The conservative nature of the magnetic field, B, is a partial view only in a region away from the source, J. It is not a universal picture, but a local picture only, describable just by the magnetic scalar potential, phi. The global picture comes from the vector potential, A, for the non-conservative nature of B.

Fig.1 Simply connected domains and non-simply connected. One subtle difference between two and three dimensions is what it means for a region to be simply connected. Any hole in a two-dimensional domain is enough to make it non-simply connected. But, in three-dimensions, a simply-connected domain can have a hole in the center, as long as the hole doesn't go all the way through the domain, as illustrated in this figure[2].

Dear P. K. Karmakar ,

A 3D vector field in any domain out of the origin z-axis must be continuously differentiable in a simply connected domain (thus no holes going through all the way along the 3D domain) this is possible only if the vector field is irrotational (i.e. vanishing curl) which static magnetic fields are obviously not since they have a no vanishing curl and therefore can not be in theory conservative by any view.

But they are in practice. Why? What is the physical explanation?

Emmanouil

[2] https://mathinsight.org/conservative_vector_field_determine

Dear @ Emmanouil Markoulakis ,

Thanks for further feedback.

It seems that your queries have a direct correlation with the factual reality, "Why electric field has only one potential (scalar); whereas, magnetic field has both scalar and vector potentials".

I don't want this conversation to turn into a semantics debate.

My question is simple.

If static magnetic fields are non-conservative in theory why it is the opposite in practice?

Are we missing a critical information here?

@Emmanouil

Your question was answered in the debate following the RG question 'Is magnetic force a conservative or non-conservative force?' in 2014:

https://www.researchgate.net/post/Is_magnetic_force_a_conservative_or_non-conservative_force

Good and interesting discussion

It is a great question, Emmanouil to straighten away aspects of field magnetic effect whether it is overall non-conservative.

Monopoles exhibit that behavior having high energy density. Duality of magnetic monopoles will tend to make it global conservative effect. This maybe the driving dynamics producing dipole & electron generating.

Best regards.

Rajan Iyer

To: Rajan Iyer ,

On which scales do magnetic monopoles exist?

To: P.K. Karmakar,

Key enquiry about magnetic monopoles scale level.

These are still ongoing investigations. Theoretically they may exist at micro level, Planck or even sub-Planck level within what our preprints refer to as vacuum energy quagmire......may be constituent of the dark matter!!

Best regards.

Rajan Iyer

time invariant are merely those laws which do not include time (as a variable) in its describing formulae.

Time is an illusion which is to divide by square-root(1-v²/c²) so that one gets the theory of relativity :))

Hi Emmanouil, The magnetic force between two steady state magnetic fields is irrotational and hence conservative. In theory there exists a magnetic scalar potential but it is hardly ever used.

The issue of the B field itself is more difficult, because modern physics has removed the physical medium through which Maxwell defined B. Mainstream defines B through the Biot-Savart Law which is extracted from the force law (Ampère’s Force Law), and it is irrotational because in this case B is a vector field measured as a point function in distance r from an origin embedded in the source of the magnetic field. It's a radial field.

The situation becomes more complicated at the source itself, which is where Ampère’s Circuital Law applies and where we have curl B = µJ. At the source, the reasoning differs. We are no longer looking at the value of B as a function of a radial distance from an origin. We are now looking at the degree to which B curls around its source electric current.

More interesting still, in order to derive the electromagnetic wave equation in what is wrongly believed to be empty space, we cannot use curl B = 0. We must use curl B = µA, where curl A = B. This is in line with Maxwell's belief that space is densely packed with tiny electric circulations. The magnetic vector potential A very obviously corresponds to Maxwell's displacement current, and it exists everywhere in space. For EM radiation to exist, Ampère’s Circuital Law must apply at every point in space. But mainstream considers Maxwell's displacement current to be a virtual phenomenon. That's the root of all the problems.

Thanks Frederick for the very interesting analysis.

Although my experimental research shows that the field of magnets is irrotatioanal anyway in origin at the net quantum level therefore conservative by origin.

That simplifies things very much and no need for funny scalar potential ψ analysis for the ideal case (not possible really) of an undisturbed isolated magnet.

Here is the conservative field proof for the above ideal case using scalar magnetic potential. H=- ∇ψ and H x ∇ = 0. Which is ridiculous of course since it is basically saying that the magnet is not a magnet when it is not interacting with anything and dormant (not a vector field) since it is playing dead therefore conservative! LOL!

A far stretched proof may I say...

The real reason as I explained is because magnetism has a quantum origin irrotational field therefore the actual vortex nature of magnetism. This quantum dipole vortex net field, decoheres (Quantum Decoherence, QDE) at the macroscopic level and is transmuted to our familiar N-S axial magnetic field.

Emmanouil

Dear @ Rajan Iyer ,

Thanks a lot for your nice and succinct reply.

In order for the magnetic "monopoles" to act as dark matter candidates, they need interact via a pure strong gravity. What about their masses then?

Warm regards,

Pralay Kumar Karmakar

Phy, TU

Hi Emmanouil, At the microscopic level, the magnetic flux density B is the curl of the magnetic vector potential A. We can only make sense out of this in a sea of tiny vortices where A is the circulating current. In the case of an individual vortex, B is an axial vector and curl B = 0. But if we are looking at the mutual alignment of B in a sea of such vortices about a source electric current J, then curl B = µJ.

Hello @P.K.Karmakar:

The question about masses-gravity with magnetic monopoles will be something to quantitatively resolved yet. Myself along Emmanouil & our science team are working to mathematics formalism that I have started on physical consolidated model:Vacuum Energy Magnetic Matter Quagmire that appears within preprints we have input to this RESEARCHGATE forum.

My projected hypothesis conceptually explains mass-like entity with monopoles since they've high energy density & equivalent "mass-like" quantity. However they can't be called as mass since they are not real scalar, but only tensor imaginary. Per Gauss theorem monopoles will not exist really!! With Emmanouil experiments & condensate observations as well as GUTs symmetries monopoles will have to also become a part of Theory of Everything algorithms equations.

We will have to continue serious knowledge investigations about monopoles to get physical picture of how energy will have large to infinite level to account for all the matter in real universe.

We hope scientific community will accept & publish our ongoing papers with promoting supporting approving financial endeavors.

Thanks with ongoing interest.

Best regards.

Rajan Iyer ongoing all through.

Dear Rajan Iyer,

Great, nice efforts, tight works in right direction.

Wish you all the best,

PK Karmakar

Tezpur University

Thank you all with encouragement.

Essentially a magnetic dipole is the connection of two opposite magnetic monopole charges ρm.

Free single magnetic monopole charge is not to be found in nature. As soon they are created they join (not merge) into dipoles.

