Dirac Expression for Vector Potential of a Magnetic Monopole Problem

[Harry134]

Hi,

Homework Statement

Consider the vector potential, $\vec{A}(\vec{x})$, below. The problem is to calculate $\vec{A}(\vec{x})$ explictly, and show that it has components $A_{r}$, $A_{\theta}$ and $A_{\phi}$

Homework Equations

$\vec{A}(\vec{x})$ = $\frac{g}{4\pi} \int_{-\infty}^{0} \frac{dz' \hat{z} \times (\vec{x} - z' \hat{z})}{\vert \vec{x} - z' \hat{z} \vert ^{3}}$

$A_{r} = 0$, $A_{\theta} = 0$, $A_{\phi} = \frac{g}{4 \pi} \frac{\tan{\theta / 2}}{R}$

The Attempt at a Solution

This is a homework problem and I have seen the solution to it. However, I do not understand any of the solution or even where to start the problem.

Thanks

 

[vanhees71]

Just evaulate the integral and check whether it coincides with what's given as the vector potential in terms of spherical coordinates.

BTW: It might be easier to work with the local form of the Maxwell equations, extended with the presence of a magnetic point charge. You only need to consider the magnetostatics equations,
$$\vec{\nabla} \times \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=g \delta^{(3)}(\vec{x}).$$
Now except at the origin you have $\vec{B}=\vec{\nabla} \times \vec{A}$. So you must find a vector potential with an appropriate singularity along a semiinfinite line (the famous Dirac string) such that it reproduces the $\delta$-distribution singularity. In terms of $\vec{A}$ the equations read
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{A})=0, \quad \vec{\nabla} \cdot (\vec{\nabla} \times \vec{A})=g \delta^{(3)}(\vec{x}).$$

Another approach is to avoid the string and use different gauges in different regions of space along the lines of the very illuminating paper

T. T. Wu and C. N. Yang. Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D, 12:3845, 1975.
http://link.aps.org/abstract/PRD/v12/i12/p3845