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Physics Lesson 19.1.6 - The Ultraviolet Catastrophe and Planck's Hypothesis

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Welcome to our Physics lesson on The Ultraviolet Catastrophe and Planck's Hypothesis, this is the sixth lesson of our suite of physics lessons covering the topic of Thermal Radiation. Photon as the Quantum of Light, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

The Ultraviolet Catastrophe and Planck's Hypothesis

All attempts of scientists during the XIX century in finding a formula that corresponds to the graph of spectral emissivity, failed. This is because they relied on the classical approach, which stresses the continuous nature of electromagnetic radiation emitted by atoms and molecules. Based on this hypothesis, two scientists - Rayleigh and Jeans - found the mathematical expression below for the emission ability of a black body:

e0 (λ) = 2π ∙ c ∙ T/λ4

The graph below shows the curve found experimentally (the solid curve) and the theoretical curve obtained through the formula of Rayleigh-Jeans.

Physics Tutorials: This image provides visual information for the physics tutorial Thermal Radiation. Photon as the Quantum of Light

From the graph is evident that for long wavelength the curve obtained from the Rayleigh-Jeans formula represents a good fit for the experimental findings but for short wavelengths, this curve deflects too much from the curve found experimentally. Moreover, the emission ability of objects points towards infinity when wavelength points towards zero.

It was an absurdity to accept the idea that object with finite dimensions could have an infinite emission ability and they are able to radiate infinitely large amounts of energy in every second. This nonsense represents a notable failure of the classical theory of thermal radiation, which is known in the history of physics as the "ultraviolet catastrophe".

In 1900, Max Planck overcame this handicap by proposing a new hypothesis. According to him, the radiation emitted from atoms and molecules do not occur in a continuous way but with interruptions and in very small portions called quanta (quantum in singular). Moreover, the energy E of a quantum of light (known as photon - the particle of light) is proportional to the frequency of radiation, i.e.

E = h ∙ f

where h (which is a pseudo-letter; it is not the traditional h of our alphabet) represents the Planck constant. It has the value of 6.626 × 10-34 J/s.

Relying on his hypothesis on the quantum nature of light, Max Planck managed to find theoretically the true shape of the spectral emissivity curve for a black body and to explain the experimental findings (laws) of Stephan-Boltzmann and Wien. The compliance with the experimental data was complete. With the Planck hypothesis, a new era of modern physics represented by the quantum physics, did start. This new branch of physics today stands at the forefront of scientific development and technology.

The above equation represents the fundamental equation of quantum theory in Modern Physics.

Example 1

Stefan and Boltzmann found experimentally the value of constant σ that bears their name. As stated earlier, it is σ = 5.67 × 10-8 J/K4m2s, while Planck found theoretically that this constant is related to the other three universal constants h, k and c (Planck constant, Boltzmann constant and light speed in vacuum respectively) through the expression

σ = 2π5 ∙ k4/15c2 ∙ h3

Based on the above formula, calculate the value of Planck's constant h giving that k = 1.38 × 10-23 J/mol·K and c = 3 × 108 m/s.

Solution 1

This simple exercise involves only calculations. Thus, after rearranging, we obtain for the Planck's constant h:

σ = 2π5 ∙ k4/15c2 ∙ h3
h = ∛ 2π5 ∙ k4/15c2 ∙ σ
= ∛ 2 ∙ (3.14)5 ∙ (1.38 × 10-23 J/mol ∙ K)4/15 ∙ (3 × 108 m/s)2 ∙ (5.67 × 10-8 J/K4 ∙ m2 ∙ s)
= ∛ 289.24 × 10-102 J3/s3
= 6.613×10-34 J/s

You have reached the end of Physics lesson 19.1.6 The Ultraviolet Catastrophe and Planck's Hypothesis. There are 6 lessons in this physics tutorial covering Thermal Radiation. Photon as the Quantum of Light, you can access all the lessons from this tutorial below.

More Thermal Radiation. Photon as the Quantum of Light Lessons and Learning Resources

Modern Physics Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
19.1Thermal Radiation. Photon as the Quantum of Light
Lesson IDPhysics Lesson TitleLessonVideo
Lesson
19.1.1What is Thermal Radiation?
19.1.2Total and Spectral Emissivity
19.1.3Radiation of a Black Body
19.1.4Stefan-Boltzmann Law
19.1.5Wien's Law
19.1.6The Ultraviolet Catastrophe and Planck's Hypothesis

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