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The Original System of Maxwell's Equations: Is it Consistent and Self-sufficient?
With gracious consent from Frederick David Tombe, I am reproducing here a private exchange I had with him a few months ago.
I hope the owner of this site will not take it as an abuse of the site for the subject and the content of the exchange may be of public interest.
I hope the owner of this site will not take it as an abuse of the site for the subject and the content of the exchange may be of public interest.
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Hi David, I hope you are doing well.
While trying to make heads and tails of the 1864 original Maxwell's equations, I have come across an interesting conundrum which I would like to share with you.
Consider with Oleg D. Jefimenko (Example 15-1.1 on p. 498 in second edition of his book “Electricity and Magnetism”) the following problem. At time t = 0, there is a charge distribution ρ0(x, y, z) in an infinite medium with dielectric and conductivity constants ε and σ. Determine how this charge distribution changes in time.
We have a full-fledged electrodynamic problem here, so one would expect that its solution will require the entire set of all 8 original Maxwell’s equations (6 vector and 2 scalar equations) to be taken into account. But, interestingly enough, the exact mathematical solution of this problem can be obtained considering only the last 4 equations, denoted as (E), (F), (G) and (H) in Maxwell’s 1864 paper:
(E) D = ε0εE
(F) J = σE
(G) divD = ρ
(H) divJ + ∂ρ/∂t = 0
Indeed, substituting Ohm’s law (F) into the continuity equation (H), we have
σdivE + ∂ρ/∂t = 0.
Applying operator div to the equation (E) (the equation of electric elasticity as Maxwell calls it), using then Gauss’ law (G) (the equation of free electricity in Maxwell’s parlance), and substituting finally divE = ρ/(ε0ε) into this equation we get
σρ/(ε0ε) + ∂ρ/∂t = 0.
Integrating and taking into account that ρ = ρ0(x, y, z) at t = 0, we obtain the solution
ρ(x, y, z, t) = ρ0(x, y, z)exp[–σt/(ε0ε)].
O. D. Jefimenko leaves it at that believing, obviously, that he got the exact mathematical solution of the problem, and there is nothing more to be said about it.
However, from the physical point of view, this formal mathematical solution is patently wrong – it flies in the face of common sense. Indeed, let ρ0(x, y, z) be zero everywhere except the interior of a small sphere where we have some distribution of free electrons. It is plain common sense that this blob of negative charges, when left to itself, will immediately start spreading outwards since the electrons repel each other. But according to the above formal solution, this blob of free charges will ultimately fade away completely at each point inside the sphere ... without the charges ever leaving the boundaries of the sphere – absolutely nonsensical physical result!
What is your take on this, David?
Best regards, Arthur.
Hi Arthur,
Jefimenko has been combining equations that were not designed to be used in the same context.
One equation is about dielectric polarization on the application of an external electric field. Another is a kind of terminal velocity analogy to the air resistance problem. That's Ohm's law. The terminal speed of a current on the application of an external electric field. One equation is Gauss's law which is the mathematical description of an irrotational radial force field. Finally one is the hydrodynamical equation of continuity.
Then he creates some hypothetical context of a charge distribution in the midst of a dielectric. It should expand under the electrostatic repulsive force, and so the charge density should decay over time. But it's got nothing to do with the first two equations on the list, or indeed even the third. Coulomb's law and the equation of continuity alone should be sufficient.
Jefimenko in general had a tendency to feed elasticity matters relating to dielectrics into Gauss's law without any justification. It's because he had the cart before the horse. Gauss's law is a component of the overall apparatus of electromagnetism. But electromagnetism as a whole is not crammed into Gauss's law.
In fact, Jefimenko’s folly is very similar to the folly of the 19th century Dane Ludwig Lorenz who came along in the mid-1860s and trampled all over Maxwell's writings, introducing the nonsense of the Lorenz gauge. Maxwell was furious, stating that Lorenz had totally missed the point.
Best Regards
David
Hi David,
I was thinking along similar lines while trying to figure out what is going on here. The equations (E) and (F)
(E) D = ε0εE
(F) J = σE
shall not be used at the same time for the same part of electromagnetic medium. Using these equations simultaneously is as unlawful as using Hook's law of elasticity and the law of plasticity for the same part of a solid body because every part of the solid body behaves like elastic medium for loads under certain level and like a plastic medium with loads over the limit of elasticity, i.e. one or the other law is to be used for each point but not both.
