2: Blackbody Radiation
- Page ID
- 6656
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This chapter briefly summarizes some of the formulas and theorems associated with blackbody radiation. A small point of style is that when the word "blackbody" is used as an adjective, it is usually written as a single unhyphenated word, as in "blackbody radiation"; whereas when "body" is used as a noun and "black" as an adjective, two separate words are used. Thus a black body emits blackbody radiation. The Sun radiates energy only very approximately like a black body. The radiation from the Sun is only very approximately blackbody radiation.
- 2.2: Radiation within a cavity enclosure
- We'll also suppose that, because of the difference in nature of the walls of the two cavities, the radiation density in one is greater than in the other. Let us open the door for a moment. Radiation will flow in both directions, but there will be a net flow of radiation from the high-radiation-density cavity to the low-radiation-density cavity. As a consequence, the temperature of one cavity will rise and the temperature of the other will fall.
- 2.3: Kirchhoff's Law
- Kirchhoff's law, as well as his studies with Bunsen (who invented the Bunsen burner for the purpose) showing that every element has its characteristic spectrum, represents one of the most important achievements of mid-nineteenth century physics and chemistry.
- 2.4: An aperture as a black body
- Pierce a small hole in the side of the enclosure. Radiation will now pour out of the enclosure at a rate per unit area that is equal to the rate at which the walls are being radiated from within. In other words the exitance of the radiation emanating from the hole is the same as the irradiance within. Now irradiate the hole from outside. The radiation will enter the hole, and very little of it will get out again; the smaller the hole, the more nearly will all of the energy directed at the hole f
- 2.5: Planck's Equation
- The importance of Planck's equation in the early birth of quantum theory is well known. Its theoretical derivation is dealt with in courses on statistical mechanics. In this section I merely give the relevant equations for reference.
- 2.6: Wien's Law
- Wien's law states that the black body radiation curve for different temperatures peaks at a wavelength inversely proportional to the temperature.
- 2.7: Stefan's Law (The Stefan-Boltzmann Law)
- The Stefan–Boltzmann law describes the power radiated from a blackbody in terms of its temperature and states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body's thermodynamic temperature T.
- 2.8: A Thermodynamical Argument
- Wien's and Stefan's laws can be derived by differentiation and integration respectively of Planck's equation. Stefan's law was derived from a simple thermodynamical argument long before the derivation of Planck's equation, and it is not necessary to know Planck's equation, let alone how to differentiate it or integrate it, in order to arrive at Stefan's law. You do, however, have to know a little thermodynamics.
- 2.9: Dimensionless forms of Planck's equation
- The Planck functions (of wavelength or frequency and temperature) can be collapsed on to dimensionless functions of a single variable if we express the exitance in units of the maximum exitance, and the wavelength or frequency in units of the wavelength or frequency at which the maximum occurs.
- 2.10: Derivation of Wien's and Stefan's Laws
- Wien's and Stefan's Laws are found, respectively, by differentiation and integration of Planck's equation.