Classical Electrodynamics · Problem 6.18

Plane Electromagnetic Waves and Wave Propagation — John David Jackson

Physics — E&M · dirac-string-vector-potential-monopole

Problem

Consider the Dirac expression

\mathbf{A}(\mathbf{x}) = \frac{g}{4\pi}\int_L \frac{d\mathbf{l}' \times (\mathbf{x} - \mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|^3}

for the vector potential of a magnetic monopole and its associated string

L

. Suppose for definiteness that the monopole is located at the origin and the string along the negative

z

axis. (a) Calculate

\mathbf{A}

explicitly and show that in spherical coordinates it has components

A_r = 0, \quad A_\theta = 0, \quad A_\phi = \frac{g(1 - \cos\theta)}{4\pi r\sin\theta} = \left(\frac{g}{4\pi r}\right)\tan\!\left(\frac{\theta}{2}\right)

(b) Verify that

\mathbf{B} = \nabla \times \mathbf{A}

is the Coulomb-like field of a point charge, except perhaps at

\theta = \pi

. (c) With the

\mathbf{B}

determined in part b, evaluate the total magnetic flux passing through the circular loop of radius

R\sin\theta

shown in the figure. Consider

0 < \theta < \pi/2

and

\theta > \pi/2

separately, but always calculate the upward flux. (d) From

\oint \mathbf{A}\cdot d\mathbf{l}

around the loop, determine the total magnetic flux through the loop. Compare the result with that found in part c. Show that they are equal for

0 < \theta < \pi/2

, but have a constant difference for

\pi/2 < \theta < \pi

. Interpret this difference.