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Engineering Physics [06PHY12] BE I Semester VTU, Belagaum Unit 1 (Modern physics)
Syllabus as prescribed by VTU
Introduction to blackbody radiation spectrum; Photoelectric effect; Compton effect; Wave particle Dualism; de Broglie hypothesis:de Broglie wavelength, Extension to electron particle; Davisson and Germer Experiment, Matter waves and their Characteristic properties; Phase velocity, group velocity and particle velocity; Relation between phase velocity and group velocity; Relation between group velocity and particle velocity; Expression for de Broglie wavelength using group velocity.
1 Introduction
The turn of the 20th century brought the start of a revolution in physics. In 1900, Max Planck published his explanation of blackbody radiation. This equation assumed that radiators are quantized, which proved to be the opening argument in the edifice that would become quantum mechanics. In this chapter, many of the developments which form the foundation of modern physics are discussed.
2 Blackbody radiation spectrum
A blackbody is an object that absorbs all light that falls on it. Since no light is reflected or transmitted, the object appears black when it is cold. The term blackbody was introduced by Gustav Kirchhoff in 1860. A perfect blackbody, in thermal equilibrium, will emit exactly as much as it absorbs at every wavelength. The light emitted by a blackbody is called blackbody radiation. The plot of distribution of emitted energy as a function of wavelength and temperature of blackbody is know as blackbody spectrum. It has the following characteristics.
• The spectral distribution of energy in the radiation depends only on the temperature of the
blackbody.
• The higher the temperature, the greater the amount of total radiation energy emitted and also
energy emitted at individual wavelengths.
• The higher the temperature, the lower the wavelength at which maximum emission occurs.
Many theories were proposed to explain the nature of blackbody radiation based on classical physics arguments. But non of them could explain the complete blackbody spectrum satisfactorily. These
theories are discussed in brief below.
2.1 Stefan-Boltzmann law
The Stefan-Boltzmann law, also known as Stefan’s law, states that the total energy radiated per unit surface area of a blackbody in unit time (known variously as the blackbody irradiance, energy flux density, radiant flux, or the emissive power), E*, is directly proportional to the fourth power of the blackbody’s thermodynamic temperature T (also called absolute temperature):
E* = σT4
The constant of proportionality σ is called the Stefan-Boltzmann constant or Stefan’s constant. It is not a fundamental constant, in the sense that it can be derived from other known constants of nature. The value of the constant is 5.6704 × 10−8 J s−1 m−2 K−4
. The Stefan-Boltzmann law is an example of a power law.
2.2 Wien’s displacement law
Wien’s displacement law states that there is an inverse relationship between the wavelength of the peak of the emission of a blackbody and its absolute temperature.
λmax ∝(1/T)
T λmax = b
where λmax is the peak wavelength in meters, T is the temperature of the blackbody in kelvins (K), and b is a constant of proportionality, called Wien’s displacement constant and equals 2.8978×10−3 mK. In other words, Wien’s displacement law states that the hotter an object is, the shorter the wavelength at which it will emit most of its radiation.
2.3 Wien’s distribution law
According to Wein, the energy density, Eλ, emitted by a blackbody in a wavelength interval λ and λ + dλ is given by
Eλ dλ =(c1/λ5e (−c2/λT)dλ
where c1 and c2 are constants. This is known as Wien’s distribution law or simply Wein’s law. This law holds good for smaller values of λ but does not match the experimental results for larger values of λ. Wien received the 1911 Nobel Prize for his work on heat radiation.
2.4 Rayleigh-Jeans’ law
According to Rayleigh and Jeans the energy density, Eλ, emitted by a blackbody in a wavelength interval λ and λ + dλ is given by
Eλ dλ = (8πkT/λ4)*dλ
where k is the Boltzmann’s constant whose value is equal to 1.381 × 10−23 JK−1 It agrees well with experimental measurements for long wavelengths This was not supported by experiments and the failure has become known as the ultraviolet catastrophe or Rayleigh-Jeans catastrophe. Here the word ultraviolet signifies shorter wavelength or higher frequencies and not the ultraviolet region of the spectrum. One more thing to note is that, it was not, as is sometimes asserted in physics textbooks, a motivation for quantum theory
2.5 Planck’s law of black-body radiation
Explaining the blackbody radiation curve was a major challenge in theoretical physics during the late nineteenth century. All the theories based on classical ideas failed in one or the other way. The wavelength at which the radiation is strongest is given by Wien’s displacement law, and the overall power emitted per unit area is given by the Stefan-Boltzmann law. Wein’s law could explain the blackbody radiation curve only for shorter wavelengths whereas Rayleigh-Jeans’ law worked well only for larger wavelengths. The problem was finally solved in 1901 by Max Planck. Planck came up with the following formula for the spectral energy density of blackbody radiation in a wavelength range λ and λ + dλ,
Eλ dλ = (8πhc/λ5)*(1/ e hc/λkT − 1)*dλ
where h is the Planck’s constant whose value is 6.626 × 10−34 Js. This formula could explain the entire blackbody spectrum and does not suffer from an ultraviolet catastrophe unlike the previous ones. But the problem was to justify it in terms of physical principles. Planck proposed a radically new idea that the oscillators in the blackbody do not have continuous distribution of energies but only in discrete amounts. An oscillator emits radiation of frequency ν when it drops from one energy state to the next lower one, and it jumps to the next higher state when it absorbs radiation of frequency ν. Each such discrete bundle of energy hν is called quantum. Hence, the energy of an oscillator can be written as
En = nhν n = 0, 1, 2, 3, ....
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