Section 34–1 Moving sources

(Retarded field / Retarded position / Retarded time)

 

In this section, Feynman discusses retarded field, retarded position, and retarded time of moving charges. He assumes that the sources are moving at a relatively high speed and a stationary observer is located far away from the sources. This section could also be titled as the “retarded field of relativistic moving sources” or “assumptions of retarded field.”

 

1. Retarded field:

“So, when we are far enough away the only terms we have to worry about are the variations of x and y. Thus we take out the factor R₀ and get (34.3) E_x = (−q/4πϵ₀c²R₀)(d²x′/dt²), E_y = (−q/4πϵ₀c²R₀)(d²y′/dt²)… (Feynman et al., 1963, p. 34–2).”

 

We may include the subscript “ret” in Heaviside-Feynman’s expression for the electric field “E = (q/4pe₀){[Ȓ/kR²]_ret + (¶/c¶t)[Ȓ/kR]_ret - (¶/c²¶t)[v/kR]_ret} (Jackson, 1999, p. 284).” Although this form of the retarded field is relatively simple, it is not used in many textbooks. Interestingly, Feynman manages to explain intuitively the radiation field (the third term) using the second derivative of the unit vector e_r′ that is pointed toward the apparent position of the charge. This term is not as simple as it may seem because there could be modifications in the symbols to show the embedded physical meanings. However, the Lienard-Wiechert form of the electric field can be separated into the near field (velocity field) varying as 1/r², and the far field (radiation field) varying as 1/r.

 

“… where R₀ is the distance, more or less, to q; let us take it as the distance OP to the origin of the coordinates (x, y, z)… (We could put it more mathematically by calling x and y the transverse components of the position vector r of the charge, but this would not add to the clarity.) (Feynman et al., 1963, p. 34–2).”

 

In the Audio Recordings [7 min: 10 sec] of this lecture, Feynman says: “x and y are the transverse components of the position r of the charge, what I mean x and y are the motion of the charge.” In chapter 28, Feynman clarifies: “[o]nly the component a_x, perpendicular to the line of sight, is important.” Instead of writing d²e_r¢/dt² » (1/R₀)d²x¢/dt², Feynman’s symbol for the retarded field may be modified as follows to illustrate three physical meanings: (1) Transverse acceleration a_^: the component a_^ can be in the direction of x, y, or a combination of x and y that is perpendicular to the line of sight of an observer; (2)  r̂_^ vector: the direction of retarded field r̂_^ is perpendicular to the unit vector r̂ that is in the direction of the apparent position of the charge with respect to the observation position, (3) t_ret: we may include the subscript ret for retarded time t_ret and retarded position r_ret because of the time delay effect. In short, we can use d²e_r¢/dt² » (a_^r̂_^)_ret/R₀ to emphasize that there is a time delay in the sidewise acceleration that is perpendicular to the unit vector r̂.

 

Note: d²e_r¢/dt² » d²(x¢/R₀)x̂/dt² » (a_x¢/R₀) r̂_^ = (a_^r̂_^)_ret/R₀

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

 

2. Retarded position:

“What we must now do is to choose a certain value of t and calculate the value of τ from it, and thus find out where x and y are at that τ. These are then the retarded x and y, which we call x′ and y′, whose second derivatives determine the field (Feynman et al., 1963, p. 34–2).”

 

Perhaps Feynman could have explained retarded position in terms of field point r (or observation point) and retarded distance |r–r_ret| using Fig 21-1 in Chapter 21 of Volume II of The Feynman Lectures. More important, a general definition of retarded position based on the formula r_ret = r(t – |r–r_ret|/c) is in principle unsolvable. Simply phrased, retarded position r_ret is defined in terms of retarded time, t – |r–r_ret|/c, but retarded time is defined in terms of retarded position in |r–r_ret|/c. On the other hand, we are unable to determine the exact path of a particle, e.g., an electron, using an experiment in the real world. Thus, it is necessary to determine the retarded position of the particle using approximation methods.

