A general problem with magnetism

[Tanja]

I'm completely confused because of a previous post on Newtons second law and magnetism! Can someone help me to find a connecting or a superior explanation for the magnetic field as it appears in its the 2 forms?

1.) In some solid due to spin alignment.
2a.) In a cirque current flow.
2b.) In electromagnetic waves and as a special potential for the Maxwell's equations.

And what would happen to the solution of Maxwell's equation if div B wouldn't be zero, claiming that there are magnetic monopols? And what if the flux of the magnetic field over any closed surface S wouldn't be zero?

Thanks

 

[pam]

Try to answer one clear specific question at a time.
Your confusion could be resolved by reading a good textbook at the
advanced UG or graduate level.

 

[Troels]

I'm completely confused because of a previous post on Newtons second law and magnetism! Can someone help me to find a connecting or a superior explanation for the magnetic field as it appears in its the 2 forms?

1.) In some solid due to spin alignment.
2a.) In a cirque current flow.
2b.) In electromagnetic waves and as a special potential for the Maxwell's equations.

I would argue on this division of the magnetic field, because "spin alignment" *is* a "cirque current flow" on the atomic scale (roughly speaking)

A more natural division is:

magnetic fields due to charges in motion, both free and bound, specified by the maxwell equation:

$$\nabla\times\mathtbf{\vec B}=\mu_0\mathtbf{\vec J}$$

And magnetic fields due to changing electric fields, as in electromagnetic waves, specified by the maxwell equation

$$\nabla\times\mathtbf{\vec B}=\mu_0\epsilon_0\frac{\partial\mathtbf{\vec E}}{\partial t}$$

But since both these fields still have $$\nabla\cdot\mathtbf{\vec B}=0$$ one cannot distinguish one from the other simply by measuring the curl and the divergence and having no knowledge of the current densities and electric fields, so the division of the magnetic field is meaningless alltogether.

And what would happen to the solution of Maxwell's equation if div B wouldn't be zero, claiming that there are magnetic monopols? And what if the flux of the magnetic field over any closed surface S wouldn't be zero?

Well first of the one is a conseqence of the other, that is:

$$\nabla\cdot\mathtbf{\vec B}=0\,\Leftrightarrow\,\oint_\mathcal{S}{\vec B}\cdot d\mathbf{\vec a}=0$$

(To verify, apply the divergence theorem). If the magnetic flux through a closed surface was non-zero, there would have to be somewhere within where magnetic field lines begun or ended: a magnetic monopole. A full treatment of maxwell equations, including magnetic monopoles has been made, but it is rather crumblesome, but does imply an even greater symmetry between electric and magnetic fields.

In the context of relativity, magnetic monopoles are quite meaningless, as here, a magnetic field is just an electric field from another point of view and vice versa, so electric charge alone can account for both types of fields. In quantum mechanics however, scientists beg for them to exist. P. A. M. Dirac proposed on a theoretical argument that the existence of magnetic charge, just a single one, anywhere in the universe, would account for the quantizitaion of electric charge.

 

[Tanja]

I would argue on this division of the magnetic field, because "spin alignment" *is* a "cirque current flow" on the atomic scale (roughly speaking)

Even in a hydrogen atom the electron doesn't rotate in a circuit around the nuclei. The most likely place to be is a sphere and there would be no definite direction of the magnetic field. As you probably know, in other atoms it's much more compleceted and I don't think that there is a case where the average electron movements can be compared to a cirque current flow. The only example I could imagine is are excitons, but they don't exist in solids with magnetic properties.

I don't understand what you mean in the middle part.

Thanks a lot for the hint on Dirac and on the full treatment of maxwell equations, including monopoles. I'll try to find more information there.

 

[Troels]

Even in a hydrogen atom the electron doesn't rotate in a circuit around the nuclei

I am aware of that. I didn't set out to give a complete quantum mechanical treatment of magnetization (we are, after all in the forum "classical physics"), which I tired to emphazise with the remark "roughly speaking". but the essential point remains: quantum mechs or not, magnetic fields are still due to the motion of electric charges OR alternating electric fields (or a combination thereof). And the two have completely identical properties as far as vector calculus is concerned, which means that you cannot tell what the source of a particular magnetic field is, unless you happen to have knowlegde of electric fields and current densities in the region.

Can someone help me to find a connecting or a superior explanation for the magnetic field as it appears in its the 2 forms?

The morale is: It is rather pointless to attempt to make a distinction between "different forms" of magnetic fields, because you won't be able to tell them apart anyway. They may have different sources, true, but a measurement of the magnetic field alone cannot tell which.