Hello Jack,

the magnetic field lines are closed ( div B = 0) and the magnetic flow is constant in a closed circle (input equal output). Applying Ampere's law, you find in the case of a horse shoe magnet a jump in the magnetic H-field. The field Hd in the air gap d is Hd =l*Hm/d (Hm - field in the magnet, l the length of the magnet). The pole is that surface where H jumps, but the flow has no jump.

A pole may not be confused with the monopole (source of a magnetic field). Real magnetic fields are source free.

Dirac's theory of magnetic monopols is very interesting and much people looked for them. But, sadly, such particle never were found.

With Regard

R. Mitdank

Hello Rudiger,

Thank you very much for your helpful and thoughtful reply.

Of course, since div B = 0 always if there are no monopoles, magnetic field lines always exist as closed loops. But if a horseshoe magnet is bent into a circle with the poles cold-welded together, there are no longer even any poles.

All the best,

Jack Denur

As I understand, your question is not about horseshoe magnet.

First of all, there is no problem with existence of circular magnetic fields even if there are magnetic monopoles (somwhere else). Such fields can be produced by electric currents. Simplest example of the circular magnetic field produced without magnetic monopoles is the field around the wire with electric current.

As for theories that require (or, at least, admit) existence of magnetis charges, their consistency with experimental data depends on the viewpoint.

For instance, there is an interesting theory of superluminal particles developed by Italian physicists (see e.g. Recami, E., & Mignani, R. (1974). Classical theory of tachyons (special relativity extended to superluminal frames and objects). La Rivista Del Nuovo Cimento Series 2, 4(2), 209–290).

According to this theory, superluminal velocity flips electric and magnetic fields, i.e. electric fields become magnetic and vice versa. Prticularly, superluminal magnetic charge will be seen as electric charge (i.e. the divergence of the electric field (div E) will be non-zero).

To understand this, consider (again) the simplest example of magnetic field around the wire with electric current. In fact, this is the case of superluminal electric charge: it's time-like component (charge density in the wire) is zero, while space-like component (electric current) is non-zero. And, as we know, this "superluminal" electric charge produces magnetic field, not electric.

Similarly, superluminal magnetic charges (if exist) will produce electric field.

According to Recami & Mignani referred above, protons can be regarded as superluminal magnetic currents inside the proton. Since these currents are superluminal, they are seen as electric charges (not magnetic).

What are the reasons to think that currents inside the proton are superluminal? Well, according to experiments on inelastic scattering of electrons on protons, the number of proton constituents (termed partons) that scatter electrons is dependent on the reference frame. The faster the observer is moving with respect to scattering particles, the more proton constituents will be observed (you can read about this here: https://profmattstrassler.com/articles-and-posts/largehadroncolliderfaq/whats-a-proton-anyway/).

But according to Special relativity, this can only happen if proton constituents are moving faster than light. As you certainly know, slower-than-light objects can be at the same place at different times. Similarly, in Special relativity faster-than-light objects can be at different places at the same time. And the "number of places" where superluminal objects can be observed at the same time depend on the reference frame of the observer.

If we accept the idea of Recami & Mignani, we obtain very nice picture of the world with both electric charges (subluminal, inside electrons, muons and tau) and magnetic charges (superluminal. inside protons). But they both look like electric charges from our frame, hence we have an impression that there are no magnetic charges in nature.

I hope this is a good example of the theory that admits magnetic charges and is consistent with our experimental observations.

Hello Murad,

Thank you very much for your very detailed and very informative reply.

As you wrote, a theory that requires (not merely allows) the existence of magnetic monopoles does NOT require that EVERY magnetic pole be a monopole. Even one magnetic monopole in the entire observable Universe suffices. Every other magnetic pole can still be an ordinary north or south pole, and circular magnets with no poles at all can still exist.

Sincerely yours,

Jack

Dear Jack,

I am assuming that by "some theories require magnetic monopoles..." you mean e.g. certain grand unified theories in which magnetic monopoles are required in regards to the early universe (quite generally spoken)?

These monopoles are not Dirac (point-like) monopoles, and they are not related to electrodynamics, but are solitonic field configurations that occur in exactly these GUT kind of gauge theories that include Yang-Mills + Higgs fields in diverse couplings. They are called 't Hooft-Polyakov monopoles.

A good starting point to read is the respective Wikipedia article https://en.wikipedia.org/wiki/%27t_Hooft%E2%80%93Polyakov_monopole, which has a reference to 't Hooft's and Polyakov's original papers.

There is therefore no mismatch with your assumptions on classical electrodynamics, as far as I see it.

Does that answer your question? Or may I have misunderstood your question?

Best regards

Oliver

Dear Oliver,

Thank you for your very thoughtful and very detailed reply. Yes, that answers my question. My question concerned any theory in general that requires (not merely allows) magnetic monopoles, not any particular theory.

All the best.

Sincerely yours,

Jack