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The Larmor Formula for the Generation of Radiation by Accelerated Charges and the Quantization of Space and Time |
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Joseph Larmor discovered a formula for the generation of electromagnetic radiation by an accelerated charge. That formula is
where E is the rate of generation of radiant energy by a charge of q accelerated at a rate of α. The symbol c stands for the speed of light.
Acceleration is the rate of change of velocity and velocity is a vector having direction as well as magnitude. Thus a charged particle traveling in a circular orbit is experiencing centripetal acceleration and thus should, according to Larmor's analysis, be radiating away energy as electromagnetic waves.
The only problem is that this apparently does not occur. On the other hand the Larmor analysis does explain the generation of thermal radiation from matter as a result of the thermal collisions taking place.
If the charge is divided up into m equal pieces each piece would be radiating at rate of (2/3)α²(q/m)²/c³ and the m pieces altogether would be radiating at a rate of
As m increases without bound the value of E goes to zero. Only if the value of α were effectively infinite there would be the possibility of a finite, nonzero value for E.
The implication that the spatial distribution of charge reduces the generation of radiation by accelerated charges to zero unless the acceleration is "infinite" is a welcom one. This eliminates the generation of radiation from particles in atoms experiencing centripetal acceleration but allows it for thermal collisons. However the argument that the distribution of charge reduces the generation of radiation to zero depends intrinsically upon the continuity (infinite divisibility) of space. If space is quantized then the generation of radiation may be reduced to some very small but nonzero value. This means that the accelerations that produce effects do not have to be infinite, just very large, as in the thermal collisions in matter. So all is right with the world and the mathematics of it.
Let ε and δ be the quanta of length and time, respectively. It is presumed that ε is equal to cδ, where c is the speed of light.
Letε³ be the volume of a quantum unit of space; . If a charge q is distributed over a volume V then the number m of equal pieces the charge can be divided into is
But charge is quantized so any charge q is made up of say μ elementary volume units each having a charge of γ. The value of γ is found by determining the volume V occupied by a known charge Q, say that of a proton. Then μ is equal to V/ε³ and γ is Q divided by μ.
The rate of generation of radiation is then
The Planck length is 1.6162×10-35 meters and this may well be the minimum spatial length. Initially, in lieu of any other estimate of the quantum length of space the Planck length will be used. This would make the quantum of volume equal to 4.22168×10-105 m³.
The charge radius of proton is approximately 0.84 fm = 0.84×10-15 m. This makes its volume 2.48×10-45 m³. Hence the number of pieces the charge can be divided into is about 5.88×1059. Thus μ is equal to 5.88×1059.
The charge of a proton is 1.602×10-19 Coulombs. This makes the charge per elementary volume unit 2.72×10-78 Coulombs. This quantity squared is 7.41×10-156. and even 5.88×1059 units of this is equal to only 4.36×10-97. The coefficient of α² in the Larmor formula is this quantity divided by c³. The coefficient of α² is thus equal to 1.61×10-122. Thus α must be very large for there to be any significant effect.
Planck time is the time required for light to traverse a distance of one Planck length. This is suggested as the quantum of time. Its value is 5.4×10-44 seconds.
The centripetal acceleration of an electron in a hydrogen atom is 9.13×1022 m/s² and this squared is 8.357×1045. This multiplied by a coefficient of 1.61×10-122 is 1.35×10-76. This is the rate of generation of energy. The amount generated in a Planck time of 5.4×10-44 seconds is 4.05×10-120 joules, practically nothing. An atom is a quantized system so its energy cannot change by just any amount. It has to change by a quantum of energy by the emission of a photon. Since there is no provision for accumulating energy no photon is ever emitted.
The average velocity of a hydrogen molecule at room temperature (293.15 K) is 1906 m/s. Consider a collision of a hydrogen molecule with the wall of a container that changes its velocity from +1906 m/s to −1906 m/s. This is a change in velocity of 3812 m/s. If it takes place in a Planck time of 5.4×10-44 s then the acceleration is 7.06×1046 m/s². This quantity squared is 4.98×1093 m²/s4. When this quantity is multiplied times1.61×10-122 the result is 8.02×10-29. This is the rate of energy generation. This rate taking place over a time period of 5.4×10-44 produces energy of 4.3×10-72 joules. This is an extremely small value. Nevertheless the computation will continue because a gas is not a quantized system and, in principle, a photon of any energy could be emitted..
The above computed energy is the energy of a photon with a frequency of
and a wavelength of
This is an impossibly small level of energy and impossibly large wavelength of radiation. It means perhaps that the quantum of time used, Planck time, was not appropriate. Actually the logic of the analysis would have each infinitesimal piece of the charge producing a photon of far smaller energy and the mulitude of these photons of infinitesimal energy would together have the energy cited above.
Note that energy generated in a time step δ is
where Δv is the change in the velocity of an H2 molecule upon collision with a wall and q is the quantum of electrical charge.
Since μ=V/ε³ and ε=cδ, where V is the volume of a known charge and c is the velocity of light
So when all dependencies on δ are taken into account the larger is δ the larger is E. For sufficiently high δ, Eδ can be made comparable to the energy of a photon associated with thermal energy. A larger delta; however also corresponds to a larger quantum of length.
The maximum the quantum of length would be the charge diameter of a proton; i.e., 1.68×10-15 m. The quantum of time is then 5.6×10-24 sec. The average velocity of a hydrogen molecule at room temperature (300° K) is about 2000 m/s. The maximum acceleration that average H2 can experience is the complete reversal of direction from a head-on collision with the wall of the container. If that collision takes place in one quantum step of time the acceleration is 7.14×1026 m²/s4. This number squared is 5.1×1053.
The rate of radiant energy generation by the proton is then
The is the rate of energy generation. The energy generated in a quantum time step of 5.6×10-24 seconds is
As will be seen below, this is not nearly enough to account for a typical photon of thermal radiation.
Wien's Displacement Law says that the most frequent wavelength λmax of radiation for a blackbody at absolute temperature (°K) of T is
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
Thus the typical photon in thermal radiation at room temperature has about 1012 times more energy than would be generated by charge accleration through the Larmor formula.
If space and time are quantized at appropriate levels the Larmor formula implies no generation of electromagnetic radiation from the centripetal acceleration in atoms and also for the accelerations involved in thermal collisions in gases. Thus the Larmor formula is irrelevant for centripetal acceleration in atoms and nuclei and also for thermal collisions in matter.
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