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Ludwig BoltzmannThe Man Who Trusted Atoms$

Carlo Cercignani

Print publication date: 2006

Print ISBN-13: 9780198570646

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198570646.001.0001

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(p.294) Appendix 9.1 The Stefan–Boltzmann law

(p.294) Appendix 9.1 The Stefan–Boltzmann law

Source:
Ludwig Boltzmann
Publisher:
Oxford University Press

As indicated in the main text, Boltzmann’s proof of the law suggested by Stefan on the basis of rather inaccurate experimental data rests on the concept of radiation pressure. Let us imagine an enclosure closed by a (slowly) movable piston with a reflecting surface. Electromagnetic waves exert a pressure on the piston which, as indicated in the text, is p = e/3. This pressure is due to the momentum which the electromagnetic field carries with it, according to Maxwell’s equations. Since this momentum density has a magnitude g = e/c, where c is the speed of light, the pressure can be computed as was done in Chapter 3 for a gas; the only difference is that the speed of the waves is constant and equal to c. The fact that we obtain p = e/3, rather than p = 2e/3 as in the kinetic theory of gases, is due to the fact that e = gc rather than e = gc/2 (light cannot be treated as a non-relativistic particle).

We can then write, if V is the volume of the enclosure and S its entropy:

Appendix 9.1 The Stefan–Boltzmann law
Since we have computed the differential of entropy, we can deduce its partial derivatives with respect to T and V:
Appendix 9.1 The Stefan–Boltzmann law
and hence, using Schwarz’s theorem on the mixed second derivatives, equate the derivative of ∂S/∂V with respect to T to the derivative of ∂S/∂T with respect to V, thus obtaining
Appendix 9.1 The Stefan–Boltzmann law
or, simplifying:
Appendix 9.1 The Stefan–Boltzmann law
and, integrating by separation of the variables:
Appendix 9.1 The Stefan–Boltzmann law
where σ is an integration constant. An easy argument also gives S = (4/3)σVT 3.