Section 41–2 Thermal equilibrium of radiation

Oscillator (idealization) / Damping (approximation) / Blackbody radiation

 

In this section, Feynman discusses a charged oscillator (idealization), its damping force (approximation), and blackbody radiation that are related to the derivation of Rayleigh-Jean’s law of radiation. Essentially, the derivation is based on the assumption of the charged oscillator in a box and the approximation method involving damping force, which results in the incorrect law for blackbody radiation. Thus, the section could be titled “A derivation of Rayleigh-Jean’s law” instead of “Thermal equilibrium of radiation.” Although Feynman’s approach differs substantially from Rayleigh’s (1900) paper titled Remarks upon the law of complete radiation, both arrive at the same result.

 

1. Oscillator (Idealization)

“Suppose we have a charged oscillator like those we were talking about when we were discussing light, let us say an electron oscillating up and down in an atom. If it oscillates up and down, it radiates light. Now suppose that this oscillator is in a very thin gas of other atoms, and that from time to time the atoms collide with it. Then in equilibrium, after a long time, this oscillator will pick up energy such that its kinetic energy of oscillation is ½kT, and since it is a harmonic oscillator, its entire energy will become kT… (Feynman et al., 1963, p. 41-3).”

 

Feynman derives Rayleigh-Jeans Law of radiation by using three classical assumptions: (1) Idealized Atom: An atom is modeled as a negatively charged electron bound to a fixed positive nucleus by a massless, perfect spring, forming a classical harmonic oscillator. (2) Ideal cavity: The radiation is confined within a cavity whose perfectly reflecting, conducting walls allow the system to reach thermodynamic equilibrium. (3) Classical Equipartition Theorem: Each classical harmonic oscillator has two quadratic degrees of freedom (one for the electric field and one for the magnetic field energy). These assumptions result in the same formula as Rayleigh-Jeans Law, which diverges at high frequencies—ultraviolet catastrophe—thereby signaling a failure of classical physics.

 

Historically, Rayleigh assumed a cubical cavity filled with electromagnetic radiation in thermal equilibrium with its walls. He treated the radiation as a system of standing electromagnetic wave and counted the number of allowed vibration modes within a small frequency interval (dn). Based on his method, he concluded that the number of modes per unit volume was proportional to the square of the frequency, N(n) µ n². Next, he applied the classical Equipartition Theorem, which states that every mode must possess an average energy of kT. Multiplying the density of modes by the average energy per mode (U(n) = N(n) ´ kT), he obtained the spectral energy density, U(n) µ n²kT. This formula, also known as the Rayleigh-Jeans Law, only agrees with experimental data at low frequencies. Compared with Feynman’s derivation, which models matter–radiation interaction using a damped charged oscillator, Rayleigh’s classical wave-based method is conceptually simpler and mathematically more direct.

 

2. Damping (Approximation)

“Thus we first calculate the energy that is radiated by the oscillator per second, if the oscillator has a certain energy. (We borrow from Chapter 32 on radiation resistance a number of equations without going back over their derivation.) The energy radiated per radian divided by the energy of the oscillator is called 1/Q (Eq. 32.8): 1/Q=(dW/dt)/ω₀W. Using the quantity γ, the damping constant, this can also be written as 1/Q=γ/ω₀, where ω₀ is the natural frequency of the oscillator—if gamma is very small, Q is very large (Feynman et al., 1963, p. 41-4).”

 

It is important to recognize that Feynman’s derivation relies on two substitutions that are conceptually different in purpose and interpretation: ω ® ω₀  and ω₀ ® ω.

 

ω®ω₀ (Replace ω by ω₀)

Feynman’s derivation employs two main approximations to simplify the mathematics and ensure the final integral tractable, followed by a third, fundamentally fatal assumption that leads to the Rayleigh–Jeans law.

A.    The Narrow Peak (Resonance) Approximation

This approximation simplifies the analysis of the scattering cross-section by using the high quality factor (Q) of the charged oscillator.

