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Magnetic Monopoles

Let us think for a moment about what MEs might be changed into if magnetic monopoles were discovered. We would then expect all four equations to be inhomogeneous:

$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$D$}$ $\textstyle =$ $\displaystyle \rho_e \quad{\rm (GLE)}$ (8.109)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$H$} - \frac{\partial \mbox{\boldmath$D$}}{\partial t}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$J$}_e
\quad{\rm\ (AL)}$ (8.110)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$H$}$ $\textstyle =$ $\displaystyle \rho_m \quad{\rm (GLM)}$ (8.111)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$D$}+ \frac{\partial \mbox{\boldmath$H$}}{\partial t}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$J$}_m \quad{\rm\ (FL)}$ (8.112)

or, in a vacuum:
$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$E$}$ $\textstyle =$ $\displaystyle \frac{1}{\epsilon_0 } \rho_e \quad{\rm (GLE)}$ (8.113)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$}- \epsilon_0 \mu_0 \frac{\partial \mbox{\boldmath$E$}}{\partial t}$ $\textstyle =$ $\displaystyle \mu_0 \mbox{\boldmath$J$}_e
\quad{\rm\ (AL)}$ (8.114)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle \rho_m \quad{\rm (GLM)}$ (8.115)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$} + \frac{\partial \mbox{\boldmath$B$}}{\partial t}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$J$}_m \quad{\rm\ (FL)}$ (8.116)

(where we note that if we discovered an elementary magnetic monopole of magnitude $g$ similar to the elementary electric monopolar charge of $e$ we would almost certainly need to introduce additional constants - or arrangements of the existing ones - to establish its units relative to those of electric charge).

There are two observations we need to make. One is that nature could be rife with magnetic monopoles already. In fact, every single charged particle could have a mix of both electric and magnetic charge. As long as the ratio $g/e$ is a constant, we would be unable to tell.

This can be shown by looking at the following duality transformation which ``rotates'' the magnetic field into the electric field as it rotates the magnetic charge into the electric charge:

$\displaystyle \mbox{\boldmath$E$}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$E$}' \cos(\Theta) + Z_0 \mbox{\boldmath$H$}'\sin(\Theta)$ (8.117)
$\displaystyle Z_0\mbox{\boldmath$D$}$ $\textstyle =$ $\displaystyle Z_0\mbox{\boldmath$D$}' \cos(\Theta) + \mbox{\boldmath$B$}'\sin(\Theta)$ (8.118)
$\displaystyle Z_0\mbox{\boldmath$H$}$ $\textstyle =$ $\displaystyle - \mbox{\boldmath$E$}' \sin(\Theta) + Z_0 \mbox{\boldmath$H$}'\cos(\Theta)$ (8.119)
$\displaystyle \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle -Z_0\mbox{\boldmath$D$}' \sin(\Theta) + \mbox{\boldmath$B$}'\cos(\Theta)$ (8.120)

where $Z_0 = \sqrt{\frac{\mu_0 }{\epsilon_0 }}$ is the impedance of free space (and has units of ohms), a quantity that (as we shall see) appears frequently when manipulating MEs.

Note that when the angle $\Theta = 0$, we have the ordinary MEs we are used to. However, all of our measurements of force would remain unaltered if we rotated by $\Theta = \pi/2$ and $\mbox{\boldmath$E$}= Z_0 \mbox{\boldmath$H$}'$ in the old system.

However, if we perform such a rotation, we must also rotate the charge distributions in exactly the same way:

$\displaystyle Z_0 \rho_e$ $\textstyle =$ $\displaystyle Z_0 \rho_e' \cos(\Theta) + \rho_m'\sin(\Theta)$ (8.121)
$\displaystyle \rho_m$ $\textstyle =$ $\displaystyle - Z_0\rho_e' \cos(\Theta) + \rho_m'\sin(\Theta)$ (8.122)
$\displaystyle Z_0\mbox{\boldmath$J$}_e$ $\textstyle =$ $\displaystyle - \mbox{\boldmath$J$}_e' \cos(\Theta) + \mbox{\boldmath$J$}_m'\sin(\Theta)$ (8.123)
$\displaystyle \mbox{\boldmath$J$}_m$ $\textstyle =$ $\displaystyle -Z_0\mbox{\boldmath$J$}_e' \sin(\Theta) + \mbox{\boldmath$J$}_m'\cos(\Theta)$ (8.124)

It is left as an exercise to show that the monopolar forms of MEs are left invariant - things come in just the right combinations on both sides of all equations to accomplish this. In a nutshell, what this means is that it is merely a matter of convention to call all the charge of a particle electric. By rotating through an arbitrary angle theta in the equations above, we can recover an equivalent version of electrodynamics where electrons and protons have only magnetic charge and the electric charge is zero everywhere, but where all forces and electronic structure remains unchanged as long as all particles have the same $g/e$ ratio.

When we search for magnetic monopoles, then, we are really searching for particles where that ratio is different from the dominant one. We are looking for particles that have zero electric charge and only a magnetic charge in the current frame relative to $\Theta = 0$. Monopolar particles might be expected to be a bit odd for a variety of reasons - magnetic charge is a pseudoscalar quantity, odd under time reversal, where electric charge is a scalar quantity, even under time reversal, for example, field theorists would really really like for there to be at least one monopole in the universe. Nobel-hungry graduate students wouldn't mind if that monopole came wandering through their monopole trap, either.

However, so far (despite a few false positive results that have proven dubious or at any rate unrepeatable) there is a lack of actual experimental evidence for monopoles. Let's examine just a bit of why the idea of monopoles is exciting to theorists.



Subsections
next up previous contents
Next: Dirac Monopoles Up: Maxwell's Equations Previous: Poynting's Theorem, Work and   Contents
Robert G. Brown 2013-01-04