James Clerk Maxwell established the equations for the dynamics of electrical and magnetic fields. In the 1890's some physicists presented an agument that those equations implied that charged particles undergoing acceleration should emit electromagneticradiation. The electrons and protons in atoms apparently traveling in orbits undergo acceleration in the sense that their momentua vectors are continually changing in their curved trajectories. This enigma has been troubling quantum physicists since that time.

The Copenhagen Interpretation (CI) resolves the enigma by denying that the charged particles of an atom have trajectories. In the CI the particles do not have a physical existence but instead only exist as probability density distributions in which they have a quasi-existence at all of their allowable states simultaneously. Somehow this resolution seems like "throwing the baby out with the bath water and replacing it with a crocodile."

Another resolution of the enigma is to maintain that the particles of an atom are organized in quantized energy states and thus a charged paricle cannot emit electomagnetic radiation unless there is a lower energy state open for it to fall into. In particular a zero energy state is not allowable.

A still another explanation of the non-appearance of electromagnetic radiation from charged particles traveling in circular orbits is that such radiation is reabsorbed by other particles as quickly as it is emitted.

The Larmor Formula

The power P radiated by a charge of magnitude q accelerating at a rate α is given by the Larmor formula

P(in cgs units) = ⅔q²α²/c³

where c is the speed of light.

This result is from J.J. Larmor's 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).

Centripetal Acceleration

If v_t is the tangential velocity of a particle in a trajectory with a radius of curvature of r then the centripetal acceleration α is

α = v_t²/r
and since v_t=ωr
α = ω²r

where ω is the angular rate of rotation in radians per second.

It might be that the analysis is valid but the result is numerically negligible.

The Case of an Electron Rotating with the Earth

For any particle of the Earth ω is equal to

2π/(24·60·60) = 7.2722×10^-5

Thus α for an electron located at the surface of the Earth at the equator is given by

α = (7.2722×10^-5)²(6.371×10⁸ = 3.37 cm/sec²

Hence

P = (2/3) (4.8×10^-10)²(3.37)²/(3×10¹⁰)³
= 6.46×10^-50 ergs per second

This is not very much and even though there are mega-gazillion electrons on the Earth's surface the total does not amount to much. The surface of the Earth is 5.1×10¹⁸ cm². A square centimeter which is 1 mm thick has a volume of 1/10 of a cubic centimeter. such a volume of water would weigh 0.1 g. One molecule of water has a mass of 3×10^-23 g. Each molecule of water contains 10 electrons. Therefore 0.1 g of water contains 3.33×10²³ electrons. Thus within an area of 5.1×10¹⁸ cm² and 1 mm thick there would be 1.7×10⁴² electrons. If all of them were radiating at the level of those at the equator it would amount to 1.1×10^-7 ergs per second. This would be doubled to take into acount the equal number of protons. It is still a near infinitesimal amount. Any radiation from charged particles in the interior of the Earth would likely be absorbed before reaching the surface. Therefore the energy which would be radiated by the charged particles of the Earth according to the Larmor formula due to their centripetal acceleration would be negligible over 10 billion years.

The Case of an Electron in a Hydrogen Atom

In a Bohr atom the angular momentum of an electron is equal to a positive integer multiple of h, Planck's constant divided by 2π. Thus for the lowest energy state

mωr² = h
and hence
ω = h/(mr²)

where m is the mass of the electron and r is the Bohr radius, which is equal to 5.29×10^-9 cm. The mass of an electron is 9.11×10^-28 g and h is equal to 1.0546×10^-27 erg-sec. Therefore

ω = 1.0546×10^-27/((9.11×10^-28)(5.29×10^-9)²)
= 4.137×10¹⁹ radians per second

The centripetal acceleration of the electron is then

α = ω²r = 9.053×10²⁸ cm/s²

The energy radiated per unit time, according to the Larmor formula, is then

P = (2/3)q²α²/c³ = (2/3)(4.8×10^-10)²(9.053×10²⁸)²/(3×10¹⁰)³
= (2/3)7×10⁷ = 4.66×10⁷ ergs per second

The energy of the lowest level electron in a hydrogen atom is about −2.2×10^-11 ergs. So the rate at which that electron would be radiating electromagnetic energy due to its centripetal acceleration would not be negligible.

