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Note for Applied Physics - Phy By vtu rangers

  • Applied Physics - Phy
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  • Visvesvaraya Technological University Regional Center - VTU
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MODULE-1 MODERN PHYSICS BLACKBODY RADIATION: A black body in principle absorbs the entire radiations incident on it and also emits all the radiations when it is heated to incandescence. The radiations emitted by such a body are known as black body radiation. 1. A black body not only completely absorbs all the radiations falling on it but also behaves as a perfect radiator when heated. 2. The radiation given out by a black body depends only on the temperature and not on the nature of the body. BLACKBODY RADIATION SPECTRUM AND ITS FEATURES Following figure illustrates the way in which the energy radiated by a black body is distributed amongst various wavelengths. The plot had following features:  There are different curves for different temperatures.  There is a peak for each curve. This means that the electromagnetic waves of wavelength corresponding to the peak is emitted to the largest extent at that temperature to which the curve corresponds.  The peak shifts towards the lower wavelength side as higher temperatures are considered. Department of Physics Page 1

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LAWS GOVERNING THE BLACKBODY RADIATION: 1. Wien’s Law 2. Wien’s Displacement Law 3. Stefan’s Law 4. Rayleigh-Jeans Law 5. Kirchoff’s Law 6. Planck’s Law WIEN’S LAW: Wien’s law gives the relation between the wavelength of emitted radiation and the temperature of the source as, E d  C15 e C2 T d Where C1 and C2 are constants and E  d is the energy emitted per unit volume in the wavelength region  and +d. This law is called Wien’s law of energy distribution in the blackbody radiation spectrum. Drawbacks of wien’s law: Wien’s law holds good only for shorter values of wavelength but fails to explain the region in which wavelengths are longer. RAYLEIGH-JEANS LAW: In 1900, Lord Rayleigh applied the principle of equipartition of energy to electromagnetic vibrations and deduced an equation for the black body raidiation. This was later modified by Sir James Jeans and came to be called Rayleigh-Jeans law. According to this law the energy density i.e. the amount of energy per unit volume of a blackbody in the wavelength ranging from  to +d is given by E d  8kT4 d Where, k is Boltzmann constant. Due to the presence of the factor λ-4 in the equation, the energy radiated by the blackbody should rapidly decrease with the increase in wavelength. Drawbacks of Rayleigh-Jeans law (or Ultraviolet catastrophe): Rayleigh-Jeans law holds good for longer values of wavelength but does not fit the experimental curves for shorter values of wavelength. Department of Physics Page 2

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This discrepancy between the theoretical conclusion and the experimental result is called “ULTRAVIOLET CATASTROPHE”. As per the equation, black body radiates enormously in the shorter wavelength side, but practically black body chiefly radiates in IR or visible range which is the longer wavelength region of the spectrum. This wrong prediction in the radiation emitted by the body is called Ultraviolet Catastrophe. PLANCK’S LAW: In order to explain the energy distribution in the complete spectrum of a blackbody radiation, Max Ludwig Planck in 1900 put forward the quantum theory of radiation. For deriving the equation Planck made certain assumption based on Quantum mechanical consideration. Assumptions of Quantum theory of radiations are, 1. The walls of the blackbody consist of a very large number of electrical oscillators. Each oscillator vibrates with a frequency of its own. 2 Energy is absorbed or emitted by a blackbody in a discrete manner, in the form of small packets called quanta. 3. Each quantum has energy that depends only on the frequency of the radiation and is given by E = nh Where h is a constant known as Plank’s constant = 6.625 x10-34 J-sec and n=0,1,2, ….. 4. An oscillator may gain or lose energy when it absorbs or emits radiation of frequency υ given by υ = ΔE/h Where, ΔE is the difference in energies of the oscillator before and after emission or absorption. Based on the above assumptions, Planck derived an equation which successfully explained the entire spectrum of blackbody radiation. The equation is given by E d  8hc  1  d ………….(1) 5  h / kT  e  1 Planck’s law explains the spectrum of blackbody radiation successfully in both the shorter wavelength and longer wavelength regions. Department of Physics Page 3

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REDUCTION OF PLANCK’S LAW TO WIEN’S LAW AND RAYLEIGH-JEANS LAW: Derivation of Wien’s law from Planck’s law  For shorter wavelengths, When λ is small, υ=c/λ is large. When υ is large, ehυ/kT is very large. i.e., ehυ/kT >> 1 ehυ/kT -1 ≈ ehυ/kT ≈ ehc/λkT (Since υ=c/λ) Substituting in Equation (1) we get 8hc 1 E d  5 hc kT d  e  E d  C15 e C T d        (2) where C1  8hc and C2  hc k . This equation is Wien’s law of radiation. 2 Derivation of Rayleigh-Jeans law from Planck’s law  For longer wavelengths, c When wavelength λ is large, υ= is small. λ ehυ/kT Since υ is very small, will be very small. Expanding ehυ/kT as power series, we have 2 ehυ/kT = 1+ hυ/kT + (hυ/kT) + …… Neglecting the higher order terms we get, ehυ/kT ≈ 1+ hυ/kT ∴ 𝐞𝐡𝛖/𝐤𝐓 -1 ≈ hυ/kT ≈ hc/λkT Substituting in equation (1), we get 8hc kT E d  5  d hc  8kT  E d  4 d        (3)  This equation is the Rayleigh-Jeans law of radiation. Department of Physics Page 4

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