The reason why div B = 0 is not because it is a spherically symmetrical inverse square law field like the electrostatic field of Coulomb's Law. In fact it's a cylindrically symmetrical field. The reason why div B = 0 is because curl A = B. B is an axial vector field, and the divergence of a curl is always zero. If this fact were more widely known, people wouldn't even be thinking about magnetic monopoles. Since the time varying induced electric field in Faraday's law is given by E = −∂A/∂t, then A is a momentum, and since curl A = B, then A is a circulating momentum and B is the associated axial vector. The magnetic dipole is therefore a circulation and one cannot separate the clockwise circulation as viewed from one side from the anti-clockwise circulation as viewed from the other side. One shouldn't be talking about magnetic monopoles at all. It's more than just that they don't exist. The idea should never have existed in the first place.

The force of the magnetic field on a charge is perpendicular to its velocity. Therefore, the magnetic field does not do work on the charge. It simply does not enter energy considerations. So it does not matter whether the magnetic field is conservative or not, we still have conservation of energy.

Dear Prof. K. Kassner,

So it is a virtual conservative vector field in this case since the general exclusive criterion for conservative irrotational vector fields F= ∇Φ, F the force and ∇Φ the gradient of a scalar potential Φ of the field does not hold (or not yet known), for the case of a single charge introduced in a stationary magnetic field.

I find this explanation unsatisfactory and by far the only case I am aware of a non-irrotational therefore non-conservative by definition, vector field case which exhibits this virtual conservative behavior.

My true meaning of my question here is: What if the cause or origin of the macroscopic classical magnetic field in our case was an underlying pure irrotational net quantum field? Therefore proving their conservative origin.

Would that not conclusively prove the irrotational conservative nature of stationary magnetic fields by origin and disregard the notion as virtual?

The logic here is that if static magnetism shows all conservative effects therefore it must be by definition an irrotational with a vanishing curl field, by origin (i.e. at quantum scale) and somehow transmutes to the familiar classical axial N-S form at the macroscopic level.

I believe our team has a specific answer to this and we are now at the process of publishing it, so that the ongoing debate of academic literature of if stationary magnetic fields are conservative or not and why will reach a conclusive end.

This will also have an important realization and theoretical confirmation, that there is actually a hidden we don't know yet, scalar potential Φ in a stationary magnetic field which its gradient describes the magnetic force F in this case.

Kind Regards,

Emmanouil Markoulakis

Research Fellow

Hellenic Mediterranean University

Reestablished and prior known as

Technological Educational Institute of Crete

Emmanouil Markoulakis

In classical mechanics, the Lorentz force is not obtained via the gradient of a potential. But it is possible to write down a Lagragian for a particle in a magnetic field which does not have the standard form L=T-V with some (possibly time-dependent) potential V(q,t) but rather contains a generalized potential V(q,q̇,t), from which the forces are not calculated via the gradient but via F = -∇q V + d (∇q̇ V)/dt. This Lagrangian is known both for the non-relativistic and relativistic cases. Its (negative) Legendre transform with respect to the velocities becomes a Hamiltonian that does not depend explicitly on time. Hence, we have energy conservation as in cases with more conventional Lagrangians.

I don't think this is the only known case of such a behaviour, it is just the most familiar one. Of course, Noether's theorem gives more general conditions for energy conservation than the presence of only conservative forces.

Thank you for your expert and very interesting analysis of the problem in hand.

However I believe that it is not really a matter of mathematical interpretation and formulation problem but there is an actual physical cause and explanation to this. An alternative physical mechanism model must be investigated.

Also I believe, the axiom of foundation restricts the above general exclusive criterion for conservative vector fields above presented, F= ∇Φ , to be circumvented and must hold in its integrity.

Noether's theorem reinforces the conclusion that magnetic fields matter phenomenon must be conservative in origin.

Nevertheless, this debate is still on in Academia with varying opinions and theoretical proofs offered but I believe quantum experimental data presented and their analysis will give an end to it.

BTW, a zero curl vector force field (i.e. our case of stationary magnetic field interacting with single charge introduced) is not an exclusive conclusive criterion and can not be used as such for characterizing the vector field as conservative.

I can show you a case analysis of a zero curl however path-dependent thus non-conservative therefore non-irrotational field, in a simply connected domain:

https://mathinsight.org/path_dependent_zero_curl

This discussion has split into two separate topics. (1) The issue of a potential energy function for the forces of attraction and repulsion between two magnets. That's one topic, and the forces are both conservative forces.

(2) Then we have the issue of the forces involved in electromagnetic induction, as in F = q[v×B − ∂A/∂t]. One component is velocity dependent while the other is time varying and non-conservative, although it acts within the confines of Lenz's law.

In the case of (1), there is a magnetic scalar potential, but it is rarely seen in textbooks. It follows on the principle of applying Coulomb's law to 'magnetic charge'.

In the case of (2), there exists a velocity dependent potential A.v which can be used in the Lagrangian analysis. This is what Dr. Kassner was talking about. A must be a momentum, and since curl A = B, then B must an axial vector field in which the axes trace out tiny circulations. This is what Maxwell was trying to tell us.

Thirdly, this topic should not get confused with the other interesting topic surrounding the curl of B and Ampère’s Circuital Law. That's a separate discussion altogether.

I agree to all of your points except the case of two macro magnets with face to face poles attracting or repelling each other depending on polarity.

In both cases the dipole interaction vector field formed between the centers of the interacting magnetic poles may have zero curl but also has a non-zero divergence when considered for the field for each single magnet (i.e. field lines from one pole do not return back to the same magnet but diverge to the pole of the other magnet therefore a non-zero divergence ∇⋅B ≠ 0 ).

The force vector is now parallel and aligned to the the velocity vector resulting to linear acceleration. When the two magnets are approached by hand always at the same points in space but by different angles than face to face and then released the work and acceleration produced is different therefore proving that this is a path-dependent vector field therefore non-irrotational and non-conservative.

Its the same as claiming that the magnetic field in 3D space of a current carrying cable is conservative. Although the magnetic field around the cable at the perimeter is conservative thus irrotational the magnetic field in the cable is not irrotational. Therefore we can say it represents a hole in the overall other else irrotational magnetic field of the current cable, running all the way through its magnetic field in 3D space therefore a non simply connected domain and thus by definition non-conservative as a net result.

As a practical example if anyone is still not convinced about the non-conservative magnetic field of a current cable is the transformer, the magnetic field produced and subsequent emf generated depends on the number of turns in the transformer coils therefore path-dependent and therefore non-conservative.

Hi Emmanouil,

We do not talk about magnetic fields in terms of whether or not they are conservative. We talk about conservative forces, as in forces that derive from a potential energy function that is position dependent. We can talk about whether or not a magnetic force is conservative.

