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.
- 2.1: Absorptance, and the Definition of a Black Body
- This page covers the concepts of absorptance, reflectance, and transmittance in relation to radiation and matter interaction. Absorptance \(a(\lambda)\) indicates the fraction of absorbed radiation at a specific wavelength, with reflectance and transmittance representing the rest. The sum of these three properties equals one. A black body absorbs all radiation, while a grey body has a consistent absorptance below one across all wavelengths.
- 2.2: Radiation within a cavity enclosure
- This page explores an experiment with two temperature-equal cavities—one with shiny walls and the other with dull ones. Opening a door between them leads to radiation flow based on density differences, altering their temperatures. It highlights that radiation density is solely temperature-dependent, contradicting the notion of a perpetual motion machine. This exemplifies a key thermodynamic principle related to radiation.
- 2.3: Kirchhoff's Law
- This page discusses Kirchhoff's law, a pivotal discovery in physics and chemistry from the mid-19th century, which links the absorption and emission of radiation. It asserts that the ratio of exitance to absorptance is dictated by temperature and wavelength. This principle has become essential for both quantitative and qualitative spectroscopy, highlighting that effective emitters are also strong absorbers.
- 2.4: An aperture as a black body
- This page explains how radiation behaves in an enclosure with a small hole, stating that radiation exits at a rate equal to that generated inside. External radiation is mostly absorbed, allowing the hole to act like a black body according to Kirchhoff's law. This principle is applied in practical scenarios, such as using warm enclosures with small holes to simulate blackbody radiation for calibrating radio telescopes.
- 2.5: Planck's Equation
- This page explains Planck's equation and its importance in quantum theory, covering different forms related to radiation exitance and detailing energy and photon emission across various metrics. It defines key constants like Planck's constant and Boltzmann's constant, providing their values. The content is essential for grasping the foundational equations of blackbody radiation within statistical mechanics.
- 2.6: Wien's Law
- This page focuses on determining the maximum wavelengths and frequencies of the four Planck functions through differentiation. It references Wien's law and provides specific equations, constants, and dimensionless numbers essential for calculations. The maximum ordinates for each function are illustrated, highlighting their dependence on temperature (T). The equations and constants together enhance the understanding of peak emission characteristics in blackbody radiation.
- 2.7: Stefan's Law (The Stefan-Boltzmann Law)
- This page addresses the integration of equations to derive total exitance across wavelengths and frequencies, culminating in Stefan's Law (M = σT^4), with σ as Stefan's constant. It also introduces a derived equation for photon number density (N = ρT^3), where ρ is linked to the Riemann zeta-function, ζ(3). Both equations confirm the consistency of integrations in thermal radiation laws.
- 2.8: A Thermodynamical Argument
- This page covers the derivation of Wien's and Stefan's laws from Planck's equation, highlighting the complexities involved. It reassures readers that a background in thermodynamics provides a simpler understanding of Stefan's law. Additionally, it presents a thermodynamical relationship linking radiation energy density and pressure, ultimately leading to the derivation of Stefan's law through the manipulation of equations related to isotropic radiation in a steady state.
- 2.9: Dimensionless forms of Planck's equation
- This page explores the Planck functions for blackbody radiation, showcasing their representation as dimensionless functions based on normalized exitance and wavelength or frequency. It includes key equations for calculating these functions and their maximum values, noting stable constants amidst evolving theories.
- 2.10: Derivation of Wien's and Stefan's Laws
- This page covers the derivation of Wien's and Stefan's Laws from Planck's equation, emphasizing the relationship between black body radiation, temperature, and wavelength. It discusses the mathematical foundations involving the sine function and the Riemann zeta function, linking them to Stefan's law. Problems presented include analyzing temperature changes in black bodies and comparing radiation energy densities in materials like silver and black carbon.