# Magnetic Monopoles & Gauss' Law for Magnetism

" Physics is not about how the world is, it is about what we can say about the world"
Niels Bohr

## Magnetic Monopoles

• In our initial discussion of magnetism we made the point that we were going to treat magnetism in a similar way to electricity.  However, whereas our discussion of electricity began with electric charges and the electric field associated with these charges, the magnetic discussion started with the existence of the magnetic field.  No mention was made of "magnetic charges", which would play the same role in magnetism as electric charges in electricity.  This is because individual magnetic charges - magnetic monopoles - are apparently impossible to isolate.
• The simplest magnetic object we have been able to isolate is the magnetic dipole.  Current loops, bar  magnets and solenoids all produce dipole fields with a characteristic magnetic dipole moment.

• The bar magnetic may be considered to be a combination of two magnetic monopoles, usually labelled North and South.  This is similar to the electric dipole comprised of equal but opposite electric charges.

• Whereas with the electric dipole it is possible to isolate the positive and negative charges, experimentally it is not possible to separate the North and South poles of a bar magnet.  Break a magnet in two and you get two magnets, each with a North and South pole.  Continuing this splitting process down to the atomic level we find that even elementary particles behave as magnetic dipoles, each with a North and South pole.  It appears that nature does not allow us to create magnetic monopoles in this way.
•   However, theoreticians developing unified quantum theories of the Universe,  so-called "Theories of Everything", are almost unanimous in the necessity for magnetic monopoles as elementary particles created shortly after the birth of the Universe.

The belief is that shortly after their creation, magnetic monopoles were "frozen out" - meaning that their interactions with the rest of the matter in the Universe is highly suppressed.  This does not prevent physicists from searching for evidence for the existence of magnetic monopoles.

## Gauss' Law for Magnetism

• So far we have discussed three basic equations describing electromagnetic phenomena - the first three of Maxwell's equations.
Gauss' Law:

Ampere's Law:

• Gauss' Law involves the flux integral for the electric field.  To complete the correspondence between electricity and magnetism we expect a fourth equation involving the magnetic flux - "Gauss' Law for Magnetism".
• The right hand side of Gauss' Law includes a summation over electric charges.  Therefore, for magnetism, we expect a summation over "magnetic charges".  But magnetic charges, North and South poles (equivalent to positive and negative electric charges) always exist in pairs, the net "magnetic charge" is thus always zero.  Gauss' Law for Magnetism must therefore take the form,

the flux of B through a closed surface is zero.

Note that the fact that the surface is closed is very important !  A magnetic flux integral  appears in Faraday's Law - in this case the surface is generally not closed.

Electric field lines begin (positive) and end (negative) on charges.  Since there are no magnetic charges magnetic field lines form closed loops.

A Princeton plasma physicist is at the beach when he discovers an ancient looking oil lantern sticking out of the sand. He rubs the sand off with a towel and a genie pops out. The genie offers to grant him one wish. The physicist retrieves a map of the world from his car an circles the Middle East and tells the genie, 'I wish you to bring peace in this region'.

After 10 long minutes of deliberation, the genie replies, 'Gee, there are lots of problems there with Lebanon, Iraq, Israel, and all those other places. This is awfully embarrassing. I've never had to do this before, but I'm just going to have to ask you for another wish. This one is just too much for me'.

Taken aback, the physicist thinks a bit and asks, 'I wish that the Princeton tokamak would achieve scientific fusion energy break-even.'

After another deliberation the genie asks, 'Could I see that map again?'

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu