F = 2 h c^{2} ^{-5} / (exp(hc/kT) - 1).

or, combining the constants:

F = c_{1} ^{-5} / exp(c_{2} /T) - 1),

where c_{1} = 2 hc^{2} = 3.7419 × 10^{-5} erg cm^{2} s^{-1} [ in cm]

and c_{2} = hc/k = 1.4288 cm °K.

In order to find the wavelength for which the emission is a maximum (at
each temperature), we must take the derivative of this function and set it
equal to 0, that is, to find the inflection point of the curve, the
point at which a line *tangent* to the curve has a slope of 0.
To do this, the first rule we will use is:

d(uv) = u dv + v du

where:

u = c_{1} ^{-5} ;

du = -5 c_{1} ^{-6} ;

v = [exp(c_{2} /T) - 1)]^{-1} and

dv = [c_{2}/(^{2} T)] [exp(c_{2} /T)/(exp(c_{2} /T) - 1)^{2}].

Combining these terms, we obtain:

0 = u dv + v du

= c_{1} ^{-5} [c_{2}/(^{2} T)] [exp(c_{2} /T)/(exp(c_{2} /T) - 1)] + [exp(c_{2} /T) - 1)](-5 c_{1} ^{-6})]

Combining terms:

5c_{1}/[ ^{6}(exp(c_{2} /T) - 1)] =
c_{1}c_{2}exp(c_{2} /T)/[ ^{7} T (exp(c_{2} /T) - 1)]^{2}

Which reduces to:

5 = c_{2} exp(c_{2} /T) / [ T (exp(c_{2} /T) - 1)] or

T =
[c_{2}/5][exp(c_{2} /T) /(exp(c_{2} /T) - 1)]

We are almost there! c_{2}/5 is easy to evaluate but the terms
containing the exponentials are a bit more difficult because they still
contain the wavelength variable. However, it is easy to see that the
term containing the ratio of the 2 exponentials should be very nearly
equal to 1 when we are near the peak of emission;

exp(c_{2} /T) exp(c_{2} /T) - 1

So let us iterate to obtain the final value for the right
hand side of the equation. As a first guess, let us take c_{2}/5,
or 0.28777 as the value of T. Using this value to evaluate the exponentials,
we find a "correction factor" of 1.00681, yielding a new guess of
0.28973 for T.
With this value for T, the ratio of the exponentials is 1.00703.
Correcting the guess again, we obtain 0.28979. With this value the ratio
of the exponential terms is now 1.00705, yielding a corrected value of
0.28979. We have apparently converged in only 3 iterations! Therefore the final equation is:

T = 0.28979.

This equation is known as the Wien Displacement Law.

Introduction to Blackbody radiation

Wien Distribution

Wien Displacement Law

Stefan-Boltzmann Law

Constants

This VRML + HTML package on Blackbody emission was constructed by Karen M. Strom. Contact your local instructor for help if it is used as part of your class.