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DrRocket
2009-Dec-09, 08:08 PM
In a related thread, now in the ATM forum and closed at this time, Tusenfem posted the following in response to my assertion that div B =0

I have to disagree here, the general equation in the presence of possible monopoles would be:

div B = 4 π ρm,

where ρm would be "magnetic charge" inside a volume.

However, as, up to now, there has never been any real confirmation of the existence of magnetic monopoles (there once was one stray observation in a superconducting thingamajik, but can't find the reference right now, which was never repeated afterward) we can safely assume that ρm=0 and thus div B = 0.

Basically, what div B = 0 means is that (loosely defined) if we have a volume V with magnetic particles in it, then there are equal amounts of field lines that go out through the surface of the volume as come in. (Actually, it is the surface integral over the surface of the volume of the normal component of the magnetic field to that surface, if you understand what I mean).

I both agree and disagree with Tusenfem. This post is intended to make clear the maintstream position with regard to monopoles within the context of classical electromagnetism.

For a concise summary of Maxwell's equations, and the necessary modifications if magnetic monopoles were to be included it the theory the reader is invited to refer to this Wiki article (http://en.wikipedia.org/wiki/Magnetic_monopole).

Maxwell's equations as currently accepted in mainstream physics, and as presented in standard text books such as J.D. Jackson's Classical Electrodynamics include Gauss's law (for magnetism), div B = 0. This simply asserts the nonexistence of point magnetic charges, or monopoles. Classical electrodynamics is sufficiently mature as to be amenable to an axiomatic treatment, and Maxwell's equations plus the Lorentz force equation are the axioms.

Tusenfem's interpretation of div B = 0 is correct, and is simply an application of the generalized Stokes Theorem (often called the Divergence Theorem in texts on electromagnetism), resulting in the integral form of Gauss's Law. Whether you prefer the point form or the integral form, the result is still the denial of the existence of magnetic monopoles in classical theory.

There are reasons stemming from quantum electrodynamics to think that monopoles might exist. They make the quantization of electric charge an elegant result rather than an ad hoc "add on". But no reliable experimental evidence has been found for the existence of magnetic monopoles.

If magnetic monopoles were to be found, then Maxwell's equations would be modified. The equation suggested by Tusenfem, and found in the table in the Wiki article, is what would replace div B = 0. That equation is the analog of div D = rho, which is Gauss's law for the electric field, for which it is known that point charges do exist (electrons and protons). One would also have to modify the Lorentz force equation to reflect the existence and effect of magnetic charge. Simply stated, physicists know how to formulate a classical electrodynamic theory that would accomodate magnetic monopoles. That theory would be mathematically consistent, and it would be an axiomatic construct using the altered Maxwell and Lorentz equations. But mathematical consistency is only part of the story. A valid theory would also have to reflect reality and the experimental data base. That data base does not at this time support the existence of magnetic monopoles, but rather denies it. This situation will change if and only if magnetic monopoles are confirmed experimentally.

The importance of this is that any claim to a proof of the existence of magnetic monopoles based on phenomena known to be very accurately described by Maxwell's equations -- currents, wires, solenoids, moving charges, antennas, etc. -- is doomed. Maxwell's equations simply forbid monopoles. Adding in the Lorentz force equation will not result in a valid argument either. The Lorentz force equation is compatible with Maxwell's equations, and simply describes the force exerted on a text particle by the fields governed by Maxwell's equations. It cannot negate div B = 0. In addition the existence of magnetic monopoles would necessitate a change in the Lorentz force equation -- see the table in the Wiki article for the form in the presence of magneetic monopoles.

So, bottom line: Mainstream classical electrodynamics is based on the established form of Maxwell's equations, including div B = 0, and monopoles are forbidden. That will not change unless and until magnetic monopoles are shown to exist in reliable experiments.

korjik
2009-Dec-09, 08:51 PM
I would argue semantics but not the conclusion.

The way I was taught was that Maxwells equations already have the monopole terms implicit in the equations, but that those terms are zero, and thus not explicitly written out.

If you look at the reference mentioned above ('http://en.wikipedia.org/wiki/Magnetic_monopole'), specifically looking at the chart of the equations with and without the magnetic charge and current terms, you will notice that they are identical if the magnetic charge and current terms are zero.