Nuclear Rotation

Aage Bohr and Ben R. Mottelson in their work on the Collective Model of nuclear structure established the existence of nuclear rotation although their conclusion was stated more cautiously. They said the spectra of nuclei were consistent with nuclear rotation. This was perhaps because Aage Borh's father, Neils Bohr, was associated with the Copenhagen Interpretation, which asserts that particles do not have trajectories. The work on nuclear rotation has continued with others such as Zdzislaw Szymanski. His work Fast Nuclear Rotation covers the fast nuclear rotation that can be induced by bombarding nuclei with a stream of particles. So nuclear rotation seems to be an established empirical fact and yet there is no evidence of the rotating nuclei emitting electromagnetic radiation.

There is also the measurements of electric dipole moments which are consistent with rotating particles.

The Nature of the Enigma

Most physicists presume the analysis upon which the Larmor formula is based is valid and it is the models of atoms which are in error or are incomplete. There is one noted physicist who has questioned the analysis.

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.

Actually the doubt concerning the Larmor formula goes back before Feynman. An old standard text on electricity and magnetism by Stephen S. Attwood, Electric and Magnetic Fields published in 1941 makes this comment concerning the Larmor formula

The problem of computing the radiant energy in terms of the acceleration (or deceleration) is really one of considerable difficulty, but the correct answer can be obtained with elementary mathematical tools if simplifying approximations are used judiciously.

Attwood then cites as a source of a derivation in the fifth edition of Sir James Jeans book, The Mathematical Theory of Electricity and Magnetism, published in 1933. On page 592, the page cited by Attwood, Jeans does in fact give something of a derivation of the Larmor formula from an equation for the force experienced by a moving electron.

Let F be the force vector acting on an electron moving with velocity vector V. The Dutch physicist, Hendrik Lorentz, in his The Theory of Electrons determined that the force acting on charge of magnitude e is

F = −⅔(e²/c³)(d²V/dt²)    (688)

Jeans says

The force given by formula (688) may be regarded as a frictional resistance opposing the motion of the electron through the ether.

Ether! In its time Jeans' work was magnificent, but ether in 1933?!

As to why and how the work done against a frictional force gets transmuted into electromagnetic waves is not explained and is probably unexplainable.

However, to go on with Jeans' derivation, the work done by the electron against any force is:

F·V = F_xV_x + F_yV_y + F_zV_z

Note that if F is perpendicular to V and hence F·V=0, there is no work done.

When the components of the force was replaced by the values found by Lorentz and the results integrated from t=0 to t=τ the result is

−⅔(e²/c³)∫₀^τ (V_x(d²V_x/dx²) +V_y(d²V_y/dy²)+V_z(d²V_z/dz²))dt
which can also be expressed as
−⅔(e²/c³)∫₀^τV·(d²V/dt²)dt

This latter expression may be integrated by parts to yield

−⅔(e²/c³)[V·(dV/dt)]₀^τ + ⅔(e²/c³)∫₀^τ (dV/dt)²dt

The second term is the value given by Larmor's formula. Jeans says that

the first term must represent changes in the energy stored in the ether.

If there is no ether then there is no radiation.

Elsewhere, page 577, Jeans 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.

What About Cyclotron and Synchrotron Radiation

Cyclotrons and Synchrotrons are particle accelerators in which a beam of charged particles is confined to a circular path by powerful magnets. Radiation is observed emanating from the beam. Such radiation is usually expained as being due to the centripetal acceleration of the charged particles. But there is also the interaction of the electric fields of the charged particles with the magnetic field of the device. The radiation from an accelerating/decelerating charge should take place when there are no external electric or magnetic fields involved. That apparently has never been demonstrated. It would also be relevant to demonstrate the existence of radiation from charged particles in circular orbits which are the result of the interaction of the charged particle with an external electric field. This also has apparently not been demonstrated.

Conclusion

Thus the physical evidence is clear that there is no radiation from the charged particles of atoms. Also, the analysis upon which the Larmor formula is based assumed a force due to the resistance of ether to the elecrical field of the charged particle being dragged through it. Thus this analysis is in doubt and must be redone eliminating the assumption of an ether and taking into account quantization.

Consider that in any gas there are charged particles within the atoms and molecules that are undergoing zillion upon zillion elastic collisions. In each of these collisions in there is instantaneous changes in velocity which according to the Larmor formula should result in the emission of sparks of electromagnetic radiation. This is not found to be occurring. Something has to be wrong with the analysis that predicts radiation from accelerating or decelerating charges.

The Explanation of Why Accelerating Charges
DO NOT Emit Electromagnetic Radiation

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. A spatially distributed charge undergoing acceleration or decejeration does not radiate. Period.