leo1

My respond to Leo Date: 03-12-22 Dear Leo, I think very laudable of your conception of spherical dipole that you took from James Wait's book. However I would be much more grateful if you analyse the derivation of the very equations by James Wait, before you compare the expressions for divergence of flux from a spherical dipole that you give in your message with the expression for divergence of vector in dynamic fields taken from my paper. Then you would find interesting things that are not seen from the final formulas. Please check attentively, and not mechanically but phenomenologically - analyse, what comes from where and why. Then you will understand the cause. Unfortunately, I have not the book by J. Wait, but I can help you with this analysis, basing on similar problem considered by A.M. Kugushev [1]. In his book he also studies the electric and magnetic field of a wire with the flowing current, whose length l << , only in the non-conducting medium, when = 0 . His final expression for E-component of the field is [1, p. 98]
	(1)
where = /c is the wave number, I_m is the amplitude of current in the wire, _a is the dielectric constant, and _a is the permeability. Comparing (1) with your expressions for E_r and E at = 0 , you can make sure, these expressions are fully similar in their structure. So the divergence taken of (1) turns to zero the same as in your case. To understand the cause, let us first pay our attention that the solution (1), just as your solutions, has been yielded by substitution of the value of vector potential into the standard system of equations [1, p. 98]
	(2)
i.e., in a roundabout way. First we find and then . With it we twice take the curl of this expression for vector potential, since we can represent (2) as
	(3)
So we twice pass (1) through the filter cutting off the longitudinal component. The cause of fact that in the resulting expression this component remains is that, additionally to this all, we lose a very important summand in the expression for electric field. Actually, if in accordance with [2, p. 41] before calculating the model we substitute the first expression from (2) to the Faraday law, we will yield
	(4)
or
	(5)
"This last equation shows that vector