Assumption: The oscillator scatters incoming radiation very strongly only when the incident frequency (ω) is extremely close to its natural resonant frequency (ω₀). Since Q is very large (~10⁸), the scattering cross-section (s_s) is sharply peaked in a very narrow interval around ω₀.

Simplification: Near resonance, the denominator in s_s can be approximated as

ω²-ω₀² = (ω+ω₀)(ω-ω₀) » 2ω₀(ω-ω₀)

Purpose: This simplifies the scattering cross-section s_s into the form of a resonance curve, allowing the integral to be evaluated analytically.

 

B.     The Smooth Spectrum (Constant Intensity) Approximation

This approximation simplifies the integration of the energy balance equation.

Assumption: The spectral intensity of black-body radiation, I(ω)—the unknown quantity being solved for—varies very slowly across the extremely narrow frequency range where the scattering cross-section s_s is significant.

Simplification: Because the resonance peak is so narrow (due to the high Q factor), the function I(ω) can be treated as a constant value, I(ω₀), and pulled outside the integral sign:

òI(ω) s_s(ω) dω » I(ω₀) ò s_s(ω) dω

Purpose: This allows I(ω₀) to be factored out of the integral and expressed it in terms of known constants (k, T, g) and a standard integral of the form ò dx/(x²+a²) = p/a.

C.    The Fatal Approximation: Classical Equipartition

Beyond these approximations lies the crucial classical assumption that ultimately invalidates the final formula.

Assumption: Every standing-wave mode (degree of freedom) of the electromagnetic field possesses an average energy kT, as stated by the classical equipartition theorem.

Why it fails:

The fundamental problem in this approximation is to assume the charge oscillator can possess a continuous range of energies, rather than being restricted to discrete energy levels.

 

Summary

The first two approximations are physically reasonable relying on resonance and constancy arguments. The breakdown arises from the third approximation—classical equipartition—which incorrectly links the possible vibrational (electromagnetic) modes to thermodynamics. This fundamental flaw necessitated Planck’s introduction of energy quantization and marked the collapse of classical physics in the description of blackbody radiation.

“Then we substitute the formula (41.6) for gamma (do not worry about writing ω₀; since it is true of any ω₀, we may just call it ω) and the formula for I(ω) then comes out I(ω) = ω²kT/π²c² (Feynman et al., 1963, p. 41-5).”

 

ω₀®ω (Why "We May Just Call It ω")

After solving the integral, Feynman obtains I(ω₀) = ω₀²kT/π²c², he then says, "since it is true of any ω₀, we may just call it ω." In a sense, this means that the oscillator with natural frequency ω₀ served as a “probe” to measure the energy density at its resonant frequency.

(In principle, one can construct oscillators with any arbitrary resonant frequency, the formula must hold for all ω.) More important, Feynman later emphasizes an consistency check on the derivation itself: “The charge of the oscillator, the mass of the oscillator, all properties specific to the oscillator, cancel out, because once we have reached equilibrium with one oscillator, we must be at equilibrium with any other oscillator of a different mass...” The complete disappearance of the oscillator’s specific properties in the intensity formula indicates the universality of the derived classical black-body radiation law. However, the functional form of Feynman’s result is identical to that derived from Rayleigh’s method—a clear indication that both share the same flaw in the classical limit. This is another reason why Feynman “did not worry” about his derivation; he could have worked backward from the known formula of Rayleigh.

 

Note: The apparent discrepancy between the Rayleigh-Jeans law and Feynman's formula arises from the different, though physically equivalent, quantities using distinct mathematical conventions. In addition, the Rayleigh-Jeans law itself appears in several equivalent forms, depending on whether it is expressed in terms of frequency or angular frequency, intensity or energy density, and per unit volume or per unit solid angle. Once these definitional and normalization differences are accounted for, Feynman’s formula is seen to be consistent with the Rayleigh-Jeans law. 

