Week 7: Sources of the Magnetic Field 229
If there is, then we would expect the field of a collection of “magneticmonopolar” chargesqm^63
to be given by:
B~(~r) =∑
ikmqmi(~r−~ri)
|~r−~ri|^3(491)
and the magnetic force between a pair of monopoles to be given by:
F~ 12 =kmqm^1 qm^2 (~r^1 −~r^2 )
|~r 1 −~r 2 |^3(492)
where I’ve introduced a magnetic force constant equivalent toketo set the scale for the units of
magnetic charge and force.
If monopoles such as these existed, clearly I could derive aGauss’s Law for Magnetism:
∮SB~·nˆdA= 4πkmQm,in S=μ 0∫
V/SρmdV (493)procedingexactlyas I did before for an isolated electrical charge! This would make thestatic
electrical and magnetic fields, at least, identical to one another, and even would suggest that there
would be a force on a magnetic charge moving in anelectricalfield that has that pesky velocity
dependent cross product in it, to maintain the symmetry even further.
As we’ll see later, we even know what the magnetic field constantkmwould have to be. In fact:km= 10−^7 tesla−meter/ampere (494)exactly(exactly because it defines the coulomb, not the other way around) as was determined and
defined by Ampere in his experiments on magnetism!
Well, dangle bait like that in front of a bunch of physicists and they’ll beharing off to the
laboratory to search for magnetic monopoles, visions of Nobel Prizes and trips to Stockholm to meet
the king dancing through their minds. For at least 60 years at this point intense effort has been
expended searching experimentally for magnetic monopoles using a variety of ingeneous methods.
Alas, noisolated magnetic monopoleshave been experimentally observed, in spite of an electro-
magnetic theory that “begs” for them. Physicists wouldlovefor at least one magnetic monopole to
exist in the Universe because if it did, quantum theory could explain charge quantization! However,
given the lack of concrete evidence for their existence at this point, it is probably safe to say that
magnetic monopoles are at the very leastrare. We express this lack of monopoles by modifying
Gauss’s Law of Magnetism to be:
∮
SB~·nˆdA= 4πkmQm,in S=μ 0∫
V/SρmdV= 0 (495)and this is just the way that you should learn it for this course.
Believe it or not, this is yet another one ofMaxwell’s equations, and we need to learn this equation
just as well as we learn its electrostatic equivalent, Gauss’s Law forElectrostatics. It actually tells
us somevery useful thingsabout the magnetostatic field. In vector differential form (something you
will learn later, if you continue on in physics) it is a key differential equation that you will need to
be able to solve field problems. Inthisclass, its implications can be summarized as:
- Magnetic field linescannotbegin or end at a point (recall that they could only end at a point
for electric field lines if the point contained anelectric charge. Nor can they cross. This leaves
only one alternative:
(^63) I meditated for quite a time what symbol to use for magnetic charge in this book. There are no particularly good
choices. The one I initially leaned towards isg, which is sort of like aqbut backwards, but this conflicts with the
gravitational field. I finally went withqm, even though this will require me to sometimes refer toelectricalcharge as
qewhen I’m discussing the two kinds of charge together. This istedious, however, in the long run, so be warned:qby
itself will generally refer to electrical charge; I will always add the subscriptmwhen discussing magnetic monopoles.