2.5: Planck's Equation
- Page ID
- 6655
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
Planck's equation can be given in various ways, and here I present four. All will be given in terms of exitance. The radiance is the exitance divided by \(\pi\).(Equation 1.15.2.). The four forms are as follows, in which I have made use of equations 1.3.1 and the expression \(h\nu = hc/\lambda\) for the energy of a single photon.
The rate of emission of energy per unit area per unit time (i.e. the exitance) per unit wavelength interval:
\[M_\lambda = \frac{C_1}{\lambda^5 \left(e^{K_1/\lambda T} - 1 \right)} \tag{2.6.1} \label{2.6.1}\]
The rate of emission of photons per unit area per unit time per unit wavelength interval:
\[N_\lambda = \frac{C_2}{\lambda^4 \left(e^{K_1/\lambda T} -1\right)} \tag{2.6.2} \label{2.6.2}\]
The rate of emission of energy per unit area per unit time (i.e. the exitance) per unit frequency interval:
\[M_\nu = \frac{C_3 \nu^3}{e^{K_2 \nu/T} - 1} \tag{2.6.3} \label{2.6.3}\]
The rate of emission of photons per unit area per unit time per unit frequency interval:
\[N_\nu = \frac{C_4 \nu^2}{e^{K_2 \nu/T} - 1} \tag{2.6.4} \label{2.6.4}\]
The constants are:
\begin{array}{c c c c c c r}
C_1 &=& 2\pi hc^2 & = & 3.7418 \times 10^{-16} \text{W m}^2 && (2.6.5) \\
C_2 & = & 2\pi c & = & 1.8837 \times 10^9 \text{m s}^{-1} && (2.6.6) \\
C_3 & = & 2\pi h /c^2 & = & 4.6323 \times 10^{-50} \text{kg s} && (2.6.7) \\
C_4 & = & 2\pi/c^2 & = & 6.9910 \times 10^{-17} \text{m}^{-2} \text{s}^2 && (2.6.8) \\
K_1 & = & hc/k & = & 1.4388 \times 10^{-2} \text{m K} && (2.6.9) \\
K_2 & = & h/k & = & 4.7992 \times 10^{-11} \text{s K} && (2.6.10) \\
\end{array}
Symbols:
\begin{array}{l}
h=\text{Planck's constant} \\
k=\text{Boltzmann's constant} \\
c = \text{speed of light} \\
T = \text{temperature} \\
\lambda = \text{wavelength} \\
\nu = \text{frequency} \\
\end{array}