Page:CunninghamExtension.djvu/9

3. The Mechanical Action of Radiation on a Bounding Surface under the Lorentz-Einstein Transformation.

M. Planck^[1] in a recent paper has shewn, from thermodynamic considerations, that the pressure of radiation in equilibrium in an enclosure must be independent of the motion of that enclosure, and obtains relations between the energy per unit volume, and the temperature of the radiation when considered at rest and in motion in turn. Inasmuch as energy and pressure are independent of thermodynamics, it seems appropriate to shew here that Planck's results for those quantities are an immediate result of the foregoing transformation.

Taking the expression for the force per unit area on a moving surface,^[2]

$p'={\frac {1}{8\pi }}\left\{2e'e_{\nu }+2h'h_{\nu }-n(ee'+hh')\right\},$

this being a vector equation, we adapt it to the case where the surface is moving in the direction of the axis of x with velocity v.

The equations of the transformation are

${\begin{array}{lll}e'_{x}=E_{X},&e'_{y}=E_{Y}/\beta ,&e'_{z}=E_{Z}/\beta ,\\h'_{x}=H_{X},&h'_{y}=H_{Y}/\beta ,&h'_{z}=H_{Z}/\beta ,\end{array}}$

Let the element of area upon which p' acts be ds, the normal having direction cosines (l, m, n), and let the transformed area be dS, with normal (L, M, N).

Then

$lds=LdS,\ \beta mds=MdS,\ \beta nds=NdS.$

Making these substitutions in p' and reducing, we obtain

$p'_{x}ds=P_{X}dS-{\frac {v}{4\pi c}}[EH]_{N}dS,$

where $P_{X}$ is the component pressure per unit area on dS at rest as given by the electric and magnetic intensities E, H.

Likewise

$p'_{y}ds={\frac {1}{\beta }}P_{Y}dS,$ $p'_{x}ds={\frac {1}{\beta }}P_{Z}dS.$

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1. Annalen der Physik, xxvi, No. 6 (1908).
2. Cf. Abraham, Theorie der Elektricität, II, p. 333. Notice that in this expression
   $e'=e+[wh]/c,\ h'=h-[we]/c,$

   so that e', h' are not the electric and magnetic intensities in the Lorentz-Einstein transformation.