Section 31–5 The energy carried by an electric wave

(Thin-layer approximation / Low-density approximation / Plane-wave approximation)

 

Feynman derives a formula of Poynting’s vector that is based on thin-layer approximation, low-density approximation, and plane-wave approximation.

 

1. Thin-layer approximation:

“All of our calculations have been made for a thin layer of material whose index is not too far from 1, so that E_a would always be much less than E_s (just to make the calculations easier). In keeping with our approximations, we should, therefore, leave out the term Ē_a², because it is much smaller than Ē_sĒ_a (Feynman et al., 1963, p. 31–9).”

 

In this section, Feynman only uses the term low-density approximation to derive a formula of Poynting’s vector, ϵ₀cE². However, he mentions that all of his calculations are made for a thin layer of material whose index is not too far from 1. We should recall that the angle between E_s and E_a is almost a right angle and it can be related to the approximation formula, e^{−iω(n−1)Δz/c} ≈ 1−iω(n−1)Δz/c. To be precise, this method of approximation is based on a “thin gas plate” in which Δz should be relatively thin in the formula. Otherwise, the decrease in E due to absorptions of light in a thicker plate would result in a significant loss of light energy. In short, we may include the term “thin gas plate approximation” or simply “thin-layer approximation.”

 

“For the first term we can write αĒ_s², where α is the as yet unknown constant of proportionality which relates the average value of E² to the energy being carried (Feynman et al., 1963, p. 31–9).”

 

In section 31-2, Feynman states: “If the source S (of Fig. 31–1) is far off to the left, then the field E_s will have the same phase everywhere on the plate, so we can write that in the neighborhood of the plate E_s = E₀e^iω(t−z/c).” He did not determine Ē_s² possibly because it involves calculations due to the decrease in E_s, that is, the absorption of light will end up as thermal energy in a material. Perhaps it is good to emphasize that the electric field E_s is attenuated by a factor of e^−kz in which k is the absorption index of the material. If the loss of energy is considered negligible, Feynman could have determined Ē_s² instead of writing it as αĒ_s². Note that in the next chapter, he changes the notation Ē_s² to < E_s² >.

 

2. Low-density approximation:

“One way of checking that our calculations are consistent is to see that we always keep terms which are proportional to NΔz, the area density of atoms in the material, but we leave out terms which are proportional to (NΔz)² or any higher power of NΔz. Ours is what should be called a “low-density approximation” (Feynman et al., 1963, p. 31–10).”

According to Feynman, leaving out terms that are proportional to (NΔz)² or any higher power of NΔz should be called a “low-density approximation.” However, it involves a low-density approximation due to smaller N and thin layer approximation due to shorter Δz. One may recall that η (= NΔz) is the number of charges per unit area where N is the number of atoms per unit volume of the thin plate. In Volume II, Feynman mentions: “…we had to restrict ourselves to finding the index only for materials of low density, like gases…. so we studied only the rarefied gas, where such effects are not important (Feynman et al., 1964).” It may be more accurate to state “rarefied gases approximation,” but physicists have developed various models of refractive index based on different atomic densities.

 

In section 32–3 Waves in a dielectric of Volume II, Feynman adds: “… if N is small enough so that n is close to one (as it is for a gas), then Eq. (32.27) says that n² is one plus a small number: n² = 1 + ϵ. We can then write n = √(1 + ϵ) ≈ 1 + ϵ/2, and the two expressions are equivalent (Feynman et al., 1964).” It should be worth mentioning that the refractive index of air is 1.0003. On the other hand, the term rarefied gas means that the pressure of the gas is much less than atmospheric pressure. Therefore, ϵ/2 is lesser than 0.0003, i.e., n is expected to be very close to 1, but small N may be known as a “rarefied gas approximation” or “low-pressure gas approximation.”

 

In Volume II, Feynman elaborates that “S = ϵ₀c²E×B, is called ‘Poynting’s vector,’ after its discoverer. It tells us the rate at which the field energy moves around in space. …… Believe it or not, we have already derived this result in Section 31–5 of Vol. I, when we were studying light (Feynman et al., 1964).” Perhaps Feynman could have said that he has derived Poynting’s vector ϵ₀cE² using the definition of energy density of electromagnetic wave and Poynting’s theorem. (It is also known as Umov-Poynting vector because Nikolay Umov, a Russian physicist, first proposed the concept of energy flux in a continuous medium in 1874.) In a sense, it is paradoxical that the formula ϵ₀cE² can be exactly proved using Poynting’s theorem, but Feynman’s proof is based on “low-density approximation.”

 

3. Plane wave approximation:

“We now go back to Eq. (30.19), which tells us that for large z E_a = NΔzq_ev(ret by z/c)/2ϵ₀c (31.26) (Feynman et al., 1963, p. 31–10).”

 

Eq. (30.19), as mentioned by Feynman, is based on the assumption of an infinite plane of constant density of oscillating charges. More importantly, the phrase “for large z” means that a spherical wave may appear like a plane wave because the curvature becomes negligible when it is significantly farther away from the source. Some may prefer the term monochromatic plane wave because this is a single-frequency spherical wave from a faraway source that would look like a plane wave after traveling a very large distance. For example, the wavefronts of light from stars are effectively parallel. It may be known as a plane wave approximation as the direction of electric fields is perpendicular to the motion of the plane wave.

 

In section 31.2, Feynman explains that in the neighborhood of the plate E_s = E₀e^iω(t−z/c) if the source S (Fig. 31–1) is far off to the left (Feynman et al., 1963, p. 31–4). That is, the source S is idealized to be infinitely far and thus the electric fields of oscillating electrons are in the same direction and have the same amplitude E₀. Specifically, the field E_s is weaker in the z-direction and it is sometimes called an inhomogeneous plane wave (Jackson, 1999). In short, the homogeneous plane wave becomes an inhomogeneous plane wave when it enters a medium. Some may prefer to describe this method as “inhomogeneous plane-wave approximation.”

 

Review Questions:

1. How would describe Feynman’s method of calculations that is based on a thin layer of material whose refractive index is close to 1?

2. Is the energy carried by an electric wave calculated by only using low density approximation?

3. Would you describe a wave of the electric field after traveling a large z as a “monochromatic plane wave” (Griffiths, 2005) or “inhomogeneous plane wave” (Jackson, 1999)?

 

The moral of the lesson: A simplified formula of Poynting’s vector ϵ₀cE² for a chromatic (or inhomogeneous) plane wave can be derived using thin-layer approximation, low-density approximation, and plane-wave approximation.

 

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

3. Griffiths, D. J. (2005). Introduction to Electrodynamics (3rd ed.). New Jersey: Pearson Education.

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