They even have the full form of the lorentz force written out just below.

Either way, the conclusion is that so far, there is no evidence that the divergence of the magnetic field is not equal to zero.

The semantics I argue is that divB=0=4 π ρm are basically the same thing, just that ρm=0. Maxwells equations dont care one way or the other wether ρm is zero or not.

This is why I dont argue the conclusion. There is no evidence that ρm is not zero. The only reason anyone still is looking is because Dirac's theory is really elegant and useful if there is a magnetic monopole somewhere in the universe. The effort to find magnetic monopoles is an effect of quantum electrodynamics, not classical.

tusenfem
2009-Dec-09, 09:42 PM
It is basically semantics and "causality," indeed we have div B = 0, and there is no evidence as of yet that this is not fulfilled anywhere in the universe. The semantical or causality discussion comes in at the point where it is said that:

div B = 0 forbids the existence of monopoles
as no monopoles are observed it follows that div B = 0

This is really nitpicking in a way, however, I adhere to point 2, as I do not believe that a mere equation can prescribe nature what it can or cannot do, an equation is always a description of the observations we have made of nature.

This said, this is really "higher physical philosophy" which does not deviate from what we know that there are no monopoles and div B remains zero.

DrRocket
2009-Dec-09, 09:46 PM
I would argue semantics but not the conclusion.

The way I was taught was that Maxwells equations already have the monopole terms implicit in the equations, but that those terms are zero, and thus not explicitly written out.

If you look at the reference mentioned above (http://en.wikipedia.org/wiki/Magnetic_monopole), specifically looking at the chart of the equations with and without the magnetic charge and current terms, you will notice that they are identical if the magnetic charge and current terms are zero.

I think we are saying the same thing, with as you say a semantic disagreement.

I am simply taking the usual form of Maxwell's equations as the embodiment of the mainstream position. One needs a baseline and I think the list of Maxwell's equations that occurs in any standard text is a proper baseline. I am also recognizing that if magnetic monopoles were to be discovered, that it would not upset the applecart and the theory is readily adapted to that situation. But lacking any empirical justification to make the change, magnetic monopoles are not part of the axiomatic logical structure of mainstream classical electrodynamics -- indeed they are forbidden. Experimental evidence to the contrary could change that, but it has not done so to date.

What is important is that the usual form of Maxwell's equations has a boat load of experimental support and it is what is used by those who employ electrodynamics as a tool to model real world phenomena. They do this regularly without any preliminary search around the laboratory and under the desks for stray monopoles. So, one is not going to find a monopole by performing any experiment or constructing any device that is known to be adequately described by the usual form of Maxwell's equations. Looking at the behavior of coils and wires, antennas, moving charges, or anything else that is well understood in the classical theory is not going to produce an observation of a monopole.

If monopoles exist, then it is an issue of quantum field theory, not classical electrodynamics. The classical theory could be modified to handle the existence of these critters, but it is basically just along for the ride.

publius
2009-Dec-10, 03:15 AM

But this subject is one of my favorites. Maxwell with monopoles would be written like this (SI, no constant):

div D = rho_e; div B = rho_m
curl E = -J_m - dB/dt; curl H = J_e + dD/dt

where the subscript 'e' and 'm represent the electric and magnetic versions of the charges and currents. Note the symmetry -- and note the minus sign on the J_m current. This is perfectly consistent and represents what I call the Maxwellian Dance (really, the relativistic space-time dance, but that's a deeper tangent we won't go off on). You can start out with electric charge, and its motion, through the dance gives you a solenoidal B. Or you can start out with a monopole charge, and its motion gives you a solenoidal D(E) field.

So far, we only know of electric charges. But the Maxwellian Dance accomodates monopoles as well.

Now, and this is a key point I like to ramble about, it goes much deeper than this. We have the concept of so-called "Duality". Consider a linear transform of the fields -- think of E and H (or D and B) as orthogonal basis vectors in an abstract vector space. Transform to a new primed basis via a simple linear transform:

E' = aE + bH; H' = cE + dH

Plug that into the homegenous Maxwell (sources = 0), and you'll find the primed fields sastify Maxwell themselves. Now, add the sources back in. You'll find they too will sastify a Maxwell set if you make the same linear transform on the sources as well as the fields.

