Wien's displacement law relates a blackbody temperature T with the wavelenght l of its maximum emission: l(max) = a/T, where a is a constant. Let's calculate a.(adsbygoogle = window.adsbygoogle || []).push({});

Spectral radiance as a function of frequency v:

R(v) = [8(pi)h(v/c)^3] * 1/[exp(hv/kT)-1].

as a function of wavelenght l:

R'(l) = (8(pi)hc/l^5) * 1/[exp(hc/lkT)-1].

You can obtain this last formula from the previous one, writing:

Integral(0;+infinity)R(v)dv = Integral(0;+infinity)R'(l)dl

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Now we derivate the radiance, to obtain the point of maximum:

dR(v)/dv = 0 --> 1-(1/3)hv/kT = exp(-hv/kT) -->

1-(1/3)hc/lkT = exp(-hc/lkT) (1)

dR'(l)/dl = 0 -->

1-(1/5)hc/lkT = exp(-hc/lkT) (2)

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Solving for l equations (1) and (2) gives two different values of l !

(1) gives (numerically, Mathcad): hc/lkT = 2.82 -->

l(max) = 5.11*10^-3/T

(2) gives (numerically, Mathcad): hc/lkT = 4.97 -->

l(max) = 2.90*10^-3/T

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The values taken for computation:

k = 1.38*10^-23 J/°K

h = 6.63*10^-34 J*s

c = 3.00*10^8 m/s

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Why two values for a?

Since the blackbody colour depends on l(max), this question could also be put:

which is the real blackbody's colour?

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# Wien's displacement law. A paradox

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