Consider the motion of a particle with mass m and electric charge qe in the field of a (...

Consider the motion of a particle with mass m and electric charge q_e in the field of a (hypothetical) stationary magnetic monopole q_m at the origin:

a) Find the acceleration of q_e, expressing your answer in terms of q, q_m, m, r (the position of the particle), and v (its velocity).

(b) Show that the speed v = |v| is a constant of the motion.

(c) Show that the vector quantity

is a constant of the motion. [Hint: differentiate it with respect to time, and prove—using the equation of motion from (a)—that the derivative is zero.]

(d) Choosing spherical coordinates (r, θ, φ), with the polar (z) axis along Q,

(i) calculate and show that θ is a constant of the motion (so q_e moves on the surface of a cone—something Poincaré first discovered in 1896)²⁴;

(ii) calculate and show that the magnitude of Q is

(iii) calculate show that

(that is: determine the function f (r )).

(f) Solve this equation for r(φ).