What you really seem to be discussing here is Ampère’s Circuital Law and why curl B is zero in space, yet equal to µJ inside a conducting wire. It's a different topic. You need to open up a new discussion on Ampère’s Circuital Law and Gauss's Law.

quote:

"What you really seem to be discussing here is Ampère’s Circuital Law and why curl B is zero in space, yet equal to µJ inside a conducting wire. It's a different topic. You need to open up a new discussion on Ampère’s Circuital Law and Gauss's Law."

The explanation to the above is quite simple:

In every irrotational force vector field everywhere the curl is zero except the center of the vortex (its apex) where the curl is non zero and there is closed loop circulation therefore a rotational field (see figure).Imagine now your cable to be the center of the vortex field shown in the figure.

What I am asking in this thread here is the following rephrased:

We all know that a stationary magnet without interacting with any external charge and no free current present and having a time invariant field has a non zero curl either around or inside the magnet. (frankly presenting the stationary not interacting magnet by stretched out mathematical trickery to appear as having zero curl as shown for example by wikipedia,

https://en.wikipedia.org/wiki/Magnetic_potential#Magnetic_scalar_potential

using the equation ∇ x H = 0 reminds me to the saying, that the moon is not there until you look at it. In case you did not understand the above equation is actually directly implying that the magnet does not exist if it is not interacting therefore not a vector field and thus has no curl. Which is of course an argument totally disconnected from physical reality. We all know that a magnet has a non-zero curl when left alone.

So, its field must be then by definition in origin non-conservative thus non-irrotational.

But its NOT! And has an irrotational force behavior when interacting with a single charge.

How can a non-conservative field in origin produce conservative forces? Or at least their effects?

Unless it is actually irrotational in origin. And by origin I mean the quantum net field of the magnet shown clearly by the quantum magnetic optic ferrolens device in real time, of the net quantum image of the N-S magnetic dipole field of a permanent magnet in the included figure.

We see clearly its two polar N-S vortices (irrotational) left and right placed back to back. The magnet is placed under the ferrolens at its side, north pole to the left and south pole to the right (see thick colored lines polar vortices).

The familiar classical macroscopic field axial N-S field imprint of the magnet is formed as a tensor field between these two back to back net quantum vortices , shown here by the thinner brown force lines.

The mechanism and physical explanation of how this net quantum dipole vortex field existing in every magnet is transmuting to the classical axial N-S field imprint at the macroscopic scale is given in our shortly coming Journal article publication.

Either way we believe this discovery will shed more light of the inner workings of matter and the connection between the Quantum world and our macroscale Universe. The experiments were done with this mesoscopic device the ferrolens https://en.everybodywiki.com/Ferrolens which is a able to project net quantum effects and phenomena to the macroscale, giving their quantum image version.

Notice here, that the Ferrolens is non-magnetohydrodynamic meaning that it is an insulator to electric currents also the antistatic surfactant coating of the magnetite nanoparticles prevents clumping or agglomeration.

copyright©Emmanouil Markoulakis Hellenic Mediterranean University 2019

(recently reestablished, prior known as Technological Educational Institute of Crete)

Emmanouil, Are you talking about the curl of B or the curl of the magnetic force between two magnets? If you are talking about the latter, what formula are you using for the force field? If it's purely radial, the curl will be zero and it will be a conservative force.

By vector calculus and gradient theorem strictly speaking, even the magnetic force of a stationary magnet acting on a single charge introduced in the field is not irrotational and therefore non-conservative.

Zero curl dues not necessary imply conservative and thus irrotational in a total simply connected domain. Although non-zero curl means 100% a non conservative field the other way around is not always true.

Nevertheless, the single case charge case definitely is irrotational I agree but proved when a different mathematical analysis is used I have in my coming publication.

As for the two permanent magnets case interacting the net result is non-conservative as I have explained for the force vector field.

Conservative nature of magnetic field is a local feature only

Its curl should vanish

quote:

"Conservative nature of magnetic field is a local feature only

Its curl should vanish"

Exactly my point! You couldn't say it any better.

That is actually what I experimentally demonstrate in our research that it actually does has a vanishing curl. Not the macroscopic magnetic field of the magnet but the origin underlying net Quantum magnetic Field (QFM) of the magnet which is a dipole vortex as shown in the figure I posted.

The macroscopic field is just a tensor field created by the two back to back polar quantum vortices shown in the figure therefore the QFM is the cause and origin of the macroscopic field which is the effect.

Therefore magnetism is pure irrotational field by quantum origin and thus conservative by nature.

The macroscopic field N-S axial field is a distraction really, nature tries to fool us with it, the real work here inside the magnet is done by the underlying net QFM which is essentially a dipole energy vortex.

I can continue the discussion all the way down to the electron If you like.

Vortex, the only flow of energy mechanism in the Universe that can create and sustain (i.e. conservative field) a point like condensation of energy thus an elementary particle.

Dear Emmanouil Markoulakis ,

Your efforts to make the points clearer are thankfully appreciated.

Hope, better explanations are yet to come.

Warm regards,

PK Karmakar

Department of Physics

Tezpur University

Hi Emmanouil, When two magnets pull together, kinetic energy is created and it came from some kind of magnetic potential energy. And the situation is reversible. The formula for the force is radial in distance so it is a conservative force.

Dear Frederick David Tombe ,

You are right. It is the case of mutual potential energy and mutual torque. On magnet executes simple harmonic oscillations in the magnetic field of the other in the small amplitude limit approximation; and vice-versa. It is a conservative case in pure isolation.

Warm regards,

PK Karmakar

Tezpur University

Dear Dr. Karmakar, You appear to be attempting to explain how magnetic force arises. That isn't relevant to the issue of whether or not it is a conservative force. But since you have brought the subject up, I can tell you that the magnetic potential energy in question, in the case of magnetic attraction, is ultimately electrostatic potential energy on a smaller scale. Magnetic lines of force are double helix alignments of rotating electron-positron dipoles.

In the case of magnetic repulsion, the potential energy is centrifugal potential energy on a smaller scale. See, Article The 1855 Weber-Kohlrausch Experiment (The Speed of Light)

for magnet attraction, and see, Article An Interpretation of Faraday's Lines of Force

Maxwell showed that lines of force behaved like vortex structures in a hypothetical fluid. then ignored the logical obvious that the magnetic lines of force are the eyes of the vortexes. Collectively they rotate magnetism in either clockwise or anticlockwise vortexes around the long axes of magnets and coils. Like spins attract......The force on conductor doesn't need an axial vector to describe its direction. The correct direction of magnetism coherently visualizes curly phenomena

Dr. Taylor, You have not reported Maxwell accurately. Here's a link to his 1861 paper and you only need to read the preamble to Part I to get the correct picture, http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf

Magnetic repulsion is caused by centrifugal force acting sideways from the lines of force.

qoute:

" Magnetic repulsion is caused by centrifugal force acting sideways from the lines of force. "

Yes, electrons are actually magnetic spinning tops and their fermionic repulsion because their like electric charges are actually cenrifugal forces of their physical spin. When two same spin (CW) electrons come close their spin repels them.

