reflection points? Assume that W_∥ = W_⊥ at the magnetic equatorial plane.

(e) Obtain numerical values for the results of items (b), (c), and (d), considering E₀ = 1 MeV and r₀ = 4R_E. Examine these results in terms of typical values for charged particles in the outer Van Allen radiation belt.

(f) Assuming that there is an isotropic population of 1 MeV protons and 100 keV electrons at about 4 R_E, each having a density n_e = n_i = 10⁷ m⁻³ in the equatorial plane, calculate the ring current density in ampere/m².