Section 30–7 The field of a plane of oscillating charges

 (Radiation field / Infinite plane / Finite charges)

 

In this section, the three interesting concepts involved in a problem are radiation field, an infinite plane of in-phase oscillating charges, and a finite amount of charges on the plane. This section is about an idealization of a plane of in-phase oscillators, but the title could be known as “The far field of an infinite plane of oscillating charges.” This section not only aptly ends with the concept of diffraction that is due to an infinite number of sources, but a formula is derived to explain the origin of refractive index for the next chapter.

 

1. Radiation field:

Suppose that we have a plane full of sources, all oscillating together, with their motion in the plane and all having the same amplitude and phase. What is the field at a finite, but very large, distance away from the plane? … (We cannot get very close, of course, because we do not have the right formulas for the field close to the sources.) (Feynman et al., 1963, p. 30–10).”

 

One may define radiation field as an electromagnetic field that provides the energy radiated by accelerated charges and it is inversely proportional to the distance from the charges. However, the term radiation field could be distinguished into two regions, namely near field and far field, that can be expressed by two different mathematical formulas. To be specific, we can define far field using the equation (28.6) in which the field produced by an accelerating charge is moving non-relativistically and it is located at a very large distance r. This problem also includes the far field of an infinite number of sources that are located at infinity. On the other hand, we do not need the right formulas here for the near field that is close to the sources.

Note: It may be confusing to some why Feynman says “… we do not have the right formulas for the field close to the sources,” but we may use near field to explain the maximum limit of refractive index of a medium (Andreoli et al., 2021).

 

“We know that the radiation field is proportional to the acceleration of the charge, which is −ω²x₀e^iωt…… Using this value for the acceleration as seen from P in our formula for the electric field at large distances from a radiating charge, we get (30.11) (Electric field at P from charge at Q) ≈ q/4πϵ₀c²ω²x₀e^iω(t−r/c)/r (Feynman et al., 1963, p. 30–10).”

 

Feynman clarifies that the formula (30.11) is not quite right because it should not only include the acceleration of the charge but its component perpendicular to the line QP. Theoretically, the component perpendicular to QP of the far field due to the charges would not be canceled out because the oscillations are all in the same direction despite having cylindrical symmetry. From a practical point of view, it is the far field that allows electromagnetic waves to propagate far distances with weak signal strength, but it can still be picked up by receivers or antennas. However, some may prefer saying the acceleration field (instead of radiation field or far field) produced by an accelerating charge varies as 1/r. This is different from the velocity field (Coulomb field) of the charge that is independent of a, but its field varies as 1/r².

 

2. Infinite plane:

“When ρ = 0, we have r = z, so the limits of r are z to infinity… Now e^−i∞ is a mysterious quantity. Its real part, for example, is cos (−∞), which, mathematically speaking, is completely indefinite (although we would expect it to be somewhere—or everywhere (?)—between +1 and −1!). (Feynman et al., 1963, p. 30–11).”

 

Feynman says that e^−i∞ is a mysterious quantity and its real part, cos(−∞), is indefinite. On the contrary, mathematicians prefer to write lim e^−iz and include the condition z ® ∞, but they consider e^−i∞ to be illegal. Furthermore, they would investigate whether a function is holomorphic using Cauchy-Riemann equations u_x = v_y and u_y = –v_x. In general, physicists may define terms involving infinity without rigor or quote a statement that is attributed to Einstein: “Two things are infinite, the universe and human stupidity, and I am not yet completely sure about the universe.” More importantly, the problem based on the infinite plane of constant density of charge is unrealistic and an infinite amount of charges on a “circular ring element” (2pr)Dr divided by the distance r that is infinity could be considered as an undefined quantity.

 

In Fig. 30–11 we have drawn the first five pieces of the sum. Each segment of the curve has the length Δr and is placed at the angle Δθ = −ωΔr/c with respect to the preceding piece (Feynman et al., 1963, p. 30–11).

