Magnetic Monopoles: Exploring Vector Potential in a Doubly Connected Space

[Tyger]

Magnetic Monopoly

Suppose you bent a long thin magnet around to make a closed loop. All the magnetic field would be contained in the body of the loop but outside the loop you would have a vector potential circulating around the body of the loop. The vector potential can circulate because the toroidal body of the magnet makes the space doubly connected. Now suppose that you opened the loop to make, not a bar magnet, but a pair of magnetic monopoles. The space would now be simply connected. What would happen to the vector potential in this case? What would the vector potential look like around an isolated monopole?

 

[Hurkyl]

There are separate potentials for magnetic charge / current.

 

[Tyger]

Originally posted by Hurkyl
There are separate potentials for magnetic charge / current.

Can you elaborate on that?

 

[Hurkyl]

The reason the vector potential exists is because the magnetic field is divergence free. There is a theorem that states:

For any vector field V, if div V = 0, then there exists a vector field A such that curl A = V.


In empty space, Maxwell's equations are (roughly) symmetric in E and B... which led Dirac to consider extending the symmetry to the case where matter is present. The symmetrized equations are (in units where the constants are 1)

div E = ρ_e
div B = ρ_m
curl E = -∂B/∂t - j_m
curl B = ∂E/∂t + j_e

In this theory, the magnetic field is no longer divergence free... but we can split the fields into two parts; one contributed by electric charge and current and one contributed by magnetic charge and current. We then form a vector potential for both parts in the usual way. (we also form a scalar potential for both partS)


Check out the text http://www.plasma.uu.se/CED/ .

 

[RKSWB]

MAGNETISM! how does this work?

OK OK... there's a lot of common knowledge about magnetism. Let's take a permenant magnet for example: there's a north and south pole, the polarity can be sharred by ferromagnetic thigns, there is an attraction and repulsion forces. But what i want to know is why this happens!

Like how is magnetism travelled, does it affect atoms a certain way? how does it travel through a vacuum? why do the lines of force bend? why do things of opposite polarities attract and same repel? There's so many questions but i can't find the answers anywhere.

Can someone please help exlpain some of these things to me?

Thanks,
-Ryan

 

[RKSWB]

and why does an electron have a negative charge? proton a positive one? the point is... what is this thing we call magnetism.

 

[Jonathan]

Magnetism ?'s from me too

I've always wondered why if you take two magnets and point two same poles in the same direction and try to get them close, they resist, while two magnetized atoms in a magnet can point the same way without flipping and demagnetizing the magnet. It seems like there's little friction on that scale, so why do they not resist?

 

[Bill Gavlas]

Mentors Warning: This post does not contain reliable information, Read for intertainment value only
Integral

The reason you cannot put the two poles (N-N) (S-S) together has to do with the output of the magnet. The electrons are expelled from the (north or south pole - my experiments indicate north) in a direction (i.e. clockwise) whereas the electrons are pulled in in a counter-clockwise direction. So if you try to put two clockwise rotating fields together (or counter-clockwise), they will not go.

If you do attempt to put two north poles together, you physically weaken a permanent magnet (permanently) so don't do it with an expensive magnet! Besides with an expensive magnet (like neodymium) it is extremely dangerous!

If you put two magnets together (N-S) you really have only one magnet! If you break a magnet in half, you have two magnets.

 

[Bill Gavlas]

Mentors Warning: This post does not contain reliable information, Read for intertainment value only

He is correct about Ben Franklin!
Integral

and why does an electron have a negative charge? proton a positive one? the point is... what is this thing we call magnetism.

As far as I can tell, the negative and psoitive charge definitions were totally arbitrary. It is similar to the story surrounding Benjamin Franklin's definition of positive and negative. It may also apply here.

We know that electricity flows from negative to positive. Franklin defined the flow incorrectly, but, out of respect for old Ben, the definition has not changed.

In normal terms, we would really think that the positive pole of a battery is actually the negative, since the electrons flow into the battery at the positive pole. The electrons flow out of the negative pole.

So the definition of positive and negative is arbritrary. In reality, the proton is a sucking force (conseqently should be labeled negative) while the electron appears to be more of a positive force (but it is labeled negative). Perhaps, scientists do this to deliberately confuse use so we cannot encroach upon their sacred ground?
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Here is what I posted in the "How Stuff Works" Forum. Perhaps it will help answer your questions?
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This is an interesting topic. One that I have studied for over 20 years.

Unfortunately, you will not find any good books or articles that explain precisely how a magnet works. The last good book I read was written in the 1930's by a scientist from General Electric. Even then, it wasn't a big book.