Essentially what elecgtric charge -e is, is the whirling magnetic dipole flux due to their physical spin proving once and for all that the origin of electricity is moving magnetism with magnetism being the prime phenomenon and electricity an effect and not the other way around.

Magnetism is not originating from moving electrons but the electron itself is created by elementary physical magnetic vortex.

Nice and informative contributions by enthusiastic Researchers.

I must appreciate.

What is meant by (a) Magnetic vortex, and (b) Magnetic vorticity?

Hi Emmanouil, More likely magnetic repulsion arises from the curl in the velocity field A that is associated with Coulomb's law of electrostatics. But to get such a curl you'd need to have an electron orbiting a positron. The radial component of A will be such that curl A = 0 while div E = ρ/ε as per Coulomb's law, while the transverse component of A will be such that curl A = B, E = −∂A/∂t, and curl E = −∂B/∂t as per Faraday's law. In other words, magnetism arises as a result of curl in the electrostatic velocity and force fields, with the transverse A being the magnetic vector potential and B the associated axial vector field.

vorticity in classical hydrodynamis is more or less analogous as effect to the curl

Field having a curl thus non-zero curl or non zero vorticity means that the net work done in a closed loop motion on a particle is not zero thus path depedent this translates that the net accelaration of the particle inside the loop is not zero and energy is consumed by the field.

Having zero curl and zero vorticity means that the force vector field does not consume energy and any change in kinetic energy of the particle in the field is translated to equal change in potential energy and vice versa therefor this field is path independent mathematical l meaning that net work done in a closed loop motion inside the field is always zero.

For stationary magnetic fields the particle introduced is an electric charge particle and the interacting EM force in this case is the Lorentz force.

The only vector field which can be conservative as above described is in geometry the form of an irrotational motion field where velocity v analogous to 1/r distance from the source/sink of the field thus a free vortex.

However by observing the magnetic flux (i.e. lines of force) on a magnet its field geometry is definitely not irrotational thus a vortex. Therefore it can not be conservative by definition however it exhibits all the characteristics of a conservative field... Why?

Our research shows that magnetic fields are pure energy vortices by quantum origin.

wh

Also an important characteristic of conservative irrottional fields is the vanishing curl, narrowing spiral motion of a particle when subjected to the field.

My discovery about the underlying QFM vortex field of magnets explains also the Lorentz force of electric charges inside the field and their spiraling motion they are subjected when they are inside a stationary magnetic field. Its not any electric charge but the moving magnetic dipole moment of the electron interacting with the stationaty magnet QFM vortex field.

As a matter of fact any nanosized matter particle or molecule which has a magnetic dipole moment will spiral in the QFM vortex field of the stationary magnet. Here is a video of the ions produced by electrolysis interacting with the magnetic field of a magnet and making the 3D imprint QFM polar vortex of the magnet:

https://www.youtube.com/watch?v=TlDGU05-nvk

This is because Quantum Decoherece (QDE) effect the very nanosized or less small particles can "feel" the quantum vortex flux of the magnet. Any larger macroscopic more condensed magnetized matter like iron filings or other magnet or ferromagnetic material can not see this field due QDE and will only see the macroscopic field N-S axial field of the magnet and when face to face with the pole of the magnet will be attracted in a straight line without spiraling helical motion. Release any magnet to another very strong magnet form a relative large distance, no spiral only straight line attraction. This is another proof that the QFM vortex field of the magnet transmutes to the familiar axial filed at the macroscale.

Why nanoparticles molecules or electrons are spiraling inside a magnetic field as Lorentz Force dictates and larger macroscale matter does not?

So Lorentz force is simply a pure quantum magnetic dipole interaction force nothing about electric charge really...

Everything can be explained with magnetism and the QFM vortex field discovered.

Finally, the macroscopic field between say two attracting magnetic poles is a linear axial field there is no radial force present only straight line parallel to the motion accelerating force path dependent and therefore this dipole interaction field can not be conservative.

Dear Emmanouil Markoulakis ,

Thanks a lot for your nice and interesting answer

For all aspects related to this kind of discussion you should read the excellent papers (also via research gate) by F.W Hehl (university of cologne) including the historical evolution of Maxwells equations. Be careful about Vortexes..this sound s like "scalar electromagnetic waves" which belong to the junk bin (simple math error in the derivation there)

Your valuable and significant contributions are really praiseworthy

The insights of Emmanuil about how fundamental is magnetism is interesting.

We do not have to forget that the spin is strictly tied to the higgs mechanism (coupling two spinors) which is so far the only reasonable explanation of how mass is of elementary particles is generated. Mass and charge come from such mechanism...

The magnetic moment is nothing but an angular momentum and is fundative due also to the fact that it is nothing but the plank constant "h"...

It all starts with curl A = B. A is a momentum density while B is an angular momentum density. The issue of being conservative doesn't arise until we are looking at the force of attraction or repulsion between two magnets.

Then when it comes to the issue of curl B, it depends on the context. For the purposes of Ampère’s Force law, we use the Biot-Savart law in which case we are considering B as a vector field with respect to distance r from a point origin. The curl of B is zero since the vector field is radial, and Ampère’s Force law is conservative.

However, if we are looking at the geometry of the magnetic field with respect to the manner in which the magnetic field lines form solenoidal rings around the source current J, then the source current J is an axial vector as per curl B = µJ. But this has no bearing on whether or not Ampère’s Force law is a conservative force.

Dear all,

"non-zero curl static time invariant" means that the field is static not irrotational..

Sufficient condition for a field to be conservative is to be irrotational which means that its rotor is zero.

This does not mean that a field which is non-irrotational is non-conservative there is a flaw in the propositional logic which I'm trying to point out!!

if a field is irrotational --> then it is conservative

the irrotationality implies the presence of a "scalar potential" hence conservativity.

(see the static gravitational field)

non-irrotational does not imply non-conservative but it is true the proposition

non-conservative --> non-irrotational...

All the non conservative fields have to be non-irrotational.

In other words the magnetic field, which is non-irrotational, does not necessarily imply that it is non-conservative...

the presence of a time invariant "Vector potential" infact, is a guarantee of the opposite...

all the energy due to circulation/circuitation in a magnetic field is stored in the magnetic field and can be given back, unless non-conservative events like emission of radiation or friction intervene in the process.

All conservative vector fields are irrotational in 3D space in a simply connected domain. This means that they have a vanishing curl (i.e. zero curl).

Therefore,

Conservative 3D space simply connected domain vector field irrotational

Since we are talking about physical fields we are not interested here in 2D space in this discussion.

Also we are not talking here primarily about the forces involved but the fields.

