Q 7.38 - An Introduction to Thermal Physics

Question

It's not obvious from Figure 7.19 how the Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the function $u\left(ϵ\right)$ for T = 3000 K and T = 6000 K (both on the same graph). Label the horizontal axis in electron-volts.

Step-by-Step Solution

Answer

The function is:

$u\left(ϵ\right)=\frac{8\pi }{(hc{)}^3}\frac{{ϵ}^3}{e^{ϵ/kT}-1}$

Step 1: Given information

The Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the function $u\left(ϵ\right)$  for T = 3000 K and T = 6000 K 

Step 2: Explanation

The photon's Planck spectrum is given as:

$u\left(ϵ\right)=\frac{8\pi }{(hc{)}^3}\frac{{ϵ}^3}{e^{ϵ/kT}-1} \left(1\right)$

This function must be plotted at temperatures of T= 3000 K and T = 6000 K, with the constants in eV supplied by:

$$\begin{gathered} h=4.136\times {10}^{-15}\mathrm{eV}·\mathrm{s} \\ \mathrm{k}=8.62\times {10}^{-5}\mathrm{eV}/\mathrm{K} \\ \mathrm{c}=3.00\times {10}^8 \mathrm{m}/\mathrm{s} \end{gathered}$$

Using python to plot the function and the code is:

The graph is: