# Distribution of energy within the spectrum

The Stefan-Boltzmann law gives the total radiation emitted by the body but tells us nothing about how it is distributed across the spectrum. If measurements are made of the energy emitted at different temperature for a black body then a series of curves like those shown in Figure 1 can be obtained. Here the vertical axis shows the energy density (that is the energy emitted per square metre per second in a small wavelength range from l to l+dl) and the horizontal axis shows the wavelength.

Some important facts can be deduced from these curves:
(a) the area between any energy-wavelength curve and the wavelength axis gives the total energy emitted by the body per unit area at that temperature
(b) the maxima of the curves moves towards short wavelengths at higher temperatures
(c) the curves for lower temperatures lie completely inside those of higher temperature
It was found that:

(a) lmT = constant where lm is the wavelength at which most energy is emitted, that is at the maxima of the curves. The constant has a value of 2.898x10-3m K.
(b) the energy emitted at this wavelength (lm) is proportional to T5.

These two results are known as Wien's laws

 Temperature (K) Wavelength lm (nm) 500 5800 750 3900 1000 3000 1750 1650 6000 480 30000 97

A range of values for lm is shown in the table. It must be remembered that these are for true black bodies only.

For an electric light bulb the bulb's filament temperature (T) = 1750 K and the peak of the energy-wavelength curve (lm) is at 1400 nm.

You can see that if lm for a black body lies in the red region of the spectrum the body will appear red hot, and as it gets hotter this peak will move towards the violet end of the spectrum. However this does not mean that the body will look "violet hot". The reason for this is that at the higher temperature all visible wavelengths will be present to some extent and so the body will appear "white hot". A white-hot body will give high emission across the whole range of the visible spectrum.

If we know the value of lm for one black body at a known temperature we can use Wien's law to calculate the temperature of another black body providing the wavelength at which maximum energy is emitted is known. This has been used extensively in astronomy for finding the temperatures of stars.

Example problem
A black body (the Sun) with a surface temperature of 6000 K emits radiation with lm = 420 nm.
Calculate the surface temperature of Sirius (the brightest star in the northern skies) if lm for Sirius is 72 nm.

lm1T1 = lm2T2
Therefore:    420x10-9x6000 = 72x10-9xT2
Finally:    Surface temperature of Sirius (T2) = 35 000 K