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2.5: Planck's Equation

Last updated

Mar 5, 2022
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- 2.4: An aperture as a black body
- 2.6: Wien's Law

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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:

$\begin{array}{l} h=\text{Planck's constant} \\ k=\text{Boltzmann's constant} \\ c = \text{speed of light} \\ T = \text{temperature} \\ \lambda = \text{wavelength} \\ \nu = \text{frequency} \\ \end{array}$

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