But now we have another problem. Maxwell pointed out specifically that there are 20 equations and 20 unknowns in his framework, and he probably considered that as an argument, if not as a proof (it couldn't be a proof anyway), that his system is self-contained, self-sufficient, and self-consistent. Heaviside’s reduction of the original Maxwell equations to a truncated framework has always bothered me precisely because of the mismatch of the counts of equations and unknowns (http://magneticuniverse.com/discussion/comment/895/#Comment_895), but, at least, I was under the impression that the original Maxwell system was free of this kind of inconsistency. Now it appears that even the Maxwell system has the same mismatch of equations and unknowns: 17 equations vs. 20 unknowns, i.e. Maxwell system seems underdetermined.
Where do we get the missing 3 equations?
A.B.
Hi Arthur,
When Maxwell talked about 20 equations, he was splitting six of his original eight into x, y, and z, components.
Of the original eight, the electromotive force equation does not appear in the modern 4, but rather appears alongside them as an extra, under the name of "The Lorentz Force". Likewise, Ohm's law, and the equation of continuity are listed separately in modern textbooks.
That leaves 5. Two of the five are then combined into one. The law of total currents is combined with Ampere's Circuital law.
The four modern equations in vector notation are fine in their own right. The problem is that the physical basis for the displacement current has been removed.
Best Regards
David
Hi David,
Yes, Maxwell's 20 equations are in components (6x3 + 2 = 20). That much is certain. But Maxwell can have 20 equations only if he counts equations (E) and (F)
(E) D = ε0εE
(F) J = σE
as equations which are valid simultaneously for each and every point in the electromagnetic medium, as Jefimenko has assumed. But, as we saw earlier, that leads to an absurd conclusion. So, in fact, Maxwell does not have 20 equations – he has only 17. My argument is that Maxwell is missing 3 equations for his system to be determinate. What are those 3 missing equations?
Try solving Jefimenko's problem for the evolution of a blob of free negative charges concentrated at time t = 0 inside a small sphere (distributed not necessarily asymmetrically) and you will find that it cannot be solved satisfactorily in Maxwell's framework of 17 equations vs. 20 unknowns for the simple reason that free negative charges have a certain inertia, and this fact is not reflected in Maxwell's system at all. What’s more, Maxwell was not (and could not be) sure that free charges have mass to begin with because the electron was discovered only after Maxwell's death.
Here is a very interesting quote from Maxwell 1864 that everyone seems has missed (pp. 500-501):
(100) The equations of the electromagnetic field, deduced from purely experimental evidence, show that transversal vibrations only can be propagated. If we were to go beyond our experimental knowledge and to assign a definite density to a substance which we should call the electric fluid, then we might have normal vibrations propagated with a velocity depending on this density. We have, however, no evidence as of the density of electricity, as we do not even know whether to consider vitreous electricity as a substance or as the absence of a substance.
But today we know that the bearer of resinous (negative) electricity – the electron – is a real substance. We know also the mass of the electron. Therefore, we may speak of mass density of an electron beam, for example. So, it seems reasonable to take Maxwell's hint on the theoretical possibility of "normal vibrations" (i.e. longitudinal waves of compression-rarefaction similar to sound waves in the ordinary material medium like air, water, etc.) quite seriously today. The really important question now is: How could such vibrations be enacted in practice and the speed of their propagation measured?
One obvious suggestion would be playing around and experimenting with an electron beam.
A.B.
Arthur,
I looked at it all again. Jefimenko doesn't know the elasticity factor for his exponential decay equation but he has assumed it by combining permittivity and resistivity, the latter coming from Ohm's law.
As regards the permittivity, that would be relevant for Coulomb's law, but then, as you say, we would need to know the mass of the charged particles before we could fully know the elasticity factor. On the other hand, if we treat the charge as a continuous compressed electric fluid, we cannot then use the permittivity factor, because the latter relates to the background dielectric and not to the charge itself.
Best Regards
David
David, it seems that you are confusing the terms, unless it was a slip of a tongue that is. Elasticity is nothing but the reverse of permittivity, i.e. elasticity = 1/(ε0ε) by its very definition.
What does not enter into Jefimenko's solution though is the permeability μ0μ (and not the permittivity or its reverse – the elasticity). And it is probably the permeability that should be associated with matters having to do with inertia in classical electrodynamics. It is probably no accident that in electric circuits the role of inertia is played by inductance, L, as evidenced from the equation e.m.f = Ldi/dt. Inductance is a close cousin of permeability. In fact, inductance of any circuit (a coil, for example) is proportional to the permeability of the medium, the coefficient of proportionality being a function of the geometry of the circuit alone by a well-known integral formula.
But in Jefimenko's problem there are no circuits – there is a 3-dimentional conductor instead. And it is not clear at all how to take into account the factor of inertia in such cases. We simply do not have a formula for inductance of a 3-dimensional conducting body.
For one, I am not even sure that the notion of inductance for a 3-dimentional conducting body can be meaningfully defined to begin with.