From what I've seen, either the information discovered about magnets is classified by the government or it doesn't exist. If it is not classified info, then magnets have been virtually ignored by the scientific community for over 70 years!

Most books and articles written about magnets are written by people who really don't understand them. The people are simply repeating what they heard from their instructor and confusing it.

Exactly how the atomic particles are aligned once iron is magnetized, I don't know. I suspect that the orbit of an electron in relation to the size of the nucleus has a lot to do with magnetism.

However, magnets are fascinating in what is going on inside and what they do.

Magnets are essentially a black hole. Unlike a black hole, however, only electrons pass through (not protons and neutrons).

Magnets feed themselves. The electrons travel from one end of the magnet (expelling end) to the other end of the magnet (sucking end). Consequently, lines of force!

Permanent magnets have small areas of influence where they are anti-magnetic as well. Iron based objects are repelled from those areas.

Magnets are unstable in that the output of the magnet is reduced by about 40% at night. Consequently, permanent magnets are not used for commercial electrical production. They have been deemed to be "unstable" as compared to electromagnets. Permanent magnets do not behave the same on a sunny day as they do at night. To the naked eye it looks the same, however.

Magnets are the only "perpetual motion" device we know of that does not physically harm us.

All iron based metals are magnetic. If you pass a compass along the side of a steel desk or cabinet, you will see the compass alternate between the north and south pole as you move along the surface of the metal. This is due to the fact that the steel was rolled and was not annealed (reheated) after the rolling process.

It is probably not a good idea to rely too heavily on the scientific community to understand what interests you. It is better to sift through the "scientific" info and determine what is really true.

Case in point. Only in the last 10 years has the scientific community classified a magnet as an electronic device. We have used magnets to produce electricity for almost 200 years and it took that long for the source of most of our electric power to be classified as such.

I guess that is how it is supposed to work?

Also, magnets have far more uses than what we currently imagine. Inter-stellar space travel is possible (at any speed) with the use of permanent magnets. Many, many more uses! But I can't get into that just yet.
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And yet more information!
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The reason you cannot put the two poles (N-N) (S-S) together has to do with the output of the magnet. The electrons are expelled from the (north or south pole - my experiments indicate north) in a direction (i.e. clockwise) whereas the electrons are pulled in in a counter-clockwise direction. So if you try to put two clockwise rotating fields together (or counter-clockwise), they will not go.

If you do attempt to put two north poles together, you physically weaken a permanent magnet (permanently) so don't do it with an expensive magnet! Besides with an expensive magnet (like neodymium) it is extremely dangerous!

If you put two magnets together (N-S) you really have only one magnet! If you break a magnet in half, you have two magnets.

 

[Jonathan]

You misunderstand, I mean the two same poles (don't matter which), are parallel. I'll try to draw it (not to scale), but it may not come out right when posted:
......_n_..._n_
. __n_..._n__.../...\.../...\
.|____|.|____|...|...|.|...|
...s...s...\_s_/...\_s_/
magnets....atoms-(my feeble attempt at spheres)

In both cases they seem like they should repel (left-right) and in the magnet case I know they do, but in the atom case they can't -or at least it must be very weak- or else magnets can't exist. I guess I've anwsered my own question, they must just be too weak to move each other much. (The dots are to keep the computer from making the pictues unrecognizable.)

 

[Jonathan]

What!?

I'm am open minded, but even I am not willing to accept the none sense you spout! Electrons don't leave one side of a magnet and enter the other! Photons do! I can prove this thus: the tube of the monitor you're staring at has a vacuum inside because air immpedes the motion of electrons. If there were electrons entering and leaving a magnet, the atmosphere around the magnet would be aglow from the tremendous currents!

 

[HallsofIvy]

The reason you cannot put the two poles (N-N) (S-S) together has to do with the output of the magnet. The electrons are expelled from the (north or south pole - my experiments indicate north) in a direction (i.e. clockwise) whereas the electrons are pulled in in a counter-clockwise direction. So if you try to put two clockwise rotating fields together (or counter-clockwise), they will not go.

OOOOOH-MYYYYYYYY-Gooood

Magnets do NOT "expel" electrons!

 

[nbo10]

Originally posted by Bill Gavlas
As far as I can tell, the negative and psoitive charge definitions were totally arbitrary. It is similar to the story surrounding Benjamin Franklin's definition of positive and negative. It may also apply here.
Edit: Having this nonsense appearing once is bad enough, let's not repeat it! :)

Integral
...

