SELF

Potential in an electromagnetic field

S.B. Karavashkin and O.N. Karavashkina

From:Alan Boswell (alan.boswell@nospambaesystems.com)

To:Re: Gradient of potential function of dynamic field

Teams: sci.physics, sci.physics.electromag

Date: 2004-02-23 05:52:20 PST

Sergey

curl grad v = 0 is a general theorem which, because it is general, also

applies to all coordinate systems and also to axisymmetric systems.

Looking at your link, I think your expression for the potential of a

current element is incomplete, and should have a factor (1-j/(kr)) that

you have omitted.

I have emailed the analysis to you separately.

Alan

From the Lorenz condition:

so if we know the vector potential A the scalar potential is:

For a z-directed dipole of length dl,

In spherical coordinates:

Alan Boswell 10-02-2004

My respond to Alan on March 1, 2004:

Dear Alan,

You raised an interesting question. Now, when we have available the conservation laws proven by us, we gradually begin to revise the mathematical tool of Maxwellian field theory. In this connection, there emerge problems which still were not seen either ignored.

You are right, scalar potential can be determined not only in the way I did it in my respond to Leo - through the equality

where V is the potential in your transcription, but also through the Lorenz condition

But I would like to draw your attention, both the first and second formulas are standard and in no way are connected immediately with our conservation theorems. The only, in paper [1, p. 241, formulas (24)-(25)] we showed that we can pass from the proven equation for divergence of vector to a dynamic field

to (2), using (1). But this concerns only the case if   satisfies (3). If you substitute your expression for to (3), you will see it does not fit. The cause is, again, not in our conservation theorems but in mismatched Maxwellian formalism, as does not fit also the standard equation

This is because is not an independent vector but only a component of dynamic gradient, as we showed it in our paper [2]. Again, in that paper we explained only the reasons of the today state. But the problem is in the very Maxwell's formalism irrespectively of our theorems.

Your derivation of scalar potential

I would some simplify by introducing

It gives some other result:

I agree, this is one more expression for the potential in the same problem, with different approaches to the solution in frames of existing Maxwellian formalism.

Here I would like to draw your attention to one more peculiarity. In my respond to Leo I emphasised the fact that the expression for vector potential is not an exact analytical regularity! The expression with which both you and Dirk agree -

- is true only for the far field in supposition that all points of the field source are equally distanced from the observation point. This does not introduce any trouble into calculations. But if we transform this solution and especially differentiate (!), we can have great problems, as for this operation not the absolute value of vector is important but its change in space. With it in expressions for divergence the non-compensated terms can appear, equalities can unfit and whatever else. Just because of it I wrote to Leo that no particular solutions yielded in frames of conventional formalism can doubt no one of our conservation theorems. And nothing unexpected that Leo yielded one result, me - another, you - third, and Dirk - fourth. Maxwellian formalism interpreted by Hertz and Heaviside is actually very mismatched. We can understand the truth, only analysing the very essence of wave processes. But formally, trying to achieve the coincidence by some transformations and substitutions, we cannot grasp this essence. Even if using for some problem some precious techniques, we can yield some particular coincidence, it will "tear" in the most unexpected way when we change the conditions. I understood this all when responded to Leo, as well as understood that the result in my respond coincided merely occasionally. But not this is important. For me, important is that the very process of thorough analysis of Maxwellian formalism gradually picks up the speed, and also important is to understand that we should not reduce our new understanding to old limits, but should understand, what will change in these old frames with our new knowledge. And this new is - to understand that Maxwellian formalism has been based on

1. conservation theorems for stationary fields while studying dynamic fields;

2. not so much accurate mathematical formalisation of effect of EM induction discovered by Faraday (his loop was not closed!);

3. great Maxwellian prediction on magnetoelectric induction that none the less still remained a prediction.

These equations are surrounded by multitude of splinters and mutually mismatched regularities, and you in your post noted a part of them. ;-) Nothing to say that such large section of EM field physics as electromechanical interaction still remained aside of these equations. Despite in some particular cases we can transit from Lorenz to Faraday's law, reverse transition is possible far from always. Just so the Lorenz force remains standing aside, not introduced into Maxwell equations and not dissolved in the dust of history of science.

You will surely agree, such amount of discrepancies within the formalism cannot provide its self-matching - such as, for example, Newton's mechanics provides. And again, not me and not our conservation theorems are guilty. They work well and have only one demerit - they are too young and still have not enough horizontal relations. This is only the matter of time. And now, though colleagues accuse me that I'm everywhere wrong, none the less, the same people, out of their wish, gradually do work these relations. See, why you used the expression for vector potential in the form (8), but not tried to derive it from the magnetic field strength? As I showed in my reply to Leo, this is also a way, people used it and tried to yield something. You could use the vector potential, twice apply the curl operation to it, yield the value of field strength, then integrate this value over the time and to yield again the vector potential whose value has to coincide with the initial value. You can check, instead it you will yield a new chain of different regularities. ;-)

The same on Russian forum on new physics, some Vallav accused me that I did nothing new in dynamic conservation laws. To prove it, he showed a very elegant way to prove the regularity (3). His derivation is worthy to show it here.

Below you can read few pieces of discussion between Gusev and Vallav on Russian forum Yandex.

************

Post # 19083. Re: vector analysis of Karavashkin's dynamic fields

Gusev, February 15, 11:40

In response to # 19034. Re: Dynamic vector analysis by Karavashkin

Vallav, February 13, 2004

…

[Gusev]

>Now I see the problem, how can we fit Div=1/c(n*dF/dt) with Maxwell's equation Div(E)=ro. It appears, in the near of pulsing charge (or dipole) there appears the charge density, while the charges are absent? This is absurd!

