Class Activity: Black Body Radiation Problem 2 b and c

In order to define a relationship between the wavelength corresponding to the peak intensity of radiation emitted from a black body (λ_max) and that black body's temperature (T), we must begin with the equation for specific intensity, B_λ(T):

The maximum of this function will be the λ such that

where the derivative is evaluated with T held constant (since we are finding λ_max for a given T). If we make a simple variable substitution, the calculation becomes much easier:

Then we need:

This is a difficult equation to solve, but if we approximate e^x to second order, then we have:

Plugging in the values for h, c, and k, we find:

where the units of λ_maxT are cm · K. Wien's law is:

For a second-order approximation of e^x, this is quite close.


Now let's consider very small photon energies, so that hv << kT. Then we can use a second-order approximation for e^x again to simplify B_v(T) for small photon energies.

Inserting this into the equation for B_v(T), we obtain:

This is the simplified form of B_v(T) for hv << kT.


Radio astronomers like to talk about "brightness temperature" instead of actual brightness because the brightness is related to the temperature, but the temperature is more helpful in discovering the black body spectrum of an object than its brightness. The black body spectrum, if the temperature is known, gives the most prominent wavelength of a black body's radiation, which is a useful value in classifying an object.


Secondary authors: Joanna Robaszewski, Lauren Gilbert