Dear Jerry,

Not necessarily the answer is no, Maxwell equation only says that there are not magnetic sources associated the magnetic field, i.e. there are always two poles for the magnetic field. What Maxwell equation doesn't tell us is where are those poles. This is what many physicists were studying for making compatible this right and experimental law with other possibilities introducing quantum mechanics and topology (Dirac, Schwinger,....).

That's a nice example of the fact that the mathematical content of equations is unambiguous while the "interpretation" resp. "associated notions" can change with changing insights:

In Maxwell's time, charged particles were unknown, so one could interpret the first equation as a definition for charge density. While Maxwell's analogy between fields and flowing incompressible fluids suggested the terms "source" and "sink" for electric charges, he was well aware of the difference between analogy and proven fact. As Daniel Baldomir wrote, in this sense the second equation just says that there are no sources nor sinks for the magnetic field. These sources and sinks were not necessarily connected with matter (electric and magnetic "fluids" were just speculation like the various aethers).

Because one prerequisite of current is the existence of (moving) charge, the second equation says indirectly also that there cannot be "magnetic currents", thus no "magnetic current density", and finally (together with the third or fourth equation, whatever the numbering is) that the curl of the electric field is associated exclusively with the temporal changing of the magnetic flux density. The notion of charged particles is not necessary so far, as can be seen from Pierre Curie's article "On the possibility of magnetic conductivity and free magnetism" (in French), 1894, published three years before the discovery of the electron. The "free magnetism" in Curie's headline denotes "austral" and "boreal" magnetism, i. e. (locally) exclusively positive divergence or exclusively negative divergence of the magnetic flux density.

After the discovery of electrically charged particles, it was generally assumed that the location of divergence of an electric field coincides always with the location of a material particle. Accordingly, austral or boreal magnetism was replaced by magnetically charged particles, the magnetic monopoles. Personally, I don't find Dirac's approach with the very long, vanishingly thin coil physically convincing, so my answer to the original question is: Yes, but the statement of the second equation is not limited to divergence coupled to particles.

To put it another way (in agreement with your penultimate sentence, Jerry): If magnetic monopoles were found experimentally, the second equation (and the one for curl E) would certainly be adapted.

Dr. Decker, The second Maxwell equation is ∇×A = B. The equation ∇.B = 0 is just a derivative. The primary equation means that B is an axial vector and that A is a circulation. This is not an equation about point sources as like ∇.E = ρ/ε which ultimately derives from the inverse square law in Coulomb’s law. It is meaningless to talk about magnetic monopoles. There is no basis for them to exist.

Dear Tombe,

You give ∇×A = B

in which A

is the magnetic vector potential of William Thomson, 1st Baron Kelvin. Maxwell equations can be written with A or B but not both in the same representation. I have used the vector field representation that is found in many references, including those references that also cover the potential representation.

Four Maxwell equations are required in vector field representation. Potential representation has only two Maxwell equations, and the one you gave is not one of them.

What you gave is the second equation that is used in relating potentials to the associated fields, in preparation for writing Maxwell equations in potential representation.

Richard Feynman did a lot to popularize the potential representation as it is much used now in quantum mechanics.

Notice that A may exist without a curl in some situations which are useful for my purpose in field effect propulsion, stress energy that is not gravitational and not electromagnetic.

Dr. Decker, Maxwell's original eight equations are listed here in this article along with references and links to Maxwell's original papers.

Article Maxwell's Original Equations

Dear Tombe,

I notice you have presented a set of eight equations which is a modern restatement with vector notations and summarizing of the Maxwell 1864 paper published in 1865. The 1861paper had 20 equations.. Neither of these are used now except as study of history. B does not occur in your equation set. H is used instead.

Here is the 1864 paper published in 1865..

Article A dynamical theory of the electromagnetic field. R. Soc. Tra...

Maxwell used some vector notation but not the notation we use. Our familiar notation did not occur until Heaviside and others produced the four equations in vector field notation in 1884.

The 1861 paper is here.

http://www.relativitycalculator.com/pdfs/On_Physical_Lines_of_Force.pdf

Here are Heaviside papers.

https://ia600207.us.archive.org/1/items/electricalpapers01heavuoft/electricalpapers01heavuoft.pdf

Volume II

https://ia600302.us.archive.org/32/items/electricalpapers02heavrich/electricalpapers02heavrich.pdf

Major review of Maxwell starts on page 429 of Volume I and continues on page 39 of Volume II. Volume II contains mostly examples of which Heaviside intended to write another three chapters when the publisher asked him to stop at page 155.

