8.2 Conic Hole in the Momentum of Electron
Here I give the full address of this «confession» of Richard Feynman:
The Feynman Lectures on Physics – FLP
6 Electrodynamics 
Chapter 27 Field Energy and Field Momentum
§4. The ambiguity of the field energy,
In order to use special advantages of discussion of the problem on the base «territories» of this and the following
Chapter 28 Electromagnetic Mass
chapters of FLP fitting it perfectly. The magnificent collection of Feynman's sketches of the well-known «holes» of electromagnetism, as well as the skillfully made sketches of the projects of their elimination or inclusion in the Theory offered by different authors there occurs. Richard Feynman was the biggest expert in the field of the analysis and classification of all these possible and impossible techniques of solution of the most difficult problems of electromagnetism, that tormented the physicists still from the times of Faraday and Maxwell. Feynman himself, though not without John Wheeler's help, during his professional progress was methodically ill with all these various hopes, once and again making sure that search of another still untried way was necessary. Feynman was one of few successful physicists of the quantum epoch, who spent a lot of time and forces on prolonged and exhausting expeditions to these still unconquered tops of classical epoch. He knew for sure from the beginning, and repeatedly made attempts to convince of it others, including numerous readers of FLP, that the deep, primordial reasons which had led to inevitable «glitter and poverty» of QED, associated with the ideas of Maxwell's theory which are not solved by and not directly associated with quantum mechanics. [28,Ch.28]
These chapters of FLP certainly contain the description and definition of concepts necessary for us here, the standard symbolism, as well as other toolkit, including even the prepared, as if for us, formulas and explaining figures. The previous (or parallel) study of the material of these chapters would help the reader to be better prepared for active perusal of this article and, perhaps, would even help to find his or her own secret footpath… Some amount of natural self-irony, reigning in FLP, – in this ideally adapted place for the physical walks to the apple-tree, – can delicately dispose the reader to the necessary tonality and to supply him with the unique hints and the mood for the successful obtaining of desirable truths. But they, staying so far somewhere in bowels of all this inevitable meanwhile heap of words, with all their apparent inaccessibility, are patiently waiting for being conquered by persevering and inquisitive researchers.
The difficulty we speak of is associated with the concepts of electromagnetic momentum and energy, when appilied to the electron or any charged particle. [28,Ch.28,§1]
According to the Pointing theorem and FLP (formulas 27.14 and 27.15):
|2u = EE + BB, S = [EB],||(1)|
Where: u – the density of field energy, i.e. quantity of energy in unit of volume of space; S – a stream of field energy, i.e. quantity of energy, passed in a unit time through a unit surface, which is perpendicular to the stream; E – a vector of an electric field; B – a vector of a magnetic field. Expressions for u and S here look easier in comparison with FLP due to the use of natural system of units of measurement, in which
|ε0 = μ0 = c = 1.||(2)|
In a similar way to how it is done in Ch. 28, §2 – The field momentum of a moving charge, let us deal with the spatial distribution of density of momentum of the electromagnetic field of the moving positive charge q. We shall consider, that speed of the uniform motion v of the charge q is much less than unit and we shall keep in the decomposition on degrees v only values up to the first order of smallness inclusive. To this approximation the electric field E(r) conserves the spherical symmetry of a static case and there appears the magnetic field B(r,θ) as the unique relativistic amendment of the first order. The densities of the electromagnetic energy u and the momentum g on distance r from the charge q accept values:
|2u = E2(r), g = [E(r)B(r,θ)] = gt,||(3)|
|g = E(r)B(r,θ) = 2·uv·sinθ,||(4)|
|B(r,θ) = [vE(r)] = B(r,θ)b, B(r,θ) = vE(r)sinθ,||(5)|
Where: θ – the angle between the direction of speed of movement v of the charge q and the vector of electric field E(r) in the point P; t – the unit vector, orthogonal to vector E(r) and coming from the point P in the direction of the vector g(θ); the vector of magnetic field B and the unit vector b are directed perpendicularly to the plane of figure from P.
It is necessary to note specially, that the multiplier sinθ has appeared in scalar quantity g(θ) of the vector of density of the momentum g(θ) as an efficient, which is included in the scalar quantity B(r,θ)=vE(r)sinθ of the vector of magnetic field B, equal to the product of the scalar quantities of vectors v and E(r) by a sine of the angle between them. The operation of projection of g(θ) on the direction of the vector of speed v has not been made yet! Obtained proportionality of the value g to a sine of the angle θ results in a very strange «circulation» of the vector of density of the momentum g(θ) along the meridians of spherical surfaces of radius r from the posterior pole at θ=π to the anterior one with θ=0, laying on different from the charge sides on an axis of its movement. A couple of such meridians laying in the plane of our figure, were displayed on it as top and bottom halves of the circumference, divided by the horizontal axis of movement and symmetry.
Therein lies the basic part of difficulty. Its other part lies in factor 2 which has appeared as a result of replacement E2(r) by 2u at the last stage of obtaining the expression for g in the formula (4). On the equator of the spherical surface which is displayed on the figure by the top point at θ=π/2 and the bottom one at θ=3π/2, the density of the momentum 2·uv is obtained, that exactly in 2 times exceeds the «desired» uv for all space, regardless of the angle θ. These difficulties of the field theory, associated with the appearance of catastrophic multipliers sinθ and 2 in the expression for field values of density of momentum g=uv·2·sinθ, «multiplied» by the extremely unpleasant meridian circulation of the density of momentum g in all space, are caused by application of Pointing expressions for the density of energy and momentum of an electromagnetic field!
Along all the axis of movement of a charge the local density of momentums g(0) and g(π) is zero. In the conic vicinity of this axis by virtue of the presence of the multiplier sinθ the value g is vanishingly small, despite of final value of the density of field energy u, which is absolutely not dependent on the angle θ. At projecting of the vector of momentum density g on the direction of charge movement one more sinθ will appear as a multiplier, whereupon the value of the contribution of a field momentum from our conic vicinity to an integrated field momentum of a moving charge will become proportional to sin2θ, i.e. even less significant. A disproportionately big contribution to a full electromagnetic momentum of a charge falls on the equatorial area of the field of a moving charge, whereas the contribution from the polar areas of the field is unified proportionally to sin2θ. This very circumstance has formed the decisive basis while choosing the heading of article – Conic Hole in the Momentum of Electron.
|The translation from Russian was made by Masha and Natasha Zazerska|
Last modifications: January 28 2003
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The Literature Quoted:
|28.||Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics, Vol.2 – Addison-Wesley Publishing Co., London, 1964|
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