﻿ The Liénard-Weichert Formula for the Radiation of Electromagnetic Waves from an Accelerated Charge
San José State University

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Thayer Watkins
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The Liénard-Weichert Formula
Waves from an Accelerated Charge

The proposition that an accelerated/decelerated charge radiates electromagnet waves 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.

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 which was published in 1905.

The Liénard-Weichert formula is best stated in terms of the velocity relative to the speed of light in a vacuum; i.e., β=v/c. Note that β is a vector. The acceleration term is then dβ/dt=α/c. Under Relativity the term

#### γ = 1/(1 − β²)½

is important. Liénard-Weichert formula is then

#### E = (2/3)(q²/c)γ6[(dβ/dt)² − β × (dβ/dt)]

As β→0, γ→1 and the Liénard-Weichert formula reduces to the Larmor formula. On the other hand, as β→1, γ→∞ and the Liénard-Weichert formula is quite different from the Larmor formula.

## Liénard-Weichert formula and a Spatially Distributed Charge

As is the case for the Larmor formula the rate of radiant energy generation in the Liénard-Weichert formula depends upon the square of the charge, q². This means that if the charge q is divided into two charges of ½q, each radiates at a rate one fourth of the rate q radiates and the two together radiate at a rate one half of the rate q radiates. If the charge is divided up into M separate pieces each one radiates at a rate (1/M²) of the rate for q and the M pieces together have a total radiation (1/M) of that rate; i.e.,

#### EM = E/M

As the charge is divided into finer and finer pieces (M→∞) EM→0.

If space is not infinitely divisible then there is limit to M which nevertheless may be very large. Consequently the radiation from an accelerated distributed charge may be insignificantly small.

## The Quantization of Length and Time

Let ε and δ be the quanta of length and time, respectively. It is usually presumed that ε is equal to cδ, where c is the speed of light. If a charge is distributed over a volume V then the number of pieces it can be divided into is V/ε³.

Some believe that the quantum of length is the Planck length of 1.6162×10-35 meters. If that is the case then the charge of a proton is distributed over a volume such that there are about 1060 pieces; i.e., M is 1060. This makes the radiation for an accelerated charge very small unless its acceleration is extremely large.

The centripetal acceleration of an electron in a hydrogen atom is 9.13×1022 m/s² and this squared is 8.357×1045. At non-Relativistic velocities this quantity is multiplied by a coefficient of 1.61×10-122. This product is 1.35×10-76. This however 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 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 Dependence of Energy Radiation on the Change in the Trajectory of the Charge

In the Larmor formula the rate of energy generation depends only on the square of acceleration; i.e., (dβ/dt)². In the In the Liénard-Weichert formula this term might seem to be reduced by an amount equal to β×(dβ/dt). The sign of this term however is not necessarily positive. For a headon collision of a molecule with a wall of the container the direction of the velocity is reversed so the sign of the acceleration term is is opposite to that of the velocity term. Thus the term β×(dβ/dt) is negative and subtracting it from (dβ/dt)² adds to the value of the combined term.

## Molecular Collisions

In molecular collisions the changes in the velocity vectors may be taking place in a quantum of time and thus producing acceleration terms of very large values. Consequently there can be radiation generated of a magnitude sufficient to account for thermal radiation. This does not occur for quanta of length and time equal to the Planck values. It doesn't even occur for quanta of length and time of 3×10-16 meters and 1×10-24 seconds. For this quantum of length the quantum of volume is 2.7×10-47 m³.

The volume of the charge of a proton is 2.48×10-45 m³. Dividing this amount by the quantum of volume gives the number of pieces the charge of a proton is divided into; i.e., 9.185×101;, essentially 102.

The net result is that such a collision produces a photon with energy comparable to that of the average wavelength of thermal radiation at room temperature.