In Volume I some preliminary work on Maxwell occurs starting on pages 195 and 416.

Readers can start to recognize familiar Maxwell vector field, while still appreciating the enormous amount of work that went into it.

Page 452 equation 36 of Volume I is the topic of my question. Does this equation say there are no magnetic monopoles?

Readers may understand that Heaviside converted 12 old Maxwell equations into the 4 new Maxwell equations we have in vector field notation. The remaining 8 old Maxwell equations , Heaviside assigned o a new category that became known as circuit analysis, and are no longer included with Maxwell equations.

The history of Maxwell equation development is also interesting and quite complicated as shone in the four linked articles. Several other researchers contributed to it for which Maxwell usually gets the credit.

Dear Jerry,

First of all it is necessary to say that, although it is useful to write Maxwell equations as differential equations as the 36 of page 452, give that we work with fields the real equations are integral. Therefore if you write the equation 36 in integral form you can see that the integral of the div B is zero doesn't means that you have magnetic sources zero. For instance if you imagine a long thin magnet with two opposite poles, the div B gives you zero if you integrate both, but if you put only an integral on the one of the poles this div B is not zero. This introduces the concept of local and global situations, which in fact is the new concept of magnetic monopole introduced by Dirac when added a forbidden string to the magnetic pole. That is to say, he added a string of singularities for the density of probability associated to manetic monopole. Nowadays this model is written a much more sophisticated form and it was shown that the magnetic monopoles appear as solution not only in Electrodynamics but in another fields too, what is strange is that they are not among us as ordinary particles with a giant mass.

Dr. Decker, Thanks. Yes, I was already aware of all that. The point is that ∇.B = 0 because ∇×A = B, therefore magnetic poles don’t enter into the analysis. B is an axial vector.

Dear Frederick,

You have a mistake, ∇.B = 0 doesn't imply ∇×A = B. That is only when you are defining the magnetic field B in a simply connected región of the space, i.e. when your space is topologically trivial.

Dr. Baldomir, You got it the wrong way around. I said that ∇×A = B implies that ∇.B = 0. A comes first, because in time varying EM induction, E = −∂A/∂t

No that is deeply wrong. A is even not a physical observable at all because it depends of the gauge. The proper and physical equation is ∇.B = 0 as Maxwell's equations say. The language of differential forms allows you to distinguish much better this subtleties of what is a closed or an exact differetial form, if you go to my papers you can find a book in Clarendon Press where there is a study devoted to this isssue.

Dear Daniel,

Integral form of Maxwell equations was preferred when I first learned it. There was concern that integration in General Relativity was not well represented at the time. The differential forms have become popular because they make no assumptions about curvature.

The long thin magnet is interesting to me allowing each pole to act separately in a metric region of GR. Long standing prediction in metrics is that the magnetic pole repels a gravity field.

Example of interest is that Earth magnetic poles are off set from rotation axis unequally in North and South. The long base line magnet they represent apparently is off set several hundred kilometers toward the north from the center of gravity and does not pass through the rotation axis. Also center of gravity is apparently off set a few kilometers to south of mass center making a pear shaped Earth. Earth is complicated preventing firm conclusion.

Neutron stars are other examples where magnetic poles are often not aligned with rotation axis.

While magnetic monopoles would be interesting, I can think of magnetic field as a polarization of vacuum space where poles might be absent except maybe at the boundary crossing from one media to another. Disrupted space could conceivably have discontinuous field lines, while Maxwell equations describe normal state of space.

Dear Jerry,

The magnetic monopole or the string of magnetic dipoles is not related with gravitation. The main interest in working with differential forms is to work in the cotangent manifold instead of the tangent one and to have informations as the Hodge operator which carries the geometrical information. In vectorial formalism these concepts are lost and you have shown how Heaviside gave a great improvement to Electrodynamics just changing the mathematical language. For me the proper one is the algebra of the differential forms instead of the vectorial calculus. In gravitation is not so good because the tensor energy-momentum plays a fundamental role and it is symmetric as the curvature of the spacetime.

Dear Daniel,

>

it seems suitable so also for thermodynamics in which what is called "inexact differential" of the heat exchanged is represented by a Cartan one-form.