That is, we have some primed source terms which are linear combinations of the original sources. Again, think of electric and magnetic monopole charge as being orthogonal. The transform is simply a transform to new basis set at an angle to the original set. That "angle" is seen to be related to the ratio of electric to magnetic charge on a source particle.

The key insight is to realize that what we now call E and B could actually be an E' and B', and what we now call electric charge is actually a primed electric charge, a linear combination of electric and magnetic charge! This is Duality.

What this means is that every particle has the same ratio of electric to magnetic charge, and that constant ratio becomes a new electric-only basis. It works either way. There is no observable difference -- observable electrodynamics works the same way as it does now, and that carries over to quantum theory as well. (And we know that in such a case, two versions/frameworks, while different "on paper" are reallly the same physically, of course).

So the question of monopoles becomes one of a question of some particle with a different ratio of the two. There's no need for it to be completely "orthogonal" to our current definition of electric charge, actually.

-Richard

DrRocket
2009-Dec-10, 03:41 AM
It is basically semantics and "causality," indeed we have div B = 0, and there is no evidence as of yet that this is not fulfilled anywhere in the universe. The semantical or causality discussion comes in at the point where it is said that:

div B = 0 forbids the existence of monopoles
as no monopoles are observed it follows that div B = 0

This is really nitpicking in a way, however, I adhere to point 2, as I do not believe that a mere equation can prescribe nature what it can or cannot do, an equation is always a description of the observations we have made of nature.

This said, this is really "higher physical philosophy" which does not deviate from what we know that there are no monopoles and div B remains zero.

Agreed.

If you look at electrodynamics axiomatically then 1 holds. But the validity of the axiom depends on 2.

If you look at it in terms of a theory the validiity of which depends one experimental verification then 2 holds. This is the way that physics works as a science. But once you have this observation established, the Maxwell equation div B = 0 is a succinct wasy to summarize the mainstream position since it is simply the mathematical embodiment of "there are no magnetic monopoles."

And if one claims that monopoles exist then one should be able to show that both 1 and 2 are wrong.

DrRocket
2009-Dec-10, 04:02 AM

But this subject is one of my favorites. Maxwell with monopoles would be written like this (SI, no constant):

div D = rho_e; div B = rho_m
curl E = -J_m - dB/dt; curl H = J_e + dD/dt

where the subscript 'e' and 'm represent the electric and magnetic versions of the charges and currents. Note the symmetry -- and note the minus sign on the J_m current. This is perfectly consistent and represents what I call the Maxwellian Dance (really, the relativistic space-time dance, but that's a deeper tangent we won't go off on). You can start out with electric charge, and its motion, through the dance gives you a solenoidal B. Or you can start out with a monopole charge, and its motion gives you a solenoidal D(E) field.

So far, we only know of electric charges. But the Maxwellian Dance accomodates monopoles as well.

Now, and this is a key point I like to ramble about, it goes much deeper than this. We have the concept of so-called "Duality". Consider a linear transform of the fields -- think of E and H (or D and B) as orthogonal basis vectors in an abstract vector space. Transform to a new primed basis via a simple linear transform:

E' = aE + bH; H' = cE + dH

Plug that into the homegenous Maxwell (sources = 0), and you'll find the primed fields sastify Maxwell themselves. Now, add the sources back in. You'll find they too will sastify a Maxwell set if you make the same linear transform on the sources as well as the fields.

That is, we have some primed source terms which are linear combinations of the original sources. Again, think of electric and magnetic monopole charge as being orthogonal. The transform is simply a transform to new basis set at an angle to the original set. That "angle" is seen to be related to the ratio of electric to magnetic charge on a source particle.

The key insight is to realize that what we now call E and B could actually be an E' and B', and what we now call electric charge is actually a primed electric charge, a linear combination of electric and magnetic charge! This is Duality.

What this means is that every particle has the same ratio of electric to magnetic charge, and that constant ratio becomes a new electric-only basis. It works either way. There is no observable difference -- observable electrodynamics works the same way as it does now, and that carries over to quantum theory as well. (And we know that in such a case, two versions/frameworks, while different "on paper" are reallly the same physically, of course).

So the question of monopoles becomes one of a question of some particle with a different ratio of the two. There's no need for it to be completely "orthogonal" to our current definition of electric charge, actually.

