Q 7.37 - An Introduction to Thermal Physics

Question

 Prove that the peak of the Planck spectrum is at x = 2.82.

Step-by-Step Solution

Answer

Hence proved that plank's spectrum is at x = 2.82.

Step 1: Given information

The peak of the Planck spectrum is at e = 2.82.

Step 2: Explanation

Set the derivative of $x^3/\left(e^x-1\right)$ with respect to x equals to zero to get the maximum, then solve for:

$$\begin{gathered} \frac{\partial }{\partial x}\left(\frac{x^3}{e^x-1}\right)=0 \\ \frac{3x^2\left(e^x-1\right)-x^3{e}^x}{{\left(e^x-1\right)}^2}=0 \\ 3x^2\left(e^x-1\right)-x^3{e}^x=0 \\ 3x^2{e}^x-3x^2-x^3{e}^x=0 \\ 3e^x-3-x{e}^x=0 \end{gathered}$$

Using matlab to solve the above equation: The code is:

Therefore, $x=2.8214$