A static magnetic B field of a permanent magnet macroscopically has curl thus non-zero curl and non-irrotational therefore by definition it should not be a conservative field. However it exhibits conservative behavior...

Why?

This is the question I ask.

The established proposition for the above is that magnets have "virtual conservative" fields. I find this answer unsatisfactory.

It should be "local conservative fields"

The final answer is:

Turns out that magnets have a quantum origin irrotational field after all.

More to come in our shortly forthcoming Journal Publication.

Emmanouil

again, the irrotationality is a "sufficient" condition not a "necessary" condition for a field to be conservative....

okay then show me a physical field (3D Space) which is conservative by strict mathematical definition thus a vector field F is conservative if and only if it has a potential function f with F=∇f but is not irrotational?

https://www.quora.com/Does-the-conservative-vector-field-and-irrotational-vector-field-imply-the-same

One subtle difference between two and three dimensions is what it means for a region to be simply connected. Any hole in a two-dimensional domain is enough to make it non-simply connected. But, in three-dimensions, a simply-connected domain can have a hole in the center, as long as the hole doesn't go all the way through the domain, as illustrated in the figure above.

The reason a hole in the center of a domain is not a problem in three dimensions is that we have more room to move around in 3D. If we have a curl-free vector field F (i.e., with no “microscopic circulation”), we can use Stokes' theorem to infer the absence of “macroscopic circulation” around any closed curve C. To use Stokes' theorem, we just need to find a surface whose boundary is C. If the domain of F is simply connected, even if it has a hole that doesn't go all the way through the domain, we can always find such a surface. The surface can just go around any hole that's in the middle of the domain. With such a surface along which curlF=0, we can use Stokes' theorem to show that the circulation ∫CF⋅ds around C is zero. Since we can do this for any closed curve, we can conclude that F is conservative.

Even using the above analysis describing the sub-condition for a simply connected domain in order a field to be conservative thus irrotational we see that that even this condition does not hold for the macroscopic stationary 3D field of a permanent magnet since its magnetic dipole moment, N-S magnetic axis of its field, represents a hole going all the way through its domain. So that the total field in this of a magnet can not be conservative since it is not a simply connected domain.

Again, why then vector field of a macroscopic magnet shows conservative behavior (i.e. energy wise) although by mathematical definition it should not be? Are the maths wrong or the physics? Or we missing something?

The answer is given in our shortly coming Journal Publication we will share here in RG,

Dear Emmanouil and Stefano,

when people call vector fields with vanishing line integral over closed curve (irrotational vector fields for short) 'conservative' they imply that they act as force fields on some material bodies in the way gravitational fields act on masses and electrostatic fields act on charges, i.e. such that the force is proportional to the field. If this scenario (which was a prominent research topic in the nineteenth century) is understood, then the field does not allow to build a perpetuum mobile and the name 'conservative' is well chosen. Not so if the field's action on material bodies is different, as is for magnetic fields! In this case 'conservative' is simply the wrong word since it suggests that total energy of a physical system comprising magnetic fields, electric charges, and permanent magnets would not be constant ('conserved'). Actually any mechanism made of frictionless turnable permanent magnets conserves energy. As Emmanouil would say: "exhibits all the effects of conservative field".

Dr. Mutze, Yes. I tried to explain that earlier. The term "conservative" ideally only applies to force fields. Having said that, since the vector field B is central to Ampère’s force law, then when ∇×B = 0, the force will be conservative. And ∇×B does equal 0 when B is defined by the Biot-Savart Law, since the focus is on the radial field and distance in r. However, when we are looking at the curl of B in relation to the source current, which is a different matter, then ∇×B = µJ but this has no bearing on whether or not Ampère’s force law is conservative.

I don't agree. The term conservative field applies to vector fields not necessarily force fields. Proof, the gravitational field which is a vector field and conservative without being a force field since according to GR gravity is not a force.

Emmanouil, A gravitational field is a force field.

Emmanouil,

of course there are mathematical texts on vector fields which treat 'conservative' and irrotational as saying the same thing. These are either not aware of the implied physical context or assume that readers will understand this context. This context is: multiplied with a charge or a mass the field a point P is a force acting on a probe particle on P. If this is given the field together with the probe particle is a energy conserving physical system (if the potential part of the total energy is suitably defined) and the name 'conservative' is OK. If one thinks of 'conservative field' as a mathematically defined entity independent of the mentioned context one walks into one of the many traps that language offers to careless scientists.

Emmanouil

I just red your contribution containing the statement:

Again, why then vector field of a macroscopic magnet shows conservative behavior (i.e. energy wise) although by mathematical definition it should not be? Are the maths wrong or the physics? Or we missing something?

It is well known that the B-field of any permanent magnet is irrotational. This is a direct consequence of the fact that it can to any desired accuracy be represented as originating from finitely many magnetic dipoles located in the interior of your macroscopic magnet. Your idea that the magnetic NS-axis (a merely imagined line!) would spoil simply connectedness and would destroy irrotationality is lacking any reason.

Dear Ulrich Mutze,

Static magnetic fields (not the magnetic force) have curl therefore non-zero curl and therefore can not be by definition conservative strictly speaking. I myself know that they are conservative but proving it is a different matter.

The reason why, is different from the explanations you give and has to do with the quantum origin of the macroscopic static magnetic fields I believe. (Although you correct stated "... any desired accuracy be represented as originating from finitely many magnetic dipoles located in the interior of your macroscopic magnet ", however the problem is that these microscopic magnets are described with the same exact model as the macroscopic bar magnet and therefore the problem remains)

So the correct answer is neither the physics nor the mathematics on this matter are incorrect but we are missing a crucial bit of physical information.

That is my opinion.

Kind Regards,

Emmanouil

quote: "Emmanouil, A gravitational field is a force field"

No it's not according to GR it is a velocity vector field and gravity is not a force but a distortion of spacetime itself. Of course I don't albeit in this description but this is a different matter.

Emmanouil, Gravity also involves a velocity field associated with the escape velocity, but normally when people talk about a gravitational field, they are talking about the gravitational force field and not the escape velocity. General relativity however focuses on the escape velocity formula. And when we are talking about a gravitational field being conservative, we are talking specifically about the force field.

Point accepted.

Dear Ulrich,

Dear Stefano,

please specify what is not clear to you.

Emmanouil,

what is wrong here:

I say "... the B-field of any permanent magnet is irrotational" and you answer

"Static magnetic fields … have ... non-zero curl …" and you ignore the obvious argument that any superposition of Dipol fields is irrotational (the mathematics is the same for magnetic dipoles and for electric ones and for static electric fields E

we know curl E = 0 from Maxwell). Of course there are static magnetic fields with curl different from 0 somewhere (those made by currents since curl H = j) but for macroscopic permanent magnets and (in order to avoid complications) restricting our interest to the space outside the material magnet we have curl B= 0. Do you agree with this statement?

