Section 32–2 The rate of radiation of energy

(Radiation field / Radiation pattern / Larmor’s formula)

 

This section is a discussion of radiation field, radiation pattern, and Larmor’s formula. Although the section is titled “The rate of radiation of energy”, another possible title is “A derivation of Larmor’s formula.”

 

1. Radiation field:

“So we know the electric field at any point, and we therefore know the square of the electric field and thus the energy ϵ₀cE² leaving through a unit area per second… Using our expression (29.1) for the electric field… (Feynman et al., 1963, p. 32–2).”

 

Feynman cites the equation (29.1) for the radiation field, but it is worthwhile to recall his explanation of equation (28.6) E_x(t) = −qa_x(t−r/c)/4πϵ₀c²r: “Only the component a_x, perpendicular to the line of sight, is important… Evidently, if the charge is moving in and out straight at us, the unit vector in that direction does not wiggle at all, and it has no acceleration. So it is only the sidewise motion which is important, only the acceleration that we see projected on the screen (Feynman et al., 1963).” However, it may not be clear why the radiation field E_x(t) is inversely proportional to r. Intuitively, one may guess a moving charge at different “retarded time” would behave like an electric current. In other words, the 1/r dependence could be related to the Gauss’ law or the additive effect of the moving charge.

 

“The quantity ϵ₀c appears quite often in expressions involving radiowave propagation. Its reciprocal is called the impedance of a vacuum, and it is an easy number to remember: it has the value 1/ϵ₀c = 377 ohms (Feynman et al., 1963, p. 32–2).”

 

Feynman says that the reciprocal of the quantity ϵ₀c is called the impedance of a vacuum. In a sense, the word vacuum is a misnomer because it is not really empty and contains quantum fluctuations, virtual particles, or zero-point energy from the perspective of quantum theory. On the other hand, the vacuum of interstellar space is permeated by the microwave background radiation or dark energy. Thus, one may explain the decay of the radiation field over a long distance is partly contributed by the so-called vacuum or permittivity of free space. It should not be surprising that the Hubble’s constant may be finetuned depending on the presence of galactic matter such as stars or blackholes that can affect the energy density of free space.

 

2. Radiation pattern:

“Using our expression (29.1) for the electric field, we find that S = q²a′²sin²θ/16π²ϵ₀r²c³ (32.2) (Feynman et al., 1963, p. 32–2).”

 

An antenna’s radiation pattern can be defined as a graphical representation of the radiation intensity of the antenna as a mathematical function of angular coordinates. Perhaps Feynman could have discussed further the radiation pattern of an oscillating charge that depends on S = q²a′²sin² θ/16π²ϵ₀r²c³. The angular structure of the radiation looks like a donut that has no hole at the center, i.e., no radiation along the direction of the acceleration (See Fig. 1). Intuitively, one may guess there is no far field at a point where sin² θ = 0, that is, along the axis of the electron’s oscillation because the charge appears to be stationary. On the other hand, there is a maximum far field at any point that is perpendicular to the direction of the electron’s oscillation where sin² θ = 1 because the movement of the electron appears as an alternating current.

 

Fig. 1: Griffiths (1999, p. 448).

“Secondly, the flux (32.2) was calculated using the retarded acceleration; that is, the acceleration at the time at which the energy now passing through the sphere was radiated. We might like to say that this energy was in fact liberated at this earlier time. This is not exactly true; it is only an approximate idea. The exact time when the energy is liberated cannot be defined precisely (Feynman et al., 1963, p. 32–3).”

 

According to Feynman, the exact time when the energy is liberated cannot be defined precisely. In general, physicists may use the word define which means measure accurately. In this case, it is difficult to measure the very weak far field because it is inversely proportional to a very long distance. Moreover, the expected measurement of liberated energy is further reduced for an exact time, that is, a very short period of time. In Volume I, Feynman suggests: “…whether or not one can define absolute velocity is the same as the problem of whether or not one can detect in an experiment, without looking outside, whether he is moving. In other words, whether or not a thing is measurable is not something to be decided a priori by thought alone, but something that can be decided only by experiment (Feynman et al., 1963, p. 16-2).” However, the total energy liberated could be inferred by considering a large number of atoms.

 

3. Larmor's formula:

“Therefore P = q²ω⁴x₀²/12πϵ₀c³ (32.6)… (Feynman et al., 1963, p. 32–3).”

 

Feynman mentions that the formulas we are now discussing are relatively advanced and they are very famous. The equation (32.6) derived is known as the Larmor’s formula which is the total power radiated spherically by a nonrelativistic oscillating charge. In Larmor’s (1897) words: “It would thus appear that when the steady orbital motions in a molecule are so constituted that the vector sum of the accelerations of all its ions or electrons is constantly null, there will be no radiation, or very little, from it, and therefore this steady motion will be permanent.” Although Larmor’s derivation of the formula is based on steady orbital motions, it can be viewed as a one-dimensional oscillation from the perspective of an observer at a point on the same plane of the orbital motion.

 

“In fact, the older books also used a system of units different from our present mks system. However, all these complications can be straightened out in the final formulas dealing with electrons by the following rule: The quantity q_e²/4πϵ₀, where q_e is the electronic charge (in coulombs), has, historically, been written as e². (Feynman et al., 1963, p. 32–3).”

 

In the Audio Recordings* [14 min: 10 sec] of this lecture, Feynman mentions the use of the “CGS system” (i.e., Centimetre-Gram-Second system of units) but this is omitted in the edited Feynman Lectures, however, it is really minor.) In essence, we can express the Larmor’s formula using the MKS units as P = q²a²/6pe₀c³, but in the older formula or using the CGS system as P = ⅔(e²a²/c³). However, in Larmor’s (1897) words, “…the rate of loss of energy by radiation is ⅔e²c^-1× (acceleration)² (p. 512)”, that is, it is different from the Larmor’s formula in the CGS system. More important, the Larmor’s formula can be used to explain why the sky is blue (see section 32.5). In addition, Feynman did elaborate that the Larmor’s formula is applicable to electrons moving in a synchrotron..., but this is not included in the edited Feynman Lectures**. (In section 34–3, he discusses synchrotron radiation using another formula.)

 

*The Feynman Lectures Audio Collection: https://www.feynmanlectures.caltech.edu/flptapes.html

 

**In the Audio Recordings [14 min: 20 sec] of this lecture, Feynman explains the Larmor’s formula as follows: “So this is the rate at which an oscillating charge will liberate energy, it is very much higher for a given oscillation the higher the frequency. It also shows that an accelerating charge radiates, as a good example, electrons going around a synchrotron radiate because they are accelerating as they are going in a curve and they lose a lot of energy on account of that. This formula is not valid for such electrons because they are moving at relativistic speed, but the formula I have written is only valid at non-relativistic speed, however, it is not hard to find the right formula for relativistic speed.” (I have edited a few words.)

 

Review Questions:

1. How would you relate the radiation field to the impedance of the vacuum?

2. How would you explain the radiation pattern of an oscillating electron?

3. How would you explain the Larmor’s formula?

 

The moral of the lesson: the Larmor’s formula can be calculated by summing the radiation intensity spherically, whereas the radiation intensity is related to the radiation field and the impedance of the vacuum that is 377 ohms.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Griffiths, D. J. (1999). Introduction to Electrodynamics, 3rd Ed. New Jersey: Pearson Education.

3. Larmor, J. (1897). LXIII. On the theory of the magnetic influence on spectra; and on the radiation from moving ions. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 44(271), 503-512.