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The Larmor Proposition that Accelerated Point Charges
Radiate Electromagnetic Waves is Irrelevant
in a World of Spatially Distributed Charges

Background

The Larmor Formula

In 1897 the Irish physicist Joseph J. Larmor published an article, "On a dynamical theory of the electric and luminiferous medium", Philosophical Transactions of the Royal Society, vol. 190, (1897) pp. 205–300. (Third and last in a series of papers with the same name). In that article Larmor derived a formula for the power R radiated by a charge of magnitude q accelerating at a rate α

R = ⅔q²α²/c³

where c is the speed of light. This version of the Larmor formula is for the centimeter-gram-second system of units.

Larmor's analysis in 1897 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. For velocities small compared to the speed of light the formula of Liénard and Wiechert reduce to the Larmor formula.

Larmor's analyses was based upon a prior analysis by Hendrik Lorentz. But Lorentz' analysis was 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 was discredited but the proposition continued to be accepted. For convenience of reference the notion that an accelerated charge emits electromagnetic radiation will sometimes be referred to as the Larmor Proposition.

A modern derivation of the proposition that accelerated charges radiate electromagnetic 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 widely believed that accelerated charges under some circumstances emit electromagnetic radiation. The issue is whether under all circumstances, macroscopic and microscopic, and in the absence of magnetic fields 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 that is construed as the derivative of a step function. 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. This is in effect the second derivative of a step function. It is surprising that such esoteric constructions as the delta function and its derivative are needed for the proof. Undoubtedly this part of Jackson's derivation can made into rigorous mathematics.

But Jackson then goes on to say

The instaneous energy flux is given by the Poynting vector

S= (c/4π)(E×B)

This means that the power radiated … is …

Jackson does not explicitly say that the power radiated from an accelerated charge is electromagnetic waves but that is what he is assuming. However from the analysis of the proof of the Poynting Theorem it is known that there is not necessarily any electromagnetic waves involved for the Poynting vector. 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. So a standard source in electromagnetic analysis has a failed proof. It may be that a rigorous proof can be constructed but it would be for a point charge.

The Skepticism Among Physicists Concerning
the Proposition that Accelerated Charges Radiate

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 … 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.

Additional Reasons to Doubt that Accelerated
Charges Emit Electromagnetic Radiation

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. Something must be wrong.

The proposition of accelerated charges radiating electromagnetic waves seems to have a nusiance value; i.e., confronting any atomic or nuclear model involving curved trajectories with the argument that they should collapse. It is always presumed that the proposition concerning accelerated charges is correct and therefore the model must be wrong.

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. So the proposition fails empirically. So something must be wrong with the proposition.

In contrast to Jackson's wholehearted endorsement of the proposition the following sample of eight 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

Stephen S. Attwood, Electric and Magnetic Fields, 2nd ed., John Wiley & Sons, 1941.

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

Henning F. Harmuth, Radiation of Nonsinusoidal Electromagnetic Waves, Academic Press, 1990.

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.

Matthew N.O. Sadiku, Elements of Electromagnetics, Saunders College Publishing, 1989.

Francis Westin Sears, Electricity and Magnetism, 3rd printing, Addison Wesley, 1970
.

The Effect of Continuously Distributed
Charges as Opposed to Point Charges

Suppose a charge of q is considered as two ½q charges. This would mean that the energy radiation is 2(q²/4)= q²/2.

If the charge of q is considered to be M charges of q/M each then the energy radiated is (1/M) of energy radiation by a charge of q. As M increases without bound the energy radiated goes to zero.

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 space is infinitely divisible and 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 if space is infinitely divisble then 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.

If Space is not Infinitely Divisible

The ancient Greeks speculated that there is a limit to how small matter can be divided which they call an a (not) tom (cut), atom. This was a couple of millennia years later proven to be true. Electric charge was also found to be quantized. It is therefore very plausible that there is an atom or quantum of length and also of time. Let ε be the quantum of length and δ the quantum of time. (It is widely presumed that ε=cδ, where c is the speed of light.)