The curl(field)=0 calculation in a volume assumes the source of the field is in the volume. That is not the case for the volumes herein considered. So, I'm having some difficulty understanding the difficulty of this question.

Dear Ulrich.

First,

quote: "...restricting our interest to the space outside the material magnet we have curl B= 0. Do you agree with this statement? "

That is not correct, the correct is,

...restricting our interest to the space outside the material magnet *in a vaccum* we have curl B= 0.

I any other surounding medium apart vaccum space the outside field has a non-zero curl therefore can not be regarded as conservative.

Second, I never said the I am interested in the special cases but as the total of the magnetic field of a permanent magnet regarded as conservative field or not.This debate is not new and subject of much discussion over the years. I am not the one who inveted this debate. Both sides have strong arguements but nothing conclusive yet about the conservative origin of the filed of permanent magnets which I am also convinced of, but I find the proofs and arguements presented not complete and satysfying.

A conservative field (not the magnetic force) should have a closed line integral (or curl) of zero . Maxwell's fourth equation (Ampere's law) can be

written here as:

(see figure)

so we can see this will equal zero only in certain cases therefore static magnetic fields can can not characterized as conservative in general.

Also I find the scalar proof presented here (https://en.wikipedia.org/wiki/Magnetic_potential#Magnetic_scalar_potential) of an isolated, undistarbed not interacting with any charges, field of a permanent magnet as being conservative not satisfying at least and quite frankly as a trick:

∇ X H =0 with H=-∇ψ, where ψ its scalar potential.

Basically saying that an isolated permanent magnet has no charges (practically menaing it is not a magnet) and therefore its field has zero curl and can be considered as conservative.

This condition is an ideal non physical condition which can no hold on pemanent magnets where their field is genrated by matter electrons inside and outside of their volume even in the vaccum outer space which we know now is not empty but full with virtual charged particles.

There was never experimentally found a zero-curl field of a permanent magnet. This is an ideal description of magnets not found in nature and applies only for the outiside field (not inside the material) of the magnet in an hypothesized ideal vaccum.

Kind Regards,

Emmanouil

There is a very real problem here which has caused a lot of confusion. The problem begins with the idea that ∇×B = 0 in free space. This is a problem which follows closely on the mystery of Maxwell’s displacement current. Ideally the correct situation is that there is no free space and that in the luminiferous medium, the equation should be ∇×B = µA where A is Maxwell’s displacement current which equates with the magnetic vector potential.

The prevailing idea that ∇×B = 0 in free space is based on the Biot-Savart Law which itself is based on Ampère’s force Law being conservative. But the B in the Biot-Savart law and in Ampère’s force Law is not exactly the same concept as the B in Ampère’s Circuital Law. In the Biot-Savart law, B is a plot of the magnitude and direction of the magnetic flux density with respect to distance r from a point origin at the source current, integrated over every source current element. Taking the curl of this B uses r as the independent variable and so we get zero. However, in the case of Ampère’s Circuital Law, the focus is on the manner in which B, as an axial vector, circulates around a source current J. The result ∇×B = µJ is arrived at by a different argument not involving any radial distance from a point origin. The latter is correct, whereas taking the curl of the Biot-Savart Law ignores the direction of B since we are only taking a partial derivative in r and ignoring the direction of J, and so the idea that ∇×B = 0 in free space, based on Biot-Savart, is not the same concept as ∇×B = µJ based on Ampère’s Circuital law.

Emmanouil,

don't become carping. You know to quantify the magnetic influence of the air in your lab and know that it is negligible for all practical purposes. You also know that it impossible in principle to show by experiment for any physical quantity that it has exactly the value 0. What is conceivable is that one establishes by experiment a bound b > 0 for which one can say with confidence that the quantity is above b (and thus established to be not 0) or below b (and thus not shown to differ from 0). The question of curl B = 0 or not outside a permanent magnet becomes interesting if by some ingenious experiment one finds a value different from zero in precisely this sense. Before that one better follows established theory which proposes the value 0 to curl B.

If you care about the virtual charged particles arround I have a further cause for concern for you: Even if you employed Euclidean geometry so far (for instance in the definition of curl) a gravitational wave may hit your lab and may invalidate all you geometry related measurements. Poor experimenter. What can he be sure of in this uncertain world?

Section IV here,

Article An Interpretation of Faraday's Lines of Force

After all this discussion one is for sure. Proving by formal classical method that the field of a permanent magnet is conservative or not is not an easy task due to the dipole nature of its macroscopic field.

The ultimate prove will come only from its quantum subfield analysis.

".... a gravitational wave may hit your lab and may invalidate all you geometry related measurements. Poor experimenter. What can he be sure of in this uncertain world? "

hahahaah!!... That was a brilliant gravitational joke made me fall off my chair! LOL!

"Before that one better follows established theory which proposes the value 0 to curl B"

Inside or outside, the fact remains that static stationary permanent magnets have fields on their entirety with curl thus non zero-curl and zero divergence. And that is the established theory for the magnetic fields of permanent magnets

the only way to have curl B =0 in all space is with B = 0 therefore not a magnet and a magnetic field.

So the title of this question thread is physically correct in all cases.

Also in practice the experimental measured value found in labs of curl B non-zero value for the outside far field depends on the size of the magnet is not due a control experiment error but it is real. For me there is a huge difference between zero and not zero in this case.

Magnets with zero curl can not exist because this defies the very definition of a permanent magnet field.

One way to produce such a field with zero curl partially, is to use Helmholtz coils where a magnetic field is produced which is uniform throughout all of space of the coils.

Emmanouil, The analysis of the forces between permanent magnets is generally side-stepped in modern physics courses. There exists no satisfactory mathematical magnetic potential energy term. The attractive force between two unlike poles looks like Gauss's law but with the role of electrostatics hidden. So you are right in saying that it won't be fully understood in the absence of knowledge regarding the deeper physical structure of magnetic lines of force. And all this follows closely on similar problems surrounding the modern textbook treatment of Maxwell's displacement current. Ultimately none of these things are going to be understood if one insists that space is empty, as the textbooks do. The textbooks preach empty space and so we end up with dubious concepts like ∇×B = 0 in the steady state, and Maxwell's displacement current being considered to be merely virtual in the dynamic state. None of this makes any sense and the elephant in the room is the fact that the medium that Maxwell used to derive his equations in the first place has since been removed from the textbooks.

Frederick,

You are absolutely right, you couldn't say it better.

That is my point the difficulty of the today's physics to explain phenomena like the conservative field of a magnet which should be child's play considering the state of our civilization in the 20th century is an indication that something is not correct in the current model they adopted from the work of Maxwell and the other great fathers of Electromagnetism. They cherry picked on Maxwell and twisted his work which was based in the assumption entirely of a medium for the EM energy.