remains the potential vector. This means, we can present it as
	(6)
where is the function of co-ordinates and time which we will call the scalar potential" [2, p. 42]. Let us pay our attention, the right-hand part of (6) is not zero but equal to some function of COORDINATES and TIME. So we have no right to equalise arbitrarily the VECTOR grad , as it is done now (see, for example, [3, p. 72]). I have fully substantiated it in [4], but even without rigorous mathematical proof it is clear that we must not neglect this term. Actually, if we think of some EM wave with the frequency of few hundredths of Hertz (such waves do exist in space and astrophysicists consider them in studying interstellar nebulae) and raise the question of presence of scalar potential in this field, then, taking into account tremendous wavelength, we have to admit its presence and its time-alternating pattern. May we neglect this term if it was the part of electric field, and just the part that exceptionally determines the potential pattern of this field? Undoubtedly, we may not. The field components that are really small or have no affection on general pattern of the field fall away themselves and require no forcible calibrations. At the same time, returning to the standard derivation on the basis of system (2), we see now that taking twice the curl of expression for scalar argument has taken from the function of electric field just grad - and this essentially distorted the result. And the second way through the Faraday law does not cancel this vector, as we do not take the curl of function! I will show it below. Let us return to (6) and finish the derivation of this expression. "In distinct from the case of electrostatics, the vector of electric field having the vortex pattern already cannot be represented as the gradient of any potential. It is expressed through the assemblage of scalar and vector potentials as follows:
	(7)
With it the second summand that connects the electric field with that magnetic expresses the Faraday law of electromagnetic induction" [2, p. 42]. So we came to the known expression which connects scalar and vector potentials with electric field in space, and surely, no component of the field was omitted. To determine on this basis the correct relationship of electric field with respect of time and co-ordinates, we have to deviate again from our course and to pay attention to the record of vector potential. According to Kugushev, the expression for vector potential on whose basis he yielded (1) for the strength of electric field is the following [1, p. 98]:
	(8)
We should mark, this expression is not an exact solution of the wave equation for vector potential. This will be very important for the below consideration. Really, as we know, the expression for vector potential is found on the basis of wave equation like [1, p.88]
	(9)
where _a is the complex permeability.
But the solution of this equation is not the function (8) which we consider but the integral function like [1, p. 89] or [3, p. 95]:
.	(10)
However, "for the region r >> l we can think all points of dipole equally distanced from the studied point of field, due to which we can re-record (10) as
	(11)
where e_₃ is the unit vector in direction with the current. As j _m= I _m /S , Sl = V and S is the cross-section of wire, we yield
	(12)
Passing to the spherical co-ordinate system in which
	(13)
we yield (8) which determines the vector potential at the distance r >> l " [1, p. 97].
Thus, (8) determines the vector potential only for far field, when according to [5], all parameters of the wave process have stabilised. So, when the researchers (Kugushev in that number) apply to near field the results obtained on the basis of this vector potential, this is illegal, as they violate the statement of problem. Therefrom we have to doubt (1). If (1) described the process in far field, there has to be no superposition of the field strengths with different attenuation characteristics and different phase shifts in this expression. This additionally corroborates the above statement that (1) has been derived incorrectly. To obtain the correct expression for the electric field strength on the basis of (7), we have to determine the scalar potential . We can find it, basing on the regularity shown in [2, p. 106]:
	(14)
Taking into account that in our problem
	(15)
we can record
	(16)
or, regarding (8),
	(17)
With it we can write down the summands in the right-hand part of (7) for the electric field strength in the following shape:
	(18)
	(19)
Thus,
	(20)
As we can see from this expression, the longitudinal and transverse components of the field have quite stabilised space attenuation with the rate of attenuation corresponding to the considered model. With it, the phase of the transverse component is shifted by 90^o as to that longitudinal. This corroborates again that we correctly stated the calibration of electric field strength to be wrong, as the potential component of the field has its own characteristics of process of space propagation different from the transverse component. Now on the basis of corrected and, the main, complete expression for the electric field strength, let us compare the divergence of strength of this field with the results of my theorem of divergence of vector in dynamic fields [6]. (It is only some strange that, having well designed your notice, you didn't mention, this paper has been published in the international journal Archivum mathematicum, 37 (2001), 3, p. 233-243. Was it correct to refer to a paper, disregarding the publication? How do you think? ;-) ) To check, let us use the standard formula from your notice:
	(21)
Given (20), the first term of the right-hand part will be
	(22)
Accordingly, the second term will be
	(23)
Whereupon
	(24)
As we can see from (24), the divergence of electric field is not zero and the phase of its time variation is shifted by 90^o as to the phase of longitudinal component. This fully corresponds to the phenomenology of process of wave propagation in space. We can easily check that the magnitude of divergence of the electric field strength vector in your problem also is in agreement with my theorem of divergence in dynamic fields. For it, let us take your (1) which you are interpreting quite right, and continue it up to the result:
	(25)
So please enjoy comparing (24) with (25). As you see, there are no real difficulties with my theorem. Only the difficulties with the existing solutions for specific models take place. ;-) Surely, if you check so the derivation of Wait's expressions, you will yield similar result satisfying my theorem. True, additionally to this derivation, you should attentively analyse the additions introduced to the expression for vector potential because of finite conductivity of medium. If you question my theorem anyway, one can disprove it only having found the error in my proof, since theorem is primary as to all solutions of modelling equations. And this is not my wish. This is the objective principle to build any theory. First basic theorems are formed (in this case they are the conservation theorems for dynamic fields), then the mathematical technique is built, then the problems modelling specific processes are solved. Maxwell had built his system so, and we have to develop the EM theory so. I would like to mark especially, realising it or not, you raised a very important and interesting question which needs to be pondered and solved. If you have a wish to dive deeper into this subject, I would be pleased to help you. Happy New year with new knowledge, Sergey Reference: 1. Kugushev, A.M. and Golubeva, N.S. The foundations of radioelectronics. The linear electromagnetic processes. Energia, Moscow, 1969, 880 pp. (Russian). 2. Levitch, V.G. The course of theoretical physics, vol.1. The State publishing of physical and mathematical literature, Moscow, 1962, 695 pp. 3. Landau, L.D. and Lifshiz, E.M. The field theory. In: Theoretical physics, vol. II. Nauka, Moscow, 1973, 504 pp. (Russian). 4. Karavashkin, S.B. On longitudinal EM waves. Chapter 1. Lifting the bans. SELF Transactions, 1 (1994), Eney Ltd., Ukraine, 118 pp. (English). http://angelfire.lycos.com/la3/selftrans/archive/archive.html#long 5. Karavashkin, S. B. and Karavashkina, O.N.Comparison of characteristics of propagation velocities of transversal acoustic waves and transversal EM waves in the near field. SELF Transactions, 3 (2003), 1, 9-17, http://angelfire.lycos.com/la3/selftrans/v3_1/contents3.html#taew 6. S. B. Karavashkin. Transformation of divergence theorem in dynamical fields. Archivum mathematicum, 37 (2001), 3, 233-243. http://angelfire.lycos.com/la3/selftrans/archive/archive.html#div

In spherical coordinate, the electric field at point P created by a dipole Il is [1]:

where
Obviously, the propagation direction of electric field is = . According to your formula [2]:
	(1)
we have

However, it can be easily verifies that

I think the problem is that (1) is wrong. Reference [2] did not consider the contribution of side-surface to the divergence of a vector when deriving (1). Reference [1] James R. Wait, "Introduction to antennas & propagation", Peter Peregrious Ltd, London, United Kingdom, 1986, pp 89, equation (5.30)- (5.31) [2] S.B. Karavashkin. "TRANSFORMATION OF DIVERGENCE THEOREM IN DYNAMICAL FIELDS".

SELF	New Year question from Leo
S.B. Karavashkin