 

3. Blackbody radiation

“It is called the blackbody radiation. Black, because the hole in the furnace that we look at is black when the temperature is zero (Feynman et al., 1963, p. 41-6).”

 

Historically, the concept of the blackbody was introduced by Gustav R. Kirchhoff in 1860, who defined it as an object that absorbs all incident electromagnetic radiation (a perfect absorber). Kirchhoff suggested:

“If a volume is enclosed by bodies of the same temperature and rays cannot penetrate those bodies, then each bundle of rays inside this volume has the same quality and intensity it would have had if it had come from a completely black body of the same temperature, and is therefore independent of the constitution and the shape of these bodies and is determined by the temperature alone (Hoffmann, 2009, p. 36-37).”

This statement summarizes Kirchhoff's Law of thermal radiation: the spectral distribution of the radiation within a closed cavity at thermal equilibrium is universal and depends only on the absolute temperature. However, the so-called blackbody becomes highly luminous when heated, glowing red, yellow, or white depending on its temperature. Astrophysical objects like the Sun are considered near-ideal blackbody radiators because their characteristic continuous emission spectrum is defined solely by their temperature, a property unrelated to their visual color. Strictly speaking, no perfect blackbody exists in nature, and the term blackbody is, therefore, something of a misnomer. Because the phenomenon fundamentally concerns the thermal radiation spectrum rather than the visual appearance of an object, many physicists today prefer the term thermal radiation or cavity radiation.

 

Note: In Griffiths’ (2017) words: “… This is called blackbody radiation. It’s a real misnomer – the Sun is, in this sense, a ‘blackbody’! It should be called ‘thermal radiation,’ because it is due to the random thermal motion of the charged particles (especially electrons) of which the object is composed.”

 

“Of course we know this is false. When we open the furnace and take a look at it, we do not burn our eyes out from x-rays at all. It is completely false. Furthermore, the total energy in the box, the total of all this intensity summed over all frequencies, would be the area under this infinite curve. Therefore, something is fundamentally, powerfully, and absolutely wrong. Thus was the classical theory absolutely incapable of correctly describing the distribution of light from a blackbody, just as it was incapable of correctly describing the specific heats of gases. Physicists went back and forth over this derivation from many different points of view, and there is no escape. This is the prediction of classical physics. Equation (41.13) is called Rayleigh’s law, and it is the prediction of classical physics, and is obviously absurd (Feynman et al., 1963, p. 41-6).”

 

The Rayleigh-Jeans Law suffered from critical failures that revealed the limits of classical physics. Firstly, it was unable to predict the characteristic peak in a blackbody's emission spectrum, contradicting the established experimental observation that radiance reaches a maximum at a specific wavelength. Secondly, its predictions deviated severely from empirical data at high frequencies, being accurate only in the long-wavelength, infrared regime. The most profound failure, known as the ultraviolet catastrophe, was its prediction that energy output would diverge to infinity at short wavelengths, implying that every object should radiate an infinite amount of ultraviolet and higher-frequency light, a result that was both physically impossible and in direct violation of the principle of energy conservation.

 

Key Takeaways:

Why the Beach is Safe (Absence of the Ultraviolet Catastrophe)

An explanation to relax safely on the beach is contributed by Planck’s quantum hypothesis in 1900, which solved the classical physics problem known as the Ultraviolet Catastrophe.

1. The Classical Prediction: The Rayleigh-Jeans Law predicted that the energy emitted by a blackbody (e.g., the Sun) should increase infinitely as the wavelength of the light decreases (i.e., moving into the UV and X-ray regions).
2. The Ultraviolet Catastrophe: If this classical prediction were true, the sun would emit most of its energy as lethal, high-frequency, short-wavelength radiation (ultraviolet, X-rays, and Gamma rays). The intense high-energy radiation would instantly incinerate all life on Earth, making the beach—and the planet—uninhabitable and unenjoyable.
3. The quantum solution: Planck’s quanta restrict the amount of high-frequency radiation that can be produced at a given temperature, allowing the black-body radiation curve to fall sharply in the ultraviolet region.