So they took the wrong turn blocked an entire possible way of scientific progress and now for the last 100 years they are still using chemical propulsion rockets to get off the ground and fly manned missions around the Earth the last 50 years , half a century, stack in a loop. There is definitely something wrong or at least incomplete or missing with the current model.

The deciding moment I believe was around 1930 close to Einstein's era and the rise of quantum mechanics where everyone got into billiards ball hunting without understanding completely the prime phenomenon in our Universe and force of EM. They should have more fully explored the nature of EM and continue Maxwell's work instead they claimed that EM was fully understood and they went on ball hunting. Without understanding the prime force of nature they went on in the quantum world and developed crazy statistical theories inconsistent with nature which will lead and already do to a dead end.

The funny thing is that denying an underlying medium consisting our spacetime and treating space as nothing was a mistake and the finally understood and last decades more and more theories pop up with undercover aether modes like loop quantum gravity vacuum foam etc. But we lost precious time. Considering the exponential growth of our civilization the last 1000 years we should have left our solar system by now instead they play merry go round around the Earth.

The last 20 years beginning of the 21st century is a medieval period for scientific theoretical and experimental breakthroughs in physics.

Emmanouil

Oh Emmanouil,

if you demonstrate curl B different from 0 experimentally with a permanent magnet (outside the material magnet, in air) you will become known as of one of the greatest contributors to modern physics. However, you can't provide more than unspecific subjective statements like "Also in practice the experimental measured value found in labs of curl B non-zero value for the outside far field depends on the size of the magnet is not due a control experiment error but it is real. For me there is a huge difference between zero and not zero in this case."

I found this paper most interesting about the magnetic field in magnetic materials:

http://bulk-sucon.eng.cam.ac.uk/amc1/BH.pdf

Frederick, bad physics text books, like bad newspapers, may show a distorted view of the world. Good textbook (like good news papers) give the clearest view we can obtain today. That even the excellent 'electrodynamics of continua' of Landau&Lifschitz (565 pages in my German edition) has no ready made formulas for forces between permanent magnets has a good reason: the only manageable case of perfect permanent magnets (where the spins are irrotatably fixed in a lattice of atoms) is trivial from a theoretical point of view (my first task in industry was to get the actual formulas for this case) and of very limited practical relevance since interacting real magnets demagnetize each other to some degree in a reversible way. If you need demagnetization effects properly taken into account many manufacturers of permanent magnets offer the service to do the calculations on their computers with their proprietary software.

Dr. Mutze, When you take the curl of the B field in the vicinity of a permanent magnet, the result ∇×B = 0 only follows if we use the Biot-Savart Law. That’s because we are taking partial derivatives in the distance r from an origin located somewhere inside the source magnet. In doing so, we are ignoring the direction of B and so we are not taking a curl in the truly physical sense in the spirit of Ampère’s Circuital Law.

The correct answer is, that in the vicinity of a permanent magnet, ∇×B = µA, where A is Maxwell’s displacement current (the magnetic vector potential). Then in the dynamic state will have the simple harmonic relationship A = −ε∂2A/∂t2, and from Faraday’s Law, E = −∂A/∂t, we can obtain Ampère’s Circuital Law in the dynamic state in the more familiar form ∇×B = µε∂E/∂t.

You can therefore now see the continuity between the steady state and the dynamic state, but it requires that space be filled with fine-grained circulating currents, A, as per Maxwell’s sea of molecular vortices. If we remove Maxwell’s sea of molecular vortices, that he used to derive his equations in the first place, we end up with the kind of confusion that has been illustrated in this thread. See, Article Wireless Radiation Beyond the Near Magnetic Field

Ulrich,

I don't understand what you are talking about.

Curl of the natural magnetic field of permanent magnets in a medium even if its air can never be zero since there are always electric charge currents present even if they are infinitesimal small compared. A small curl value in air can not equated to zero. This condition theoretically exists only in free vacuum space but never verified experimentally. Magnetism can never exist and propagated without virtual photons produced by electron interactions. So even in free space which is full of virtual particles the zero curl condition will never be reached in physical reality. It is just an ideal case.

Macroscopic static permanent magnets 3D magnetic dipole fields as a whole have non-zero curl always, that is the case we are discussing here and not the curl of the magnetic field 1 meter away from the magnet in a 10X10 cm 2D sheet area.

The question in this thread is valid, thus how a non-zero curl filed thus it has no vanishing curl and therefore non-irrotational can exhibit energy wise conservative behavior ?

The answer is to be found in the quantum scale of matter and not via any macroscopic analysis. That's all I am saying here.

Emmanouil

Emmanouil, When you say that we can never have ∇×B = 0 in physical reality, your statement is true providing that we accept the existence of Maxwell’s sea of molecular vortices. But for mainstream physicists who reject Maxwell’s sea of molecular vortices, they can have ∇×B = 0. They can have it either by taking the curl of the Biot-Savart Law, in which case the result is no longer Ampère’s Circuital Law because it neglects the orientation of the B field, or they can have it by applying Ampère’s Circuital Law, ∇×B = µJ, on the basis that J = 0 in space. Either way, it is totally misleading. Empty space also empties out the displacement current ε∂2A/∂t2 which is crucial in the dynamic state and so the texts books have to find another way of justifying Maxwell’s displacement current, which they can’t do without contradicting Ampère’s Circuital Law.

As regards however the conservative Ampère Force Law from which the Biot-Savart law is extracted, none of this has any bearing on it because taking the curl of the Biot-Savart Law does actually lead to zero, even if the result is no longer Ampère’s Circuital Law.

Emmanouil,

you don't understand what I was talking about. Probably it surmounts my mental capabilities to change since. I'll not continue to try. Good luck.

The curl property represented by geometrical shapes as simply I could, so that I make sure that we speak the same language.

And here is the real physical reason, for the conservative energy wise behavior of macroscopic static dipole magnetic fields, thus their quantum origin irrotational polar vortex fields as shown in the above photograph in real time of a magnet by the quantum optic device Ferrolens ( https://en.everybodywiki.com/Ferrolens ).

These N-S polar vortices are totally masked out and not detectable at the macroscopic level due the Quantum Decoherence (QDE) effect. These dipole quantum net irrotational vortices are the actual reason for the conservation in magnets and responsible for the creation of the macroscopic magnetic dipole tensor field we call classical field of magnets.

Magnets have conservative fields because at the quantum level their net polar fields are truly irrotational vortices with a vanishing curl thus curl zero.

Don't believe it. No problem. Wait for the publication.

copyright©Emmanouil Markoulakis Hellenic Mediterranean University (HMU) 2019

Emmanouil, This article deals with all the issues that have been raised in this thread, especially the issue that curl B never equals zero.

