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An Analysis of the Argument that Accelerating or Decelerating Charged Particles Emit Electromagnetic Radiation |
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The original analyses by Hendrik Lorentz and Joseph Larmor on the proposition that accelerating or decelerating charges emit electromagnetic radiation are flawed by a dependence upon there being luminiferous ether prevading space. Supposedly the effect arises because of the forces resulting from a charged particle dragging its electric field through the ether. The notion of an ether has been discredited but the proposition continues to be accepted.
This material is an examination of a modern derivation of the proposition. The modern derivation utilized is the one 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.
Jackson then procedes to develop a formalism which is to be the basis of further analysis. Joseph Larmor published the first analysis, in 1895, conerning radiation by an accelerated charge based upon the derivation by Hendrik Lorentz of the force experienced by a particle due its charge field being dragged through the ether. It was not appreciated at the time but Larmor's analysis presumed that the velocity of the charge was small relative to the speed of light in a vacuum. 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. Liénard and Wiechert based their analyses on the vector and scalar potentials of the electric field of a particle.
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. Such esoteric constructions as the delta function and its derivative may not be rigorous mathematics.
Thus while Jackson's derivation makes no reference to ether it utilizes something equally dubious, the derivative of the so-called delta function.
After having derived a formula Jackson then goes to section entitleld
Total Power Radiated by an Accelerated Charge--
Larmor's Formula and its Relativistic Generalization
The initial text of that section is
If a charge is accelerated but is observed in a reference frame where its velocity is small compared to that of light, then in that coordinate frame the accelerated field reduces to
E_{a} = (e/c)[(n×(n×(∂β/∂t))/R]_{ret}
In this formula e is the particle charge, β is the velocity of the particle relative to the speed of light, v/c. The vector n is the unit normal to the potential surface.
Jackson then goes on to say
The instaneous energy flux is given by the Poynting vector
S= (c/4π)(E×B) = (c/4π)|E_{a}|²n
Jackson does not explicitly say that the power radiated from an accelerated charge is electromagnetic waves but that is the impresssion he leaves. However from the analysis of the proof of the Poynting Theorem it is known 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.
It is relevant that a major difference between classical field theory and quantum field theory is that in classical field theory there are particles with singularities in their potential functions whereas in quantum field theory no such singularities exist.
A charged point particle would have infinite energy; therefore not one could be created. There is not enough energy in the entire universe to create even one.
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. But the Larmor formula and its derivation are there, so something must be wrong.
There is no provision in the conventional analysis for netting out of positive and negative charges. The result has the emission of radiation depending upon the square of the charge. Therefore the electron and the proton rotating about their center of mass in a hydrogen atom should both be emitting radiation. But also the electrons and protons in a hydrogen molecule subject to the elastic collisions in a gas should be emitting radiation due to their deceleration in one direction and acceleration off in another direction.
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.
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.
Here is another case of skepticism. In an online discussion concerning a seeming paradox concerning radiation from accelerated charged particles Marcel Urban of the French National Centre for Scientific Research says
It may be that we are wrong about the fact that an accelerated charge should emit photons. It may be that an accelerated charge does not radiate...period! In that case there is no more to worry about.
The Larmor Effect is derived for point particles and depends upon the square of the charge. If a charge of Q is distributed over M points then the M points radiate an amount proportional to (Q/M)² for a total of Q²/M. If M goes to infinity, as it would for a spatially distributed charge no matter how small the region of distribution, then the radiation goes to zero.
Thus there is and has been a good deal of skepticism among physicists concerning radiation by accelerated charged particles. The derivation of the proposition is not without questions concerning rigor and there is no evidence that the proposition holds at the quantum level.
The proposition holds only for a charged point particle and no such particle exists in the real world. A spatially distributed charge does not radiate. Period.
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