One frequently hears of Foucault currents, but have you ever seen anyone trying to figure out what paths exactly those Foucault currents take in a 3-dimentional conducting body? I have not.
The 3 equations that are missing in the Maxwell's framework in order to clinch thing together may have something to do with all this.
A.B.
Hi Arthur,
I was aware of the linkage between inductance and permeability, and the analogy with inertial mass. In fact, I often laid out on the blackboard for students the analogy table between an LCR circuit and an oscillating mechanical spring, as in,
L is to m
C is to 1/k (k is the spring constant)
R is to air resistance
And I meant what I said regarding elasticity. And I was fully aware that elasticity in the case of a dielectric is 1/permittivity.
I said that Jefimenko doesn't know what the elasticity is in the case of a blob of liquid charge, and that if instead he prefers to use discrete particles, then he has left out the inertial mass.
I don't think permeability is relevant in this scenario, even though it is closely connected with density. But it's more about the density of magnetic field lines, rather than density in general.
Best Regards
David
Thanks for the clarification, David, now I see your point.
Now, let us consider a simplified version of Jefimenko's problem. At time t = 0 there is a distribution ρ0 of free negative charges (electrons) in an infinite medium of dielectric constant ε0 (pure ether). The Ohm’s law is irrelevant now, so we can exclude it from the list of 8 original equations of Maxwell. Determine how this charge distribution changes in time.
We may simplify the problem farther by assuming that the charges are distributed uniformly over a spherical shell: ρ = ρ0 = const for R1 < r < R2, and ρ = 0 everywhere else. Thus, we get a simple one-dimensional problem.
However, this problem – no matter how simple it might look from the mathematical point of view – cannot be solved in the framework of 7 equations of Maxwell (1 scalar and 6 vector equations) which we are left with after discarding Ohm’s law. To make the problem physically meaningful, and mathematically determinate, we need to add one more vector equation to the framework. And that equation, I suggest, is the second law of Newton for the gas of electrons that possess certain inertia – an inertia that, as you have correctly pointed out, has nothing to do with the permeability or any other physical property of the ether.
A.B.
Hi Arthur,
Yes, it's inertial mass that is missing from the pack, as regards solving that particular problem.
Maxwell of course was never working on that particular kind of problem. The closest that Maxwell came to the concept of mass, was density. The concept of "density of free electricity" features in his writings. There was nothing lacking in Maxwell's work for the purpose with which he set out. It's easy to adjust the original equations into a more dynamical friendly format simply by replacing Maxwell's electromotive force by the modern electric field.
I don't know whether or not you ever read my article on "Maxwell's Original Equations". It's here,
http://gsjournal.net/Science-Journals/Essays-Mechanics/Electrodynamics/Download/3889
Displacement current is the most subtle aspect in the whole topic, and even Maxwell himself didn't quite pin it down, although he got closer than most, and left us with plenty of clues. The answer was staring him in the face, but he never quite saw that the vector A, that we call the magnetic vector potential, and which he referred to as Faraday's electrotonic state, is in fact the displacement current.
Best Regards
David
David, yes, I have your article, and I still have to study it closely.
A.B.
Speaking of Einstein's special and general theories of relativity, check out my discussion thread Common Sense vs Einsteinianity, in particular what Tesla had to say about them.
It is worth mentioning that he attempts to explain the displacement current notion in more physical terms, rather than vector equations of abstracted physical reference.
The issue of not being contained or complete is a mathematical procedural one., but it highlights how procedure can alert one to missing phenomenal modelling .
The aether having a density, and indeed having a substantial existence and/or nature begs the question: what is its dynamic? When the question is considered the most fundamental answer is : rotational .
While the coordinate directions are assumed by simple physicists/ mathematicians to be Rectilineal , they are better represented as curvilineal. This is why Maxwell endorsed Hamiltons Quaternion algebra . However , Lord Kelvin put academic pressure on him to rescind , and so the quaternionic solution to his equations were buried.
What you call an indeterminacy is most likely a Quaternion solution. . Such a solution would imply rotationl dynamic., and in particular trochoidal dynamic.
The rotation problem and Hamilton's discovery of quaternions I
The rotation problem and Hamilton's discovery of quaternions II
The rotation problem and Hamilton's discovery of quaternions III
The rotation problem and Hamilton's discovery of quaternions IV
As a side note: Norman Wildberger made an interesting assertion elsewhere that "physics is also largely a quadratic subject" which seems corroborated nicely by the beautiful Bernoulli’s argument for V^2 in Vis-Viva controversy: See Appendix to the following article by S. Gavin and S. P. Karrer
http://rhig.physics.wayne.edu/~sean/Sean/Course_information_files/VisViva.pdf