-Richard

I haven't checked what you have said, a matter of some simple algebra, but I bellieve it and I follow what your are sayiing -- with one exception.

This abstract vector space with bases E and B does not seem to have any natural inner product, and therefore is just a vector space, not a Hilbert space. Everything that you say goes through without any need for the notion of orthogonality.

If there is some natural inner product that lends itself to this application, I don't see it immediately. But it does not seem to be necessary anyway.

One other thing, and again I have not checked this out. I would expect that if you went through the usual special relativistic dance of transforming the E and B fields between reference frames in relative motion that this transformation would also apply to the electric and magnetic charges and they would appear as observer-dependent quantities just as with the associated fields assuming that the ratio of magnetic charge to electric charge is not zero.

It certainly appears to be an elegant formalism. All that is missing is at least one monopole to make it physical.

publius
2009-Dec-10, 04:19 AM
Doc,

Jackson goes into the Duality transforms a bit, I think. A quick Google on "electromagnetic duality" turns a bunch of links to papers. Schwinger did a QED treatment with the notion of "dyons" -- you might look for that as well.

There is one hint that monopoles might just exist in this universe, and that is the curious case of the quantization of electric charge. There is no real reason for it in an electric charge-only universe.

It turns out the product of electric and magnetic charge(or flux, a coulomb-weber in SI units) has units of action/angular momentum. That is, a coulomb-weber is a joule-second. Any system of electric and monopole charge thus has angular momentum (even if the charges are stationary -- bringing them together requires a torque, which goes in the EM field -- the static EM field of a monopole/electric charge combo has angular momentum stored). That angular momentum (and action) must be quantized, and to do that requires that both electric and magnetic charge must be quantized themselves.

Dirac was the first to discover this, IIRC. Look up the "Dirac quantization condition" or some such.

-Richard

korjik
2009-Dec-10, 04:26 AM
So, how many of the viewers of this thread actually understand the discussion?

:D

publius
2009-Dec-10, 04:35 AM
Oh, and about the SR transforms. Remember that charge is an invariant (now charge densities and currents are frame dependent, but total charge is invariant). This would apply to monopole charge as well. Thus, the ratio of electric to monopole charge on particles would be an invariant as well.

The way to think of this in the more elegant 4-vector formulation of EM is as an abstract type of coordinate change. It would be similiar to transforms of the inertia tensor to different axes. If all particle have the same ratio of the two charges, you might liken the electric-charge only version as the "principle axes" transform. The field strength tensor would change components based on the new abstract basis. But this is not a transform of the space-time coordinates, but one in this abstract E-H space.

The take-home message from Duality is, I think, that what we call "electric" and "magnetic" are really arbitrary, just coordinates so to speak. The invariant physics is in the Maxwellian Dance itself.

-Richard

Nereid
2009-Dec-10, 04:44 AM
So, how many of the viewers of this thread actually understand the discussion?

:D
I will hazard this guess: if any viewers of this thread are presenters of any ATM ideas (in the ATM section), then the answer is, among that set of viewers, a goose egg*.

* with only one exception (the sweet quaternion guy) but I doubt he has visited us for well over a year now ...

steve_bnk
2009-Dec-10, 05:10 AM
Oh, and about the SR transforms. Remember that charge is an invariant (now charge densities and currents are frame dependent, but total charge is invariant). This would apply to monopole charge as well. Thus, the ratio of electric to monopole charge on particles would be an invariant as well.

The way to think of this in the more elegant 4-vector formulation of EM is as an abstract type of coordinate change. It would be similiar to transforms of the inertia tensor to different axes. If all particle have the same ratio of the two charges, you might liken the electric-charge only version as the "principle axes" transform. The field strength tensor would change components based on the new abstract basis. But this is not a transform of the space-time coordinates, but one in this abstract E-H space.

The take-home message from Duality is, I think, that what we call "electric" and "magnetic" are really arbitrary, just coordinates so to speak. The invariant physics is in the Maxwellian Dance itself.

-Richard

if I toss a capacitor with a charge on it to a passing ship traveling at a different speed, I'd assume they will measure the same voltage.

With charge giving rise to an elecric field how would one see the same electric field in a second frame traveling at a differnt velocity? I would think Maxwell's Equations being non-invariant would preclude getting the same measurent of an electric field.