I can't decide if this is the funniest or dumbest thing I've ever read

 

[Jonathan]

I know! I though maybe it was just a typo, but I replyed to another post of his because he had repeated it! He should be on the Theory Development page, though I proved in that reply that magnets can't possibly expel electrons (ionization of the air would be obvious, if the electrons had enough energy to get through all that air). Some thing else too- what would cause them to curl and enter the other side? You should see the rant in his other post!

 

[Jonathan]

I think it's both, and sad too, because this guy is or will be voting!

RKSWB-You have too many questions and they are too vague for me to anwser, sorry, just don't believe Bill.

 

[Integral]

http://home.comcast.net/~rossgr1/magetism.pdf is a PDF scaned from Elementary Theory of Electric and Magnetic Fields by Warren B. Cheston

This is a pretty big file ~400k so if you are on a dialup, be warned.

Bill I am with the rest here, if you wish to learn about the behavior of Magnets and Magnetic fields all you need to is understand Maxwell's equations. To do this you need to know Partial Differential Equations and Multivariable Vector Calculus. Anything short of that is arm waveing. IMOH, arm waving makes for a poor fan and even worse basis of understanding.

Edit:
I have merged the various magnetism threads into this one, simply because they all seemed to be asking about the same questions.

 

[Tyger]

Originally posted by Hurkyl
The reason the vector potential exists is because the magnetic field is divergence free. There is a theorem that states:

For any vector field V, if div V = 0, then there exists a vector field A such that curl A = V.


In empty space, Maxwell's equations are (roughly) symmetric in E and B... which led Dirac to consider extending the symmetry to the case where matter is present. The symmetrized equations are (in units where the constants are 1)

div E = ρ_e
div B = ρ_m
curl E = -∂B/∂t - j_m
curl B = ∂E/∂t + j_e

In this theory, the magnetic field is no longer divergence free... but we can split the fields into two parts; one contributed by electric charge and current and one contributed by magnetic charge and current. We then form a vector potential for both parts in the usual way. (we also form a scalar potential for both partS)


Check out the text http://www.plasma.uu.se/CED/ .

We can make a line integral, which is the equivalent of moving a charge and watching the phase change, through the body of the magnetic loop, out into space as far as we like and back again to get a specific amount of phase change because of the vector potential associated with the loop. But when you open the loop to make monopoles the vector potential is no longer well defined because the region is now simply connected. And I suspect that is why we don't see any magnetic monopoles, that the magnetic field is defined as the curl of the vector potential, where the potential merges together there is no curl and no magnetic field.

The idea of keeping separate books for electric and magnetic poles seem pretty fishy to me. Another phony bookkeeping trick that causes trouble somewhere down the line.

 

[Integral]

Originally posted by Tyger
Suppose you bent a long thin magnet around to make a closed loop. All the magnetic field would be contained in the body of the loop but outside the loop you would have a vector potential circulating around the body of the loop. The vector potential can circulate because the toroidal body of the magnet makes the space doubly connected. Now suppose that you opened the loop to make, not a bar magnet, but a pair of magnetic monopoles. The space would now be simply connected. What would happen to the vector potential in this case? What would the vector potential look like around an isolated monopole?

Been thinking about this.

It is not possible to differentiate between the field around a long circiulaly wound solenoid and a bar magnet. Now it can be proven that the field surounding a solenoid is weakest (lower flux density) near the middle and stronger (higher flux density) at the ends, through the center of the solenoid the field is constant.

Now if you take a solenoid and and wrap it into a torus (this is called a Rolland ring) the field out side the ring VANISHES, while the field inside remains constant and uniform.

Why would a toroidal bar magnet behave differently?

 

[Tyger]

Originally posted by Integral
Been thinking about this.

It is not possible to differentiate between the field around a long circiulaly wound solenoid and a bar magnet. Now it can be proven that the field surounding a solenoid is weakest (lower flux density) near the middle and stronger (higher flux density) at the ends, through the center of the solenoid the field is constant.

Now if you take a solenoid and and wrap it into a torus (this is called a Rolland ring) the field out side the ring VANISHES, while the field inside remains constant and uniform.

Why would a toroidal bar magnet behave differently?

What I'm trying to do, of course, is explain why there aren't any monopoles. And what I'm saying is if every great once in a while, instead of the loop opening to make a bar magnet, it makes a pair of monopoles. You can start with the pair of monopoles in empty space and ask the very same question about what happens to the vector potential, but the magnetic loop makes it much easier to see the double connectivity that is such an important part of the magnetic field, and its role regarding the vector potential.