 

The first term of e^iθ = e^−iωz/c has the most contribution to the total field because z is the shortest distance from the plane to P, but ωz/c is relatively small compared to z and thus the angle of the first arrow (θ) should be small. Similarly, the remaining each segment of the curve has the length Δr and it is placed at the angle Δθ = −ωΔr/c with respect to the preceding piece. One should realize that the real part (x component) starts to decrease in Fig. 30–11 when we add the fourth and fifth term, but this may not really correspond to the physical situation. In other words, not only e^−i∞ is a mysterious quantity, but it is remarkable that e^−iωz/c starts to oscillate right at the beginning. This suggests that the idealized model based on e^−iωz/c has a serious limitation and thus needs tweaking.

 

In Surely You’re Joking, Mr. Feynman!, Feynman (1997) challenged Paul Olum to give him an integral that most people could evaluate with only contour integral, but he could use other methods: “One time I boasted, ‘I can do by other methods any integral anybody else needs contour integration to do.’ So Paul puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration! He was always deflating me like that. He was a very smart fellow (pp. 195-196).” Perhaps Feynman could have explained how the physical situations result in another Cornu spiral or contour integral based on Fermat’s principle of extremum path.

 

3. Finite charges:

“In any real situation the plane of charges cannot be infinite in extent, but must sometime stop... If, however, we let the number of charges in the plane gradually taper off at some large distance from the center (or else stop suddenly but in an irregular shape so for larger ρ the entire ring of width dρ no longer contributes), then the coefficient η in the exact integral would decrease toward zero. (Feynman et al., 1963, p. 30–11).”

 

Feynman suggests tweaking the problem by decreasing the number of charges in the plane that are farther from the center, i.e., the coefficient η in the exact integral would decrease toward zero. We can also include the projection of the acceleration on the plane perpendicular to the line PQ, but we would feel unlucky to even approximate the integral involving an additional 1/r. More importantly, some may argue whether it is mathematically legal to let e^−i∞ equal to zero due to various physical situations. For instance, the coefficient η could be specified as a function that varies with 1/r or e^−r because there is no infinite amount of charges in the real world. Alternatively, Feynman could have discussed whether including the projection of the acceleration on the plane perpendicular to the line PQ would definitely improve the formula, however, he claims that the formula (30.18) or (30.19) is correct at any distance z.

 

It is interesting to note that (iωx₀e^iωt) is just equal to the velocity of the charges, so that we can also write the equation for the field as Total field at P=−ηq²ϵ₀c[velocity of charges] at t−z/c (Feynman et al., 1963, p. 30–11).”

 

Feynman concludes the chapter by saying the formula derived is fortunately rather simple and it is valid for distances far from the plane of oscillating charges… If he assumes that η is a function that varies with 1/r or e^−r, then the integration method has to be changed and it would result in a different formula. The so-called simple formula is a result of idealizations and approximations because we have avoided realistic models such as including the projection of the acceleration on the plane perpendicular to the line PQ (cosine factor). In a sense, this problem beautifully shows the daily life of a physicist in cheating (idealizing models) and tweaking (approximating formulas). However, Feynman could end this chapter by explaining the importance of the formula derived because it relates the delay in phase of the field to refractive index.

Note: It may be worth mentioning this section was delivered at the beginning of the lecture of chapter 31 on refractive index.

 

Review Questions:

1. How would you define the radiation field at a distance far from a plane of oscillators (Idealization)?

2. Should we assume the problem to be based on an infinite plane of constant density of oscillating charges (Limitation)?

3. Is it mathematically legal to let e^−i∞ equal to zero such that we can obtain a reasonable approximate answer (Approximation)?

 

The moral of the lesson: We can understand diffraction further by idealizing (cheating) a plane of in-phase oscillating charges, approximating (tweaking) the density of charges on the plane, and confessing the limitation of the model. In a sense, this is an unfortunate problem, but it comes out—fortunately a rather simple formula by “cheating” and “tweaking” (creating luck).

 

References:

1. Andreoli, F., Gullans, M. J., High, A. A., Browaeys, A., & Chang, D. E. (2021). Maximum Refractive Index of an Atomic Medium. Physical Review X, 11, 011026.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

3. 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.