DrRocket
2009-Dec-10, 05:28 AM
There is one hint that monopoles might just exist in this universe, and that is the curious case of the quantization of electric charge. There is no real reason for it in an electric charge-only universe.
...

Dirac was the first to discover this, IIRC. Look up the "Dirac quantization condition" or some such.

-Richard

Yes. I am aware of Dirac's work. One monopole anywhere in the universe and electric charge quantization drops out. No need to put it in by hand in an ad hoc manner. Very elegant. Unfortunately the monopole is still hiding.

DrRocket
2009-Dec-10, 05:45 AM
So, how many of the viewers of this thread actually understand the discussion?

:D

I think the participants understand it.

Nereid is probably right about ATM proponents.

Rubberneckers are on their own . Some may understand others may not.

If John David Jackson is lurking, he understands. I hope he is smiling.

publius
2009-Dec-11, 06:23 AM
Doc (and all),

I'm gonna continue with this, just not tonight, as I'm busy installing wads of software back on my new system. Patience..... :)

-Richard

publius
2009-Dec-14, 04:23 AM
if I toss a capacitor with a charge on it to a passing ship traveling at a different speed, I'd assume they will measure the same voltage.

With charge giving rise to an elecric field how would one see the same electric field in a second frame traveling at a differnt velocity? I would think Maxwell's Equations being non-invariant would preclude getting the same measurent of an electric field.

Sorry for taking so long to reply. This question gets rather complex. Best to do circuit theory in the rest frame of the circuit in question. :)

If you charge a capacitor and toss it to some moving frame, they will measure the same voltage, because it switched frames and is now at rest with them when you toss it. However, a relativistically moving capacitor will look different.

The scalar potential V, and vector potential A are not invariant, but are the time-like and space-like components of the 4-potential, (V, A) (with a factor of c thrown in on V or whatever to make it work out). A likewise the charge density and current density are not invariant but are the components of the 4-current, (rho, J) (again, with a factor of c thrown in there somewhere).

So this means our notion of voltage is not invariant (and throw EMF, not just scalar potential in the meaning of "voltage).

Even the notions of capacitance and inductance are going to be frame dependent. Lorentz contraction is going to change the dimensions of our capacitors and coils. But consider this. Suppose we have an LC oscillator in our moving frame. The frequency of that is going to slow by the gamma factor, and that will independent of however our circuit is oriented with respect to the line of motion.

If the capacitor plates are perpendicular to the line of motion, the plates will be closer together. If parallel, the plates will be the same distance apart, but the surface area will change. And likewise for the coil. But no matter, the end result will be an LC circuit will have the value of sqrt(LC) reduced by gamma.

I think you can see this can get complicated quickly. :) Again, best to do circuit theory in the rest frame.

But there are some relatively simple example that can help show what's going on. Consider an infinitely long wire carrying a current I, with a charged particle moving parallel to the wire, such that the currents are in the same direction. In this frame, the wire is neutral and there is only a magnetic field. The charged particle is attracted to the wire magnetically.

Now, go into the rest frame of the charged particle. There will still be a magnetic field, but in this frame the particle isn't moving, so v X B = 0. So how can the particle be attracted to the wire?

In that frame, there is an electric field, and that's what's responsible for attracting the particle to the wire. But where does that electric field come from? The wire was neutral in the first frame. The answer is the wire is not neutral in the moving frame!

In the first frame, the 4-current was of the form (0, J). But in the moving frame, the transform results in a charge density component (rho', J').
How can that be? Lorentz contraction of the lattice of the wire. You've got electrons moving in the wire. In the moving frame, the positive charge density on the lattice contracts more than the electrons, resulting in a loss of neutrality.

How is the consistent with conservation of charge, you may ask? Well, we're dealing with an infinite wire, and in infinite amount of total positive and negative charge, and this is what happens with infinite quantities. It's funny how infinite things like this can help simplify things, but there's always some funny business that pops up somewhere with infinities.

Similiar stuff happens to finite circuits, but it gets more complex. For example, a polarization in the rest frame will transform to a magnetizaiton in a moving frame, and vice versa. A permanent magnet will appear to have a charge separation to a moving frame.

And if you think this is comlicated, you ought to see what curved space-time and full GR do to Maxwell and stuff like this. :lol:

-Richard