7.15 Radiation from Hot Bodies
227
Fig. 7.7 Planck’s
Distribution. The vertical axis
is the intensity of light per
unit frequency, divided by the
peak intensity at 8000 K
1
0
0
3
6
9
1 2
Relative Intensity
15
18
8000 K
7000 K
Frequency / (10 14 s –1 )
6000 K
Wien’s Law,
λ max =
0.00289776855 m · K
T
;
can be derived from Planck’s distribution. It gives the wavelength for the greatest
light intensity per unit wavelength. Wien’s relation shows that the wavelength at the
peak intensity increases inversely with the temperature.
Light intensity from a blackbody at all frequencies follows by integration of the
Planck expression, giving Stefan’s Law:
I = ε
∞
0
2f 2
c 2
hf
exp (hf/(k B T )) − 1
df
(7.16)
= ε
2πh
c 2
k B T
h
4 ∞
0
x 3
exp (x) − 1
dx
(7.17)
= εσ T
4
(7.18)
where σ , ‘Stefan’s constant’, has the numerical value 27 5.6704 × 10 −8 W per square
meter per Kelvin −4 . The constant ε is the surface emissivity of the material emitting
light through thermal motion of its molecules.
27 Fortuitously for those who often use Stefan’s law, the digits of Stefan’s constant in the mks
system of units follow the sequence 5678, with the 8 being the negative power of ten.