Introduction to Electrodynamics · Problem 5.45

Magnetic Fields in Matter — David J. Griffiths

Physics · magnetic-monopole

Problem

Consider the motion of a particle with mass mm and electric charge qeq_e in the field of a (hypothetical) stationary magnetic monopole qmq_m at the origin: B=μ04πqmr2r^.\mathbf{B} = \frac{\mu_0}{4\pi}\frac{q_m}{r^2}\hat{r}. (a) Find the acceleration of qeq_e, expressing your answer in terms of qq, qmq_m, mm, r\mathbf{r} (the position of the particle), and v\mathbf{v} (its velocity). (b) Show that the speed v=∣v∣v = |\mathbf{v}| is a constant of the motion. (c) Show that the vector quantity Q≡m(r×v)−μ0qeqm4πr^\mathbf{Q} \equiv m(\mathbf{r} \times \mathbf{v}) - \frac{\mu_0 q_e q_m}{4\pi}\hat{r} 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)(r, \theta, \phi), with the polar (zz) axis along Q\mathbf{Q}, ((i)) calculate Q⋅ϕ^\mathbf{Q} \cdot \hat{\phi}, and show that θ\theta is a constant of the motion (so qeq_e moves on the surface of a cone—something Poincaré first discovered in 1896); ((ii)) calculate Q⋅r^\mathbf{Q} \cdot \hat{r}, and show that the magnitude of Q\mathbf{Q} is Q=μ04π∣qeqmcos⁡θ∣;Q = \frac{\mu_0}{4\pi}\left|\frac{q_e q_m}{\cos\theta}\right|; ((iii)) calculate Q⋅θ^\mathbf{Q} \cdot \hat{\theta}, show that dϕdt=kr2,\frac{d\phi}{dt} = \frac{k}{r^2}, and determine the constant kk. (e) By expressing v2v^2 in spherical coordinates, obtain the equation for the trajectory, in the form drdϕ=f(r)\frac{dr}{d\phi} = f(r) (that is: determine the function f(r)f(r)). (f) Solve this equation for r(ϕ)r(\phi).

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