 

“In the first approximation, this delay is R₀/c, a constant (an uninteresting feature), but in the next approximation we must include the effects of the position in the z-direction at the time τ, because if q is a little farther back, there is a little more retardation. This is an effect that we have neglected before, and it is the only change needed in order to make our results valid for all speeds (Feynman et al., 1963, p. 34–2).”

 

It is surprising that Feynman considers the time delay R₀/c to be an uninteresting feature in the first approximation, but it is important from the perspective of gauge invariance. Historically, the introduction of the retarded potentials, which is defined in terms of retarded position and retarded time, as well as gauge transformation should be attributed to L. V. Lorenz (1867) instead of H. A. Lorentz. This gauge condition could be known as Riemann-Lorenz condition because Riemann was the first to suggest the inclusion of the time delay effect due to the finite speed of light (Kragh, 2016). However, Lorenz’s gauge was not agreed by Maxwell because it contradicted Maxwell’s derivation of the electromagnetic wave equation and Coulomb gauge.

 

3. Retarded time:

“What we must now do is to choose a certain value of t and calculate the value of τ from it, and thus find out where x and y are at that τ... Thus τ is determined by t = τ+R₀/c+z(τ)/c (Feynman et al., 1963, p. 34–2).”

 

Feynman mentions that τ is determined by t = τ+R₀/c+z(τ)/c, however, it can only be solved by using approximation methods. We may represent the first approximation by z₀ » R₀ and define R₀ as the constant “average distance” from the position of the moving source q to the observer. In addition, the second approximation may be represented by x′₀ << R₀ (or y′₀ << R₀) by assuming the observer is located far away from the source q. In other words, x′₀ is negligibly small as compared to the distance R₀ because Feynman says that “we shall still assume that the detector is very far from the source (p. 34–1).” To determine the retarded time, we need to identify the nature of accelerated motion, e.g., circular motion that is discussed in this chapter.

 

“We can summarize all the effects that we shall now discuss by remarking that they have to do with the effects of moving sources. We no longer assume that the source is localized, with all its motion being at a relatively low speed near a fixed point (Feynman et al., 1963, p. 34–1).”

 

The concept of retarded time is in the context of a “stationary observer” and a source moving at a relativistic speed. Note that Maxwell’s electromagnetic theory (including magnetic field) is already relativistic before Einstein’s relativity. Furthermore, there is conceptual circularity in any definition of speed of light in terms of time, and vice versa. Specifically, the measurement of one-way speed of light presupposes the knowledge of times (or simultaneity) at two distant locations (Salmon, 1977), but the clocks at the two locations are coordinated using the speed of a light signal. Essentially, the use of c = |r – r_ret|/(t_ret – t) in defining retarded time means that we have also adopted the speed of light as a convention* to synchronize the observer’s present time and the source’s retarded time.

*Poincaré–Einstein synchronization convention: In Einstein’s words, “I maintain my previous definition nevertheless, because in reality it assumes absolutely nothing about light. There is only one demand to be made of the definition of simultaneity, namely, that in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand is indisputable. That light requires the same time to traverse the path A –> M as for the path B –> M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity.” 

Review Questions:

1. Would you explain radiation field by modifying some of Feynman’s symbols or using the term transverse components of the position vector r of the charge?

2. Do you agree with Feynman that the constant delay R₀/c is an uninteresting feature in defining retarded position?

3. How would you define the concept of retarded time?

 

The moral of the lesson: the concept of retarded position is defined in terms of retarded time, but retarded time is defined in terms of retarded position, that is, we need approximation methods to determine retarded position and retarded time.

 

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. Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons, New York.

3. Kragh, H. (2016). Ludvig Lorenz, Electromagnetism, and the Theory of Telephone Currents. arXiv preprint arXiv:1606.00205.

4. Lorenz, L. (1867). XXXVIII. On the identity of the vibrations of light with electrical currents. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34(230), 287-301.

5. Salmon, W. C. (1977). The philosophical significance of the one-way speed of light. Noûs, 253-292.