2.6: Wien's Law
( \newcommand{\kernel}{\mathrm{null}\,}\)
The wavelengths or frequencies at which these functions reach a maximum, and what these maximum values are, can be found by differentiation of these functions. They do not all come to a maximum at the same wavelength. For the four Planck functions discussed in Section 2.6 (Equations 2.6.1- 2.6.4), the wavelengths or frequencies at which the maxima occur are given by:
For Equation 2.6.1:
λ=W1/T
For Equation 2.6.2:
λ=W2/T
For Equation 2.6.3:
ν=W3T
For Equation 2.6.4:
ν=W4T
Any of these equations (but more usually the first one) may be referred to as Wien's law.
The constants are
Wn=hckxn,(n=1,2)
Wn=kxnh,(n=3,4)
where the xn are the solutions of
xn=(6−n)(1−e−xn)
and have the values
x1=4.965114
x2=3.920690
x3=2.821439
x4=1.593624
The Wien constants then have the values
W1=2.8978×10−3 m K
W2=3.6697×10−3 m K
W3=5.8790×1010 Hz K−1
W4=3.3206×1010 Hz K−1
The maximum ordinates of the functions are given by
Mλ(max)=A1T5
Nλ(max)=A2T4
Mν(max)=A3T3
Nν(max)=A4T2
The constants An are given by
An=2πk6−nynh4c3,(n=1,2)
An=2πk6−nynh2c2,(n=3,4)
where the yn are dimensionless numbers defined by
yn=x6−nnexn−1
That is,
y1=21.20144
y2=4.779841
y3=1.421435
y4=0.6476102
The constants An therefore have the values
A1=1.2867×10−5W m−2K−5m−1
A2=2.1011×1017ph s−1m−2K−4m−1
A3=5.9568×10−19W m−2K−3Hz−1
A4=1.9657×104ph s−1m−2K−2Hz−1