A magnetic monopole is defined (if one exists) by a magnetic field singularity of the form...

A magnetic monopole is defined (if one exists) by a magnetic field singularity of the form B = br/r3, where b is a constant (a measure of the magnetic charge, as it were). Suppose a particle of mass m moves in the field of a magnetic monopole and a central force field derived from the potential V(r) = −k/r.

(a) Find the form of Newton’s equation of motion, using the Lorentz force given by Eq. (1.60). By looking at the product show that while the mechanical angular momentum is not conserved (the field of force is noncentral) there is a conserved vector

 

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(b) By paralleling the steps leading from Eq. (3.79) to Eq. (3.82), show that for some f(r) there is a conserved vector analogous to the Laplace–Runge–Lenz vector in which D plays the same role as L in the pure Kepler force problem.