﻿ Nuclear Models and the Proposition that Accelerated Charges Radiate Electromagnetic Waves
San José State University

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Nuclear Models and the Proposition
Electromagnetic Waves

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:

• The proposition was originally formulated by the Irish physicist Joseph Larmor. His formula for the rate E at which energy is radiated is

E = (2/3)α²q²/c³

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.

• Larmor's analysis in 1895 was followed by a more comprehensive analysis by Alfred-Marie Liénard in 1898 and, independently, by Emil Wiechert in 1900. Their analyses were compatible with Einstein's Theory of Special Relativity published in 1905.
• Others developed alternate derivations of the Larmor formula. One such modern derivation is given in the venerable text, Classical Electrodynamics by John David Jackson. Jackson opens his chapter on Radiation by Moving Charges with a very strong statement
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.

• In contrast to Jackson's wholehearted endorsement of the proposition the following sample of six texts on electricity and magnetism have no reference to accelerated charges or the Larmor formula in their indices:

R.H. Atkin, Theoretical Electromagnetics, John Wiley, 1962.

Carl H. Durney and Curtis C. Johnson, Introduction to Modern Electricity and Magnetism< McGraw-Hill, 1969.

Lev Landau and E. Lifshitz, The Classical Theory of Fields, Addison-Wesley, 1957/ Paul Lorrain and Dale R. Corson, Electromagnetic Fields and Waves, Taiwan, 1970.

R. Murugeshan, Electricity and Magnetism, 9th ed., S. Chand, New Delhi, 2007.

Walter E. Rogers, Electric Fields McGraw-Hill, 1954.

Francis Westin Sears, Electricity and Magnetism, 3rd printing, Addison Wesley, 1970.
• But the fact remains that nuclei rotate and yet there is no radiation emitted and they do not collapse. See for example Fast Nuclear Rotation by Zdzislaw Szymanski. Aage Bohr and Ben R. Mottelson in their Collective Model given in their two volume study Nuclear Structure conclude cautiously that the empirical evidence is consistent with nuclear rotation. For more on the enigma of the absence of electromagnetic radiation from charged particles in atoms see Enigma.
• Sir James Jeans in his book, The Mathematical Theory of Electricity and Magnetism, published in 1933, says
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. But when Richard Feynman, whom many say was the finest mind in physics in the 20th century, second only to Albert Einstein, says there is something wrong with the Larmor formula then that should be taken seriously.

• Here is a general explanation of why there is no radiation from charged particles in atoms and nuclei. The quadratic dependence on charge in the Larmor formula presents a real problem for a spatially distributed charge. Suppose the charge of q is considered as two charges of q/2. Then radiation from one is one fourth of the radiation from a charge of q and the two together give radiation of one half that of a charge of q.

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.

E = limitm→∞ (2/3)α²q²/m = 0

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.

• It is a relatively simple task to generalize the division of the charge into nonequal portions so long as the maximum size portion goes to zero as the division becomes finer and finer. Let ajm for j=1,…,m be the proportions of the charge for the m-th division with a1m being the largest portion and amm being the smallest portion.

Then

E ≤ (2/3)α²(a1mq)²/a1m = (2/3)α²(a1mq²)

Therefore if limm→∞a1m is zero then limm→∞E is also equal to zero.

• There is one more objection to the proposition that accelerated charges radiate electromagnetic waves. In the Theory of General Relativity there is what is called The Equivalence Principle. This is the assertion that there should be no difference between a body at rest experiencing a uniform gravitational field and a body experiencing uniform acceleration. There is no reason to expect a charged particle resting in a uniform gravitational field to be emitting radiation. Therefore, according to the Equivalence Principle there should be no radiation from a charged particle experiencing uniform acceleration.
• The existence of cyclotron radiation and betatron radiation does not contradict what is given above because magnetic fields are involved in both the cyclotron and the betatron.

Conclusion

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.