Dr. Baldomir, Gauge theory doesn't come into it when we start with A as a momentum, as Maxwell did. We may not know the numerical value of A but it is only the principle that is important. Numbers stay out of it until the speed of light is introduced to the equations through the 1855 Weber-Kohlrausch experiment. Maxwell used ∇×A = B in his original list of equations. The equation ∇.B = 0 is just a differential form of a differential form. When we know that we will not even be talking about magnetic monopoles. B is an axial vector,s and indeed it’s pretty obvious just from looking at iron filings patterns around a bar magnet, that the magnetic field is solenoidal. So to even talk about magnetic monopoles indicates a lack of understanding of the nature of magnetism.

See the list here of Maxwell’s original equations in his 1865 paper,

Article Maxwell's Original Equations

A is not a physical observable in classical or quantum electrodynamics. It is not a question of numbers!!! This is a very basic knowledge.

The vector A is essential for understanding the linkage between E and H in electromagnetic induction. That is, E = −∂A/∂t where ∇×A = µH.

Faraday's and Ampere's laws do it much better employing both fields in a gauge independent transformation.

Dr. Baldimor, As in you mean ∇×E = −µ∂H/∂t and ∇×H = ε∂E/∂t? These are only differential equations and they don’t tell us the precise relationship between E and H. They imply a symmetry of some kind but mask the fact that we are specifically dealing with cylindrical symmetry. If however we bring A into it, then we know that H is an axial vector and that A circulates around H, and that E is a transverse acceleration and hence in effect, an angular acceleration. You miss out on a lot of information about the EM wave carrying medium by ignoring A.

Dear Daniel,

Magnetic monopoles have been included in gravity of general relativity since about 1965 when Kerr-Schild found new ways to express EM potentials. The topic has fallen out of favor because of much research that was done not finding monopoles.

Here is a 1993 university study in Mexico and UK.

https://rmf.smf.mx/pdf/rmf/39/1/39_1_57.pdf

Also a reference is RG is given, with references starting in 1974.

Article Magnetic monopole in the general theory of relativity

Consider a long thin dipole magnet of which one pole has passed inside an event horizon. The remaining pole is momentarily the closest physical case to a magnetic monopole. A super massive black hole is intended for this example to give a fairly small change of gravity along the magnet. This example is probably not what Maxwell intended.

Much research is done on long thin jets coming from poles of super massive objects. One possible explanation is that the gravity at poles can be weakened by strong magnetic fields, and the magnetic fields focus the jets.

In GR the electric fields and magnetic fields combine to weaken gravity. in a short range using the Kerr-Schild method. This has been in publications for decades and is one reason the search for magnetic monopoles began.

Dear Jerry,

I understand you, but let me to tell you my opinion about these papers:

1. Magnetic monopoles are not possible in Einstein's gravitation because there are not magnetism at all.

2. They extend this word "magnetic monopole" for studying a topological solution of these equations, i.e. the one associated to the first Chern number.

3. This analogy is suitable because Einstein's equations are also a gauge theory with an inner symmetry associated to the Lorentz group. The mechanism that they follow is the t'Hooft-Polyakov non Abelian model.

4. This is just one analogy but never a magnetic monopole and less if you want to put it in the context of Maxwell equations of electromagnetism.

Dear Frederick,

First of all it seems that you didn't understand me. This equation is not the Ampere-Faraday's one

∇×H = ε∂E/∂t

If you don't write the density of currents J , certainly you are going to have difficulties for fixing the fields. Respect to Faraday's is the one necesary for introducing the vector potential A that you seem to employ from the sky.

In any case, this is basic electromagnetism and quite far of the magnetic monopole solution as the thread asks. For me the discussion with you is finished if no new inside is given in such direction.

Sorry, I wonted to write Ampere-Maxwell's equation and no Ampere-Faraday

Dr. Baldomir, As regards the full Ampère-Maxwell equation ∇×H = ε∂E/∂t + J, I deliberately left out the J because it is not important in space where we combine the two curl equations to derive the EM wave equation. What is important is that when we derive the EM wave equation, that E refers to −∂A/∂t in both of the two curl equations.

If you ignore the vector A, you are ignoring the details of the inter-relationship between E and H, and so you lose the full physical significance of ∇.B = 0. Those who go on a search for magnetic monopoles are on a wild goose chase because they have failed to grasp the basics of the inter-relationship between E and H, which involves A. And A does have a physical meaning. Maxwell saw it as an electromagnetic momentum.

Dear Frederick,

Without sources you only have electromagnetic waves and in such a case you don't need potentials at all. Have a good day!!!

