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:

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:

Stefan-Boltzmann Law: In 1879, Stefan found experimentally that the total intensity (power per unit surface area obeys)

is a constant that depends on the material for a black-body for a non black-body Boltzmann derived this theoretically in 1884. 1.ii.3 Attempts to derive the blackbody spectrum Wien's Law Wien extended the Stefan-Boltzmann law to obtain

This only works for large frequencies . and are free parameters that have to be fixed by experiment. Rayleigh-Jeans Law Rayleigh considered the statistical mechanics of radiation and derived

This only works for small . Ultraviolet catastrophe: Total intensity diverges for high frequency.  Implies that the cavity contains infinite energy Planck Distribution Planck obtained

which agrees with experiment for

.

is called Planck’s constant: A new fundamental constant of nature introduced by quantum theory.

The following graph compares the three distributions.

1.ii.4 The Quantum Postulate To derive his distribution, Planck assumed that the energy of the radiation emitted at frequency by the oscillating electrons in the walls of the cavity could only come in integer multiples of .

This is the quantum postulate. For a derivation of the Planck distribution from this posulate, see http://disciplinas.stoa.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf 1.ii.5 Wien's Displacement Law Wien’s displacement law says that the maximum energy density of the blackbody spectrum is at for some constant . We can derive this by finding the maximum of the Planck distribution expressed as a function of wavelength  that we found in the in-class activity. Setting 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

with

.

In Class Activities

Derive the Rayleigh-Jeans law from the Planck formula for small .

Derive Wien’s law from the Planck formula for large .  What are  and in terms of , , and ?

The energy density for frequencies between and is , where .   Derive the energy density as a function of wavelength using .