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Physics LibreTexts

2.6: 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.

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}