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# 2.5: Planck's Equation

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

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}