In short, the Ultraviolet Catastrophe does not occur because energy is quantized (Planck's Law). Importantly, the Sun's energy peak is within the visible and infrared spectrum—not the high-energy UV range—a crucial fact that permits life to flourish and people to enjoy the beach.

 

The Moral of the Lesson:

In his autobiography, Feynman mentions about enjoying the beach in the afternoon in Brazil...

In Feynman’s (1985) words: When I got to the center, we had to decide when I would give my lectures--in the morning, or afternoon. Lattes said, "The students prefer the afternoon."

“But the beach is nice in the afternoon, so why don't you give the lectures in the morning, so you can enjoy the beach in the afternoon.”

"But you said the students prefer to have them in the afternoon."

"Don't worry about that. Do what’s most convenient for you! Enjoy the beach in the afternoon."

So I learned how to look at life in a way that's different from the way it is where I come from. First, they weren't in the same hurry that I was. And second, if it's better for you, never mind! So I gave the lectures in the morning and enjoyed the beach in the afternoon. And had I learned that lesson earlier, I would have learned Portuguese in the first place, instead of Spanish……

 

One day, about 3:30 in the afternoon, I was walking along the sidewalk opposite the beach at Copacabana past a bar. I suddenly got this treMENdous, strong feeling: "That's just what I want; that'll fit just right. I'd just love to have a drink right now!" I started to walk into the bar, and I suddenly thought to myself, "Wait a minute! It's the middle of the afternoon. There's nobody here, There's no social reason to drink. Why do you have such a terribly strong feeling that you have to have a drink?"--and I got scared. I never drank ever again, since then. I suppose I really wasn't in any danger, because I found it very easy to stop. But that strong feeling that I didn't understand frightened me. You see, I get such fun out of thinking that I don't want to destroy this most pleasant machine that makes life such a big kick. It's the same reason that, later on, I was reluctant to try experiments with LSD in spite of my curiosity about hallucinations (p. 204).”

 

This passage from Feynman's autobiography contains at least two distinct lessons, both contributing to a central moral about self-awareness, intellectual integrity, and optimizing one's life experience. In essence, the moral is: Live intentionally. Whether it's a societal expectation or a sudden craving, pause and ask "Why?" Choose your actions based on understanding and reason, not on unexamined habit or impulse. This disciplined self-awareness is what allows you to preserve your greatest asset—your conscious, thinking self—and maintain your freedom to enjoy life fully.

 

Feynman was enjoying the beach by “absorbing” Vitamin D without ultraviolet catastrophe.

Malibu, 1950 (Source: No Ordinary Genius: The Illustrated Richard Feynman, p. 88)

 

Is a Physicist's Beach Day an Act of Faith?

One might ask why a physicist, who understands the sun's damaging UV radiation, would willingly bask in it. The answer is a calculated balance of biological need and physical trust.
The compelling reason is Vitamin D. The body uses UV-B photons as a catalyst to produce this crucial hormone, supporting skeletal and immune health. Without it, no amount of sunscreen or caution matters for long-term wellness.

The permission comes from Planck. Our field tells us the solar spectrum is benign at its core, governed by quantum mechanics that prevent an ultraviolet energy overflow. We enjoy the beach with faith that the physics is sound and the risk manageable.

In short, the physicist enjoys the beach with informed intentionality: to absorb a precise band of solar radiation for biological benefit, while aware of the dangers of its higher-energy UV components.

 

References:

Feynman, R. P. (1985). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

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.

Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.). Pearson.

Hoffmann, D. (2009). Black Body. In Compendium of Quantum Physics (pp. 36-39). Berlin, Heidelberg: Springer Berlin Heidelberg.

Rayleigh, L. (1900). Remarks upon the law of complete radiation. Phil. Mag, 49, 539.