My question is caused by one curious fact of the electrodynamics.
It is commonly accepted opinion that the electromagnetic fields can at least formally be decomposed onto the rotational and irrotational components. For the electric field of the classical charge being in arbitrary motion is seems to be obvious, namely, the radiated fields are transverse and therefore the rotational, the bound fields are longitudinal and therefore the irrotational.
But:
1. There is no example of such a decomposition of the E field of the classical charge if its law of motion is known;
2. The analogue of Helmholtz's proof of the theorem isn't extended to the electrodynamics. At least, no proof of this theorem is given in the textbooks.
Regarding p. 1, it is easy to give the counter-example, when the decomposition of the E field is impossible (the attached file).
But what is a reason of the absence of the analogue of Helmholtz's proof of the theorem isn't extended to the electrodynamics?
I suggest that because this theorem belongs to the mathematics, some mathematical obstacles should exist to prevent the extention of the theorem to the electrodynamics. What obstacles?
Dear Vladimir,
Your question is not an easy one. I agree that it is particularly difficult to find a good treatment of the electromagnetic fields produced by accelerated charges. I am sure it has got enough attention in the past (like the work by Lienart and Wiechert). For a number of cases practical solutions have been found, but they have as far as I know mostly the character of a workaround. I am afraid we have to live with that conclusion for now.
Your question itself is highly interesting, but I cannot answer it. I can only give some comment. One remark is very formal. Helmholtz''s theorem requires the em fields involved to be smooth and differentiable. For a charged point particle accelerated by an external force, this is correct apart from the place holding the particle itself. There the field definitely has a singularity. In the example you gave yourself, you found the discrepancy between Schott's solution and the result of your eq. (1) along the z-axis, so it includes the singularity itself. That is the only mathematical issue I can see for now.
I have to trust you with respect to the derivation in your example, but are you really sure that E-rot vanishes because of symmetry arguments?. I suppose you exert a force on the particle in the z-direction, hence there will be motion in the z-direction, but your symmetry is around z=0. As soon as the particle starts to move, its new position is no longer symmetric with respect to z=0. It may look subtle, but radiation forces are no first order phenomena.
Kind regards, C.M.J. Wijers
Dear Christianus,
Thank you for your comment.
I agree that motion of the charge in the z axis is the exceptional case. But I made some calculations of the decomposition of the E field created by the elementary dipole (so called Hertz's solution). For this E field curl E =/= 0 despite the field cannot be decomposed in accordance to the Helmholtz theorem too.
I don't understand a reason yet why it is so.
And I suggest that the presence of the singularity is essential because the proof of the Helmholtz theorem is based on the relation
\nabla^2 (1/R) = 4 pi delta(R)
where delta is the singular function.
With the best wishes,
Vladimir
Dear Vladimir,
There are a number of papers that deal with the Helmholtz Theorem for electrodynamics as opposed to electrostatics or magnetostatics. Probably the best one I've seen is by A.J.P. Davis from 2006 (pdf attached). The classical Helmholtz Theorem describes the behaviour of a field at a given instant in time, and does not describe how the field evolves with time since there are no time derivatives in the original theorem, only spatial ones (div and curl). It is therefore necessary to extend the Helmholtz Theorem in the manner that Davis describes in his paper to account for time-dependence. This results in a more generalised form of the theorem that involves a time derivative as well as div and curl spatial derivatives. One recovers the original form of the Helmholtz Theorem from this generalised form when the time dependence is removed. In addition, one must ensure that causality is satisfied which requires the use of retarded quantities in the generalised formulation. This is essential to accurately represent the fields generated by moving, and in particular, accelerated charges. The works of the late Oleg Jefimenko are also closely related to this topic.
Regards,
Ray Simpkin
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