The quantum of spatial volume would then be ε³. If V is the volume of a quantum of charge then the number of pieces M that charge would be divided into is given by

M = V/ε³

The charge of a proton is distributed over a sphere of radius 0.84 fermi (8.4×10-16 meter). The volume occupied by the charge is then 2.483×10-45 m³. Some believe that the quantum of length is the Planck length of 1.6162×10-35 meters. This would make the quantum of volume equal to 4.222×10-105. If that is the case then

M = 2.483×10-45/ 4.222×10-105 = 4.222×10-105
= 0.588×1060

This would mean that a quantum of charge is divided up into about 1060 pieces; i.e., M is 1060. This makes the radiation for an accelerated charge very small unless its acceleration is extremely large.

For the Bohr model of a hydrogen atom the tangential velocity of an electron in the lowest orbit is 2.19×106 m/s, which is less than 1 percent of the speed of light.

The centripetal acceleration of an electron in a hydrogen atom is 9.13×1022 m/s² and this squared is 8.357×1045 m²/s4 or 8.357×1049 cm²/s4 The charge of an electron is 1.602×10-19 coulombs. The rate of radiant energy generation due to the centripetal acceleration of the electron according to the Larmor formula (in cgs units) modified to take into account the spatial distribution of charge into 0.588×1060 pieces is

R = [(2/3)(8.357×1045)(1.602×10-19)²/(3×1010)³]/0.588×60
= 9.0×10-84 ergs/sec = 9.0×10-91 joules/sec

This however is the rate of generation of energy. The amount generated in a Planck time of 5.4×10-44 seconds is 4.86×10-134 joules, practically speaking nothing. An atom is a quantized system so its energy cannot change by just any amount. It has to change by the emission of a photon equal in energy to a quantum of energy of the system. Since there is no provision for accumulating energy no photon is ever emitted.

The Results for Other Values of
the Quanta of Length and Time

The largest the quantum of length could be is the charge diameter of a proton, which is 1.68×10-15 meters. This would make M equal to 1 and the quantum of time equal to 5.6×10-24 sec. This would make the rate of radiant energy generation equal to 5.3×10-31 joules/sec. The energy generated in one quantum of time would then be 3.0×10-54 joules, again nothing, practically speaking.

The Situation for Acceleration Spikes
Due to Collisions of Molecules

The previous analysis above demonstrated the irrelevance of the Larmor proposition for the centripetal acceleration of electrons in atoms. There remains the possibility that a high level of deceleration or acceleration due to the collisions of molecules might overcome the other factors in the Larmor equation to give numerically significant results. Such is not the case as will be demonstrated later. If the quantum of time is small the magnitude of the acceleration resulting from a reversal of velocity is large. But if the quantum of time is small then so is the quantum of length and hence the quantum of volume. The number of pieces M that a charge is divided into is large. The effect of M more than offsets the increase in the magnitude of acceleration due to a smaller time step. But before the numerical details are presented it is necessary to compute the energy of photons involved in thermal radiation.

The Wavelength of Thermal Radiation
from H2 at Room Temperature

Wien's Displacement Law says that the most frequent wavelength λmax of radiation for a blackbody at absolute temperature (°K) of T is

λmax =b/T

where b is equal to 0.0029 mK. Thus for a temperature of 300° K the most frequent wavelength of thermal radiation is about 10-5 meters. The energy of a photon of this radiation is

photon energy = hc/λmax
= (6.626×10-34)(3×108)/(10-5)
= 2.0×10-21 joules

Determination of the Radiant Energy
Generated by the Collision of A Charge

As present previously, the Lamor formula in cgs units is

R = (2/3)α²q²/c³

where α is acceleration in cm/s², q is charge and c is the speed of light in cm/s. This is the rate of energy generation.

The acceleration generated in a collision is the change in velocity divided by the time involved in the collision.

