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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=(6n)(1exn)

and have the values

x1=4.965114

x2=3.920690

x3=2.821439

x4=1.593624

The Wien constants then have the values

W1=2.8978×103 m K

W2=3.6697×103 m K

W3=5.8790×1010 Hz K1

W4=3.3206×1010 Hz K1

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=2πk6nynh4c3,(n=1,2)

An=2πk6nynh2c2,(n=3,4)

where the yn are dimensionless numbers defined by

yn=x6nnexn1

That is,

y1=21.20144

y2=4.779841

y3=1.421435

y4=0.6476102

The constants An therefore have the values

A1=1.2867×105W m2K5m1

A2=2.1011×1017ph s1m2K4m1

A3=5.9568×1019W m2K3Hz1

A4=1.9657×104ph s1m2K2Hz1


This page titled 2.6: Wien's Law is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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