Deleted research item The research item mentioned here has been deleted

Frederick. Right you pin pointed the heart of the problem using Maxwell complete and not cherry picked, original theory.

There is a huge difference in physical meaning between a small curl and curl zero. You can not say because in a cherry picked area 2 meters away from the magnet you measured a very small curl even if this is infinitesimal small that the curl is zero not even as an approximate and then gneralize this for the whole outside field of the magnet saying it has curl zero. That is twisting the physical property of magnetism. Even at very homogeneous field of Helmholtz coils the field has eventually to curl to return to the other pole.

So the statement of magnetic field with zero curl is deceiving even in a spaecific area of the field there is aways a non zero curl value present. It is against the very nature of magnetism. Magnetism can not exist with zero curl.

Virtual photons as literature refers them which the magnetic flux lines consist of are made due the interaction of electrons. There is no propagation or generation of light in space without them (i.e. electrons) not even for a mm of travel. So displacement current is always present even at deep vacuum space which now science refers to as being full of virtual particles popping constantly in and out of existence in our specetime domain. Displacement current is essential to the very existence of magnetism and thus the curl of magnetic fields can never be zero, zero means no magnetism.

You go even deeper in your article describinng "empty" space full of quantum vortices aetherial medium distortions responsible for the displacement current. I agree, these are essentially the virtual photons literature refers and is indirect aknowldeging that 3D space is actually a finite medium thus a matter-energy object. And in my opinion our Universe has at least two spacetime domains in superposition. A superluminal spacetime domain or else called historically Aether where energy entities are in superluminal speeds and when strongly interacting are loosing speed and falling to subluminal speeds and are falling into our 3D spacetime domain forming our ordinary eneragy-matter we perceive as our Univrerse.

The acknowledgement of the existence of a superluminal spacetime domain is essential for the very decription of the speed of light as a boundary or else the speed of light limit looses its meaning. Limit in what? That is what logic dictates and is the answer for the non-locality (a funny term to describe superluminal interactions) nonsense of the quantum world. Everything has a perfect logic explanation. I refuse to accept science without logical explanations forced into my throat.

Emmanouil

Also know this, NASA has concealed many years now information from the public.

Vacuum space is full of free electrons and by that I don't mean the so called virtual particles but literally.

Dear Emmanouil,

let's go back to your question:

"non-zero curl static time invariant magnetic fields therefore non conservative by definition"

If you take as the definition of "conservative" the property of being irrotational then a time invariant M field is not *conservative* for sure. The property of being Irrotational is directly connected with a scalar potential, (like the gravitational field), positional, there is a dependence only on the radial position of a test body set in a position referred to a center of forces.

The property of being Irrotational in Physics is in general stronger than the property of being conservative. Conservative is in general something which does not have dissipation in term of heat or radiation, in other words the process can be reversed, no energy flies away..it is better described by thermodynamics...

The Vector potential A, used to describe the magnetic fields let the quantity of circuitation being conserved, by running around the close path in the opposite way the energy is given back.

Dear Stefano,

I think I made my point clear in my last posts and experimental data presented about the quantum origin irrotationality vanishing curl of magnets therefore definitely conservative fields by origin and definition.

The macroscopic classical N-S axial field of magnets is just there to deceive, a product of quantum decoherence effect at the macroscopic scale. The real generator field responsible for the macroscopic filed of magnets is the underlying net quantum dipole vortex field of magnets (QFM) present on every magnet and masked out at the macroscale.

Nature plays tricks on us and has hidden secrets :)

I refer ultimately to conservative behavior term energy wise for a permanent magnet thus its tendency to retain its magnetism and energy when interacting with charges. Of course due thermodynamics and entropy their is a loss of magnetization over time but this is a very slow process for permanent magnets.

Kind Regards,

Emmnaouil

Also in must inform you all that the Ferrolens quantum magnetic optic device is non-magnetohydrodynamic meaning it is an insulator to electric currents and its surfactant coating of the 10nm magnetite Fe3O4 nanoparticles makes it also antistatic.

https://en.everybodywiki.com/Ferrolens

Also as I said I don't trust NASA, they have many little dirty secrets.

https://www.youtube.com/watch?v=c-C1jdoe8eY

Dr. Quattrini, Consider Ampère’s Force Law, F = µ/4π[I1I2∮∮dl1×(dl2×r̂)/r2] . This is the magnetic force between two closed electric current circuits. It's conservative and it has an associated scalar potential based on the solid angle.

But having concluded that ∇×B = µA and not zero in space, does this have any impact on the argument, bearing in mind that B is extracted from the Ampère’s Force Law formula?

Yes I see the conflict.

Emmanouil, The answer will be that when ∇×B = 0, it is no longer Ampère’s Circuital Law, but it will be indicative of an irrotational force. The real Ampère’s Circuital Law in space will be ∇×B = µA and we will be using a polar origin placed within each individual tiny vortex rather than at the source of the magnetic field.

Theories, thoughts and ideas are generally accepted, dismissed, maybied, hard-basketted . later-basketted etc., according to ones schooling. This process rarely but drastically stumbles when the proposition contravenes what was learnt in schooling;

Scholars took over 100 years to unlearn Ptolemy and Aristotle and accept Copernicus: decades to accept Einstein; and in my case decades so far to unlearn that magnetism doesn’t irrationally encircle its current but flows absolutely logically parallel to it.

Emmanouil, OK, so now that we’ve got the details cleared up, the answer to the question in the heading of this thread is, that you are not comparing like with like. There are two different scales of magnitude involved in each case.

(1) When you say that the magnetic field has a non-zero curl, as in ∇×H = J = A, we are on the micro-scale. For the purposes of taking the curl, we are using a point of origin inside one of the tiny vortices that makes up the magnetic field. J = A is the circulation within the tiny vortex. This is Ampère’s Circuital Law, and in the dynamic state it is significant in conjunction with displacement current and Faraday’s law of Induction for the purposes of deriving the electromagnetic wave equation. A is displacement current.

(2) When you say that that the magnetic field is conservative, this is based on taking the curl of H on the large scale with a point origin inside the source magnet. The result, ∇×H = 0 is not Ampère’s Circuital Law. It is not the same concept as in (1). In the case of (1), we are looking at a physical curl within a vortex. But in the case of (2), we are taking partial derivatives in distance r of a radial field. It’s not the same thing. It’s all explained in this article,

Article Ampère's Circuital Law and Displacement Current

Basically ∇×H = 0 refers to an isolated from any charges magnet including its own material. Thus an isolated magnet is not a magnet and therefore we can say , ∇×H = 0 because H = 0 !! That's a trick and basically BS.

Also forget the subject I don't want further discuss it, became tiresome and I made my point.

In general the current formalism in describing macroscopically magnetism has some holes which can only explained correctly by using quantum magnetism.