[Vallav]

Well, you will never fit. Karavashkin's formula (if n is the unit vector) is true only in two cases:

1. For transverse EM waves, but in this case the result is zero.

2. For plane longitudinal wave. Here the result is not zero, but long ago known.

[Gusev]

>>>Please give us an example of real physical field which is not included into this class.

[Vallav]

>>Any field except three:

1. Wave from an infinite plane source

2. Wave from an infinite line source

Particular case - standing wave F=F0*sin(wt)*cos(kr)

[Gusev]

>We have to think about it.

[Vallav]

Addition. In cases 2 and 3 for longitudinal wave, Karavashkin's formula does not work, as the condition div(F0)=0 is violated.

[Vallav]

>>>>And it is long ago known that such expression for divergence of longitudinal wave exists. All the novelty is that Karavashkin has concealed it.

[Gusev]

>>>I have a doubt that everyone knew long ago. Me and Zinovy did not know. Epros, as far as I could understand, does not know and disagrees.

[Vallav]

>>You are saying it for nothing.

[Gusev]

>>>Please, refer us to an accessible textbook where this has been stated. (And if not difficult, quote a relevant place).

[Vallav]

>>What for? We can easily calculate it by frontal attack:

>>div(F0(r)*F1(wt-kr))=div(F0(r))*F1(wt-kr)+F0(r)*grad(F1(wt-kr))

> > grad(F1(wt-kr))=(dF1(s)/ds)*grad(-kr)=-(dF1(s)/ds)*k

> > =-(k/|k|)*dF1(wt-kr)/dt*(|k|/w)=-(1/c)*dF1/dt*n

[Gusev]

>Excellent! 3 lines substitute 3 pages of Karavashkin's text.

[Vallav]

>>Or a signed and stamped certificate is necessary?

[Gusev]

>What for to misrepresent so? We are speaking not of "a signed and stamped certificate". You publicly argued Karavashkin's priority - it means, you have to corroborate it, referring to the source. I'm repeating, please provide the reference.

[Vallav]

Why have I to corroborate that it came to mind to someone long before Karavashkin - to differentiate the formula for longitudinal plane wave? Seriously?

[Gusev]

>And one more. You are writing, "he concealed it". For nothing are you saying it. He, just as me and others, could be not aware.

[Vallav]

Do you believe, it even did not come to his mind - to calculate the divergence of longitudinal plane wave? It means, div(F)=d(F0*sin(wt-kx))/dx? And to compare with his "result"? With such detailed introduction to the problem?

Uncitation

*************

In the quoted place of that post, Vallav first represents the dynamic vector as a product of some stationary vector into some scalar value ₀    dependent on the distance and time:

Basically, if speaking of the merely mathematical approach to the problem, for linear spaces this is admissible. Further he behaves in a very simple way, breaking down the divergence of product of vector into scalar in standard way:

In order to transit from (10) to our expression (3), he breaks down the gradient of scalar in the right-hand part of (10):

True, this derivation is not without defects of principle, which I just have avoided in my complete proof of theorem of divergence of vector in dynamic fields. In particular, what was his grounds to break down the gradient of scalar function as the product of derivative with respect to direction into the gradient of (- )? If Vallav supposed dynamic gradient described in our paper, it is broken down otherwise. If he supposed spatial gradient in its common sense, it also is broken down otherwise. Second, is not it strange that he substituted the derivative with respect to direction by that to time, excluding the second term of variable (- )?

Whereupon he yielded a trivial juggling grounded on his confidence that this transformation has been already corroborated by our theorem of dynamic gradient. But it is not corroborated by this theorem. And the only thing that can draw our attention in this derivation is Vallav's striving to establish the horizontal relations in the new mathematical formalism. True, Vallav attempts to prove in this way the opposite, but this is already not the issue. We clearly see his progress in understanding, how important the time variable is to describe the vector operations in EM field theory, isn't it? ;-)

And one more point takes here our attention. In the quoted post, Vallav confines the validity of my divergence theorem to the condition that the spatial divergence of spatial component of vector in (9) has to vanish! It means, if this component was not zero, my theorem is not true. I replied him that in (16) of [1], the first integral over the spatial variable has been equalised to zero on the grounds of divergence theorem for stationary fields. If he has this integral non-zero, it naturally will be present in the solution. But if this integral is non-zero, then irrespectively of the additions which we have introduced into the conventional conservation laws, we can simply delete all the vector algebra together with Maxwellian formalism. So, when I read Vallav's limitations, I most of all was surprised that he himself did not understand this simple thought. Though on the other hand, having established this limitation for Maxwellian formalism, he has in this way to lift the limitation for my theorem.

Thus you see, Alan, not everything is so simple and clear in the conventional formalism of field theory. During the century and a half there was cut so much firewood that we should move now very carefully, to cut less new. In this viewpoint, I was pleased seeing your attentive attitude to the transformations. We all are imperfect, make errors and have difficulties, but the main is to select the rational and to try to create the building of new physics, not dogmatising the authorities but understanding, they have done their utmost that was possible at that level of development.

Respectfully,

Sergey

Reference:

1. S. B. Karavashkin. Transformation of divergence theorem in dynamical fields. SELF Transactions, archive,

http://angelfire.lycos.com/la3/selftrans/archive/archive.html#div  

2. S. B. Karavashkin and O.N. Karavashkina. On gradient of potential function of dynamic field. SELF Transactions, 4 (2004), 1, 1-9        http://selftrans.narod.ru/v4_1/grad/grad01/grad01.html