## Small Group Activity: Energy radiated from one oscillator

Contemporary Challenges 2021
This lecture is one step in motivating the form of the Planck distribution.
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## Classical prediction

\begin{align} \left<\text{energy rate}\right> &= \frac23 \frac{q^2}{4\pi\epsilon_0 c^3}\frac{k_BT}{m}\omega_0^2 \end{align} Draw a graph with $\left<\text{energy rate}\right>$ (the optical power output) on the vertical axis, and the oscillator frequency (or alternatively $\hbar\omega_0$) on the horizontal axis. Sketch the relationship when
1. $T=300\text{ K}$
2. $T=600\text{ K}$
Assume the mass and charge of the the oscillators are constants.

## Quantum prediction

When we account for energy quanta, the expression changes to \begin{align} \left<\text{energy rate}\right> &= \begin{cases} \frac23 \frac{q^2}{4\pi\epsilon_0 c^3}\frac{k_BT}{m}\omega_0^2 & \text{when }\hbar\omega\ll k_BT \\ \frac{\hbar\omega^3}{4\pi^2c^2}e^{-\frac{\hbar\omega}{k_BT}} & \text{when }\hbar\omega\gg k_BT \end{cases} \end{align} Sketch the relationship on your same graph, when
1. $T=300\text{ K}$
2. $T=600\text{ K}$
Assume the mass and charge of the the oscillators are constants.

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