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Nuclear Models and the
Proposition that Accelerated Charges Radiate Electromagnetic Waves |
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Whenever a model of nuclear structure is proposed that involves rotation it is immediately confronted with the proposition that charged particles in orbits are subject to centripetal acceleration and accelerated charges radiate electromagnetic waves. Hence particles lose energy and the nucleus would collapse.
It is presumed that the proposition is valid and therefore the model cannot be valid. Here it is argued that proposition is not valid for real particles. The bases for this assertion are:
where α is acceleration, q is charge and c is the speed of light. It is important to note that this radiation is asserted to exist even in the absence of a magnetic field.
Larmor based this formula on previous work of Hendrik Lorentz. Lorentz presumed that a charged particle dragged its field through the lumineferous aether that pervades the universe. This lumineferous aether was found to not exist. So the original justification of the proposition was invalid.
It is well known that accelerated charges emit electromagnetic radiation.Perhaps it would be more proper to say that it is well known that accelerated charges under some circumstances emit electromagnetic radiation. The issue is whether under all circumstances, macroscopic and microscopic, accelerated charges emit electromagnetic radiation.
The analysis that Jackson bases on the Liénard-Wiechert potentials depends intrinsically on the so-called Dirac delta function. This is a "function" that is everywhere zero except at a point where it is infinite. It is a spike. Mathematically the delta function is not a function; it is called a distribution. But Jackson's derivation not only utilizes the delta function but the derivative of the delta function; a positive spike combined immediately with a negative spike. It is surprising that such esoteric constructions as the delta function and its derivative should be require for the proof of the Larmor formula. No doubt this mathematics can be made rigorous.
Thus while Jackson's derivation makes no reference to aether it utilizes something equally dubious, the derivative of the so-called delta function. It also makes use of the Poynting Theorem. Jackson does not explicitly say that the power radiated from an accelerated charge is electromagnetic waves but that is the impression he leaves. However from the analysis of the proof of the Poynting Theorem it is know that there is not necessarily any electromagnetic waves involved. The electric and magnetic fields may move and in taking their energy with them generate an energy flow. Thus Jackson fails to prove that an accelerating charge radiates electromagnetic waves.
Carl H. Durney and Curtis C. Johnson, Introduction to Modern Electricity and Magnetism<
McGraw-Hill, 1969.
Paul Lorrain and Dale R. Corson, Electromagnetic Fields and Waves, Taiwan, 1970.
R.H. Atkin, Theoretical Electromagnetics, John Wiley, 1962.
Henning F. Harmuth, Radiation of Nonsinusoidal Electromagnetic Waves, Academic Press, 1990.
R. Murugeshan, Electricity and Magnetism, 9th ed., S. Chand, New Delhi, 2007.
Lev Landau and E. Lifshitz, The Classical Theory of Fields, Addison-Wesley, 1957.
It must be added that the new dynamics referred to in Section 620 (Quantum Theory) seems to throw doubt on this formula for the emission of radiation. Many physicists now question whether any emission of radiation is produced by the acceleration of an electron, except under certain special conditions.
The venerable Richard Feynman in his Lectures on Gravitation says "we have inherited a prejudice that an accelerating charge should radiate." He argues that the Larmor formula giving the power radiated by an accelerating charge as proportional to the square of the acceleration "has led us astray." Feynman maintains that a uniformly accelerating charge does not radiate at all. He argues that it is the rate of change of acceleration that results in electromagnetic radiation from charged particles. This assertion does not entirely take care of the problem of the absence of radiation in atoms. For electrons in non-circular orbits there is variation in acceleration.
If the charge is divided up into m equal portions the effect of one portion is (1/m²) of the effect of a charge of q. Altogether the effects of the m portions is 1/m of the radiation of a charge of q. Thus if the charge is distributed over space like a ball or a sphere there would be an "infinite" number of infinitesimal pieces and their total radiation would be zero.
Hence the radiation of electromagnetic waves by accelerated charges would be valid only for point particles and real particles are not point particles. They have charge distributions over space.
Thus no matter how large is the acceleration the radiation generated by a spatially distributed charge is zero. Likewise no matter how large is the charge or how small is the scale so long as the particle is not a point particle the electromagnetic wave generation is zero.
Then
Therefore if lim_{m→∞}a_{1m} is zero then lim_{m→∞}E is also equal to zero.
There is no real enigma concerning the absence of electromagnetic radiation for charged particles in atoms and nuclei. The Larmor formula would apply only to point particles and real particles are not point particles. They have charge distributed over a spatial region which may be small but it is not a point.
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