Dr. Baldimor, Your response made absolutely no sense, and on multiple counts. You said "without sources you only have electromagnetic waves". It's not clear at all what this phrase means. And it's not clear how it relates to the context of the discussion. It's then even less clear what you mean when you say "in such a case you don't need potentials". We were only talking about one potential, as in the magnetic vector potential A, and it is inherent in the inter-relationship between E and H. So we do need A if we are to understand the physical nature of the E and H waves and their connecting mechanism.

This discussion began because some people have been on a wild goose chase hunting for so-called magnetic monopoles without even the remotest idea of what a magnetic monopole is supposed to look like. Theoretically it's like looking for a clock that only has a clockwise front but no anti-clockwise reverse. This futile search would end once people get it clear in their heads that magnetism is about rotation and cylindrical symmetry. You can't mix magnetic monopoles and gravitation in the same analysis. It makes no sense. And it helps even less when you want to cut out important links in the electromagnetic relationships such as

∇×A = B and E = −∂A/∂t. Have a read at this article which explores the physical nature of magnetic lines of force. Maxwell's equations are listed in a summary at the end,

Article An Interpretation of Faraday's Lines of Force

Dear Daniel,

I am mostly in agreement with you, one possible exception being that electromagnetic stress energy tensor routinely enters GR field equations as a component of total stress energy. This is not a magnetic monopole although some researchers have tried to include a monopole.

My feeling is that magnetism arises as a polarization of vacuum space, and the absence of magnetic monopoles represents a continuity of space.

I can conceive of situations when space might be momentarily discontinuous in a catastrophe. Maxwell does not speak to such cases, like much of physical science.

Dr. Decker, Be careful not to fall into the trap of equating magnetization with linear polarization. There is however a continuum between them at an antenna where cable waves transition into wireless waves. Cable telegraphy waves are indeed tied up with linear polarization of the so-called vacuum, but magnetization is something different, albeit very closely related. See this article,

Article Cable Telegraphy and Poynting's Theorem

Dear Jerry,

It is true that one of the sources of the energy-momentum tensor can be purely electromagnetic but this cannot induce a singularity. Your question goes in fact further and it was one important discussion in gravitation if it were possible to obtain a "magnetic monopole" solution implicit within the space-time metric. And historically this was made by the metric Taub-NUT (one of the solutions of Einstein's gravitational equations). In this metric are introduced electric and magnetic sources (magnetic poles) as the source of the energy-momentum tensor and therefore there are magnetic singularities as could be the magnetic monopole. But notice that they are introduced explicitally in the energy-momentum tensor (not only the usual electric and magnetic fields of the energy-momentum tensor). I attach you a link for this metric

https://en.wikipedia.org/wiki/Taub%E2%80%93NUT_space

Paul Dirac in 1931 argued that magnetic monopoles should exist to explain the equal and opposite quantum charge of electron and proton, but the attraction force between two magnetic monopoles should be (137.2)2 larger than force between electron and proton, 4692.25 making the observation of a monopole unlikely. Original paper is here.

http://users.physik.fu-berlin.de/~kleinert/files/dirac1931.pdf

Dear Jerry,

The magnetic monopole was the issue of my phD many years ago. It is a very interesting subject because it was the first to introduce topology as a fundamental tool in Physics for showing how the electric charge was quantized. But let me to say that the Dirac model has some dificulties.

First of all, the mass of the monopole is not calculable as the source is put in by hand, making mass a free parameter.

Secondly, the monopole creates a singular vector potential along

the nodal line. The singular vector potential can be managed within the framework of differential geometry, as we will see for the Wu–Yang monopole which shows that the singularity can be removed by a gauge transformation (this was my main contribution working in Principal Fiber Bundles and showing how the singularities belong special topological classes of the gauge group intrepreted as the fiber).

This means that the Coulomb force between two magnetic monopoles of unit topological charge (n = 1) is (137/2)2 ≈ 4700 times stronger then that of two electrons, making magnetic monopoles very hard to separate and in fact it corresponds to a dimensionless coupling constant around 34.25 instead of the usual fine constant 1/137 for the electrons. This may be one of the reasons why magnetic monopoles are so hard to find experimentally,

Before focussing more on magnetic monopole, one should ask a fundamental question that why magnetic lines form closed loop? Or in other word why they have finite curl? Will magnetic monopole allow a finite curl associated with magnetic lines?