The charge diameter of a proton is approximately 1.68 fermi = 1.68×10-13 cm. If this is taken to be the quantum of length then the quantum of time equal to 5.6×10-24 seconds.

The average velocity of an H2 molecule at room temperature (300deg; K) is about 2×105 cm/s. In a head-on collision with a container wall the direction is reversed. If this takes place in one quantum of time the magnitude of the maximum acceleration is

αmax = (4×105/5.6×10-24)
=7.14×1028 cm/sec²

. This quantity squared is 5.1×1057 cm²/sec4.

The charge of the proton is 1.6×10-19 coulombs. This squared is 2.56×10-38. The value of R is the rate of energy generation; the quantity of energy generated in one quantum of time δ is then

Ecol = δR = (5.6×10-24)[(2/3)(5.1×1057)(2.56×10-38)/(2.7×1030)
Ecol = 1.8×10-35 joules

The energy of a photon generated at room temperature by a head-on collision of an H2 molecule with a container wall of 6.74×10-33 joules is a far cry from the energy of 2.0×10-21 joules which is the average photon energy of the thermal radiation from H2 at room temperature (300° K). The acceleration/deceleration photon energy is about a hundred trillion times smaller than the thermal radiation photon energy at room temperature.

Thus if the transformation of thermal kinetic energy into thermal radiation due to collisions is involved in thermal radiation it is quantitatively insignificant. However there are other reasons to believe the accelerated charge effect is even less significant than the above computation indicates. The spatial distribution of charge results in the charge being divided up into a myriad of small pieces having near infinitesimal total effect.

Cyclotron and Synchroton Radiation

A beam of charged particles in a cyclotron or a synchroton is held in a circular orbit by a powerful magnetic field. Such beams do radiate electromagnetic waves in the visible range. But radiation from a charge traveling in a magnetic field is a different matter from the Larmor effect which is supposed take place even in the absence of a magnetic field. Therefore cyclotron and synchroton radiation do not constitute empirical verification of the Larmor effect.

Conclusion

The original analyses of Larmor, Liénard and Weichert upon which the proposition that an accelerated point charge radiates electromagnetic waves were invalid because they were based upon the existence of luminiferous ether. The modern derivation of J.D. Jackson is also invalid because it relies upon a misinterpretation of the Poynting vector of the Poynting Theorem.

But even if a valid derivation exists it would be for a point charge. When a charge is distributed over space the effect is divided by the number of pieces the charge can be partitioned into. If space is infinitely divisible the effect is reduced to zero.

If quanta of length and time exist the effect may be vanishingly small quantitatively. Therefore the proposition that an accelerated charge radiates electromagnetic waves may be irrelevant for a physical world of spatially distributed charges.

Computation were carried out using two values for the quanta of length and time. One set of computations uses the Planck length and time and the other uses the charge diameter of a proton as the quantum of length with the quantum of time being the time required for light to travel that distance. Those computations of the energy generated from the Larmor effect in one quantum of time due to the centripetal acceleration of an electron in a hydrogen atom show the effect to be vanishingly small.

Another set of computations were made for the accelerations and decelerations due to molecular collisions of H2 gas at room temperature. The energy of the photons generated by such collisions would be vanishingly small compared to the average photon energy due to thermal radiation from that gas.

Thus it is not surprising that there is no evidence for the existence of the Larmor effect at the atomic level. At the macro level the existence of cyclotron and synchroton radiation is often taken as evidence of the Larmor effect. But cyclotron and synchroton radiation involve charged particles traveling through a magnetic field. That is the only way circular orbits can be achieved for charged particles. Thus those phenomena are not evidence for a pure Larmor effect.

Therefore the proposition that an accelerated point charge should radiate electromagnetic waves is irrelevant for the spatially distributed charges of the physical world.

If the charge is not a point particle but instead a charge distributed over space then it might appropriately be considered an infinite number of infinitesimal charges and hence there would be no electromagnetic waves radiated.

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