158 



temperature, or E = k(T* — T*). The law has been amply verified for a 

 black body by the experiments of Lvimmer and Pringsheim'', but is not strictly 

 true for other bodies. Boltzmann deduced the law theoretically in the fol- 

 lowing manner. 



According to the electromagnetic theorj- of light, light exerts a pressure 

 on any perfectly reflecting plane surface which is perpendicular to the direc- 

 tion of the light numerically equal to the energy density of the radiation. 

 When light is incident in all directions we may assume one-third of it travel- 

 ing in each of three mutually perpendicular directions, and so the pressure 

 exerted upon the walls of a perfectly reflecting vessel filled with radiation 

 would be equal to one-third of the energy density. Let AC (Fig. 1) be a 

 cylinder of unit cross-section and length a, having perfectly reflecting sides 

 and a perfectly reflecting piston P, but the end AB is a perfectly black body 

 at a temperature T. Then the space between AB and P will be filled with 

 radiation of energy density (/ corresponding to the temperature T. When 

 equilibrium is established replace AB bj' a pei-fectly reflecting plate, and 

 push the piston P in from a point distant x from .\B to a point distant (.r — dx). 

 The total amount of energy su])plied. ilQ, is ecjual to the increase of the in- 

 ternal energy, ill', plus the external work perfoiined, dW. Therefore 



4 

 d(2 = (ir + d\V = flfxii) + ]) <1\- = X du + -■>. dx. 



If '^ is the entroj)y, then 



dQ X 4u d? d;T) 

 d? = = — du -1 dx = — du H dx. 



T T :rr du dx 



X d? 4u d? <i-'. d fxl d J 4u 



''t du' :VV dx' dudx dxlTj du'sTJ 



Since T is indcpcnilent of .r 



1 4f 1 udT I du 4<ri' 



T sIt TMuJ' u T 



(1) u = kT'', where A- is a constant. 



Suppose that instead of the case above considered we take the case of a 

 small radiating body at the center of a hollow sphere having perfectly re- 

 flecting walls. Then ur- will be constant, where ;• is the distance from the 



center of the sphere. 'I'lien I' = 4x I ur-dr = 4-r''u. ii being in this case the 



•^ o 

 energy density at the surface of the sphere. The radiation pressure on the 



'.\nn. (Icr Physik, 0.3, p. .395, 1807. 



