Heating metal causes it to radiate heat and light. The color/frequency of radiation changes as we increase temperature.

The following video shows how the color of the radiation changes as a block of iron is heated up.  When current is passed through the heating element it heats up, but also becomes magnetized like a solenoid because it is a coil.  A block of magnetized iron is used that floats above the heating element so that you can see its color more clearly, but any block of metal would radiate in a similar way.

Video credit: Evolving Sciences

A blackbody is an idealized object that absorbs all of the radiation that hits it (none is reflected so it looks black in ordinary lighting).

A metal cavity with a small hole in it is an approximate blackbody.

The following YouTube video shows what happens when you heat up a metal cavity.

In thermal equilibrium, a blackbody emits a characteristic spectrum of radiation that only depends on its temperature, e.g. the temperature of the cavity walls for an approximate blackbody made from a metal cavity.

This simulation shows how the spectrum depends on temperature: 

PhET Blackbody Spectrum Simulation

Have a play with it.   Note that infra-red radiation causes heat.  How does the efficiency of a light bulb compare with the efficiency of the sun at producing visible light?  Why does the light from a light bulb look yellower than the light from the sun?

For those who do not like interactive things, here is a graph of the blackbody spectrum for various temperatures:

Image Credit: Samuel J. Ling, Jeff Sanny, William Moebs, University Physics Volume 3 (OpenStax, 2016)

Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction

The blackbody spectrum has the following features:

• There is a well-defined, continuous energy distribution that only depends on the temperature of the cavity walls.
• Wien’s displacement law: There is a maximum at a particular wavelength: \[\boxed{\lambda_{\max} \propto \frac{1}{T}.}\]
• Stefan-Boltzmann Law: In 1879, Stefan found experimentally that the total intensity (power per unit surface area obeys) \[\boxed{I = a\sigma T^4,}\] where

  • $a$ is a constant that depends on the material
  • $a=1$ for a blackbody
  • $a<1$ for a non blackbody
  • $\sigma = 5.67 \times 10^{-8}\,\text{Wm}^{-2}\text{K}^{-1}$
  • Boltzmann derived this theoretically in 1884.

• Wien extended the Stefan-Boltzmann law to obtain \[\boxed{u(\nu,T) = A\nu^3 e^{(-\beta\nu/T)}.}\]
• This only works for large frequencies $\nu$.
• $A$ and $\beta$ and are free parameters that have to be fixed by experiment.

• Rayleigh considered the statistical mechanics of radiation and derived \[\boxed{u(\nu,T) = \frac{8\pi \nu^2}{c^3} k T.}\]
• This only works for small $\nu$.
• Ultraviolet catastrophe: Total intensity $\int_{-\infty}^{\infty} u(\nu,T)\,\mathrm{d}\nu$ diverges for high frequency.  Implies that the cavity contains infinite energy

• Planck obtained \[\boxed{u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1},}\] which agrees with experiment for \[\boxed{h=6.626\times 10^{-34}\,\mathrm{m}^2\mathrm{kg}\mathrm{s}^{-1}.}\]
• $h$ is called Planck’s constant: A new fundamental constant of nature introduced by quantum theory.

The following graph compares the three distributions.

Image Credit: Wikimedia Commons

To derive his distribution, Planck assumed that the energy of the radiation emitted at frequency $\nu$ by the oscillating electrons in the walls of the cavity could only come in integer multiples of $h\nu$.

\[\boxed{E(\nu)=nh\nu\qquad\qquad\mbox{for}\qquad\qquad n=0,1,2,3,\cdots.}\]

This is the quantum postulate.

Since not everyone taking this class has studied statistical mechanics yet, the derivation of the Planck distribution from the quantum postulate is beyond the scope of this course, but see this link for a derivation.

• Wien’s displacement law says that the maximum energy density of the blackbody spectrum is at $\lambda_{\max} = b/T$ for some constant $b$.
• We can derive this by finding the maximum of the Planck distribution expressed as a function of wavelength $\tilde{u}(\lambda, T)$. Expressing the Planck distribution in terms of wavelength is one of the in-class activities.
• Setting $\frac{\partial \tilde{u}}{\partial \lambda}=0$ yields a transcendental equation (i.e. an equation that has no closed-form solution so we have to solve it numerically).  Solving this numerically gives \[\boxed{\lambda_{\max} = \frac{b}{T},}\] with \[\boxed{b = \frac{hc}{4.965 k}.}\]

1. Derive the Rayleigh-Jeans law $u(\nu,T) = \frac{8\pi \nu^2}{c^3} k T$ from the Planck formula $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$ for small $\nu$.
2. Derive Wien’s law $u(\nu,T) = A\nu^3 e^{(-\beta\nu/T)}$ from the Planck formula $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$ for large $\nu$.  What are  $A$ and $\beta$ in terms of $h$, $c$, and $k$?
3. The energy density for frequencies between $\nu$ and $\nu + \mathrm{d}\nu$ is $u(\nu, T)\mathrm{d}\nu$, where $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$.   Derive the energy density $\tilde{u}(\lambda,T)$ as a function of wavelength using $\lambda = \frac{c}{\nu}$.