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14 Classical Mechanics and the Old Quantum Theory
E X A M P L E 14.2
Show that the wavelength of maximum spectral radiant emittance is inversely proportional
to the absolute temperature and find the proportionality constant.
Solution
To find the maximum, we set the derivative of the function of Eq. (14.4-5) equal to zero:
dη
dλ
2πhc 2 (hc/λk B )Te hc/λk B T − 5(e hc/λk B T − 1)
λ 6 [e hc/λk B T − 1]
0
This expression can vanish only if the numerator vanishes, which is equivalent to
hc
λ max k B T
5
1 − e −hc/λ max k B T
or, if we let x hc/λ max k B T , then
x 5
1 − e −x
This equation must be solved by numerical approximation. The result is that x 4.965, so
that
λ max
1
4.965
hc
k B T
2.898 × 10 −3 mK
T
(14.4-6)
This result agrees with Wien’s law and with the experimental value of the proportionality
constant.
E X A M P L E 14.3
Use the definite integral
∞
0
x 3
e x − 1
dx
π 4
15
(14.4-7)
to derive the Stefan–Boltzmann law, Eq. (14.4-1). Calculate the theoretical value of the
Stefan–Boltzmann constant.
Solution
The total energy flux is
∞
0
η(λ) dλ
∞
0
2πhc 2
λ 5 (e hc/λk B T − 1)
dλ
We make the variable change
x
hc
λk B T
so that
∞
0
2πhc 2
λ 5 (e hc/λk B T − 1)
dλ 2πhc 2
k B T
hc
5 hc
k B T
∞
0
x 3
e x − 1
dx