498 The Internal Energy of the Lorentz Electron. 



electron with a dilemma, in that the idea of a continuous 

 flux of energy is inconsistent with the view that the energy 

 o£ the field is everywhere electromagnetic (i. <?., is of the 



form — - ). For, unless denial is made of the con- 



tinuity through space of the flux, the circuit must be 

 imagined to be closed through the nucleus, e.g., as indicated 

 by the dotted lines. We can exclude, as practically iden- 

 tical with that of the discontinuity of the flux, the supposition 

 that it traverses the nucleus with an infinite velocity; conse- 

 quently there must be a finite energy density in the nucleus, 

 in spite of the fact that E and H are zero there. Thus 

 accepting the continuity of the Poynting flux it is necessary 

 to postulate the presence of non-electromagnetic energy in 

 the nucleus. The equations (4) and (5) of an electro- 

 magnetic field are in this respect inconsistent with each 

 other inside the nucleus, and one of them must be supple- 

 mented or modified. There is, however, no advantage to 

 be gained by sacrificing both. If we make the supposition 

 that Foynting's expression forms a valid measure of the 

 energy flux in every part of an electromagnetic field, we 

 not only retain as much of the original theory as is possible, 

 but we can deduce a value of the internal energy of the 

 electron which is in complete accord with that necessitated 

 by the mechanical theory. The supposition implies that 

 in the interior of the nucleus, since P vanishes along with 

 E and H, the energy present (whatever its nature) is in a 

 condition of zero flux through a plane fixed in space — or, in 

 other words, is (virtually) at rest *. 



It is evident from the figure that the whole ivu flux which 

 passes through the moving equatorial section of the nucleus 

 must be equal and opposite to the whole flux through the 

 moving equatorial plane of the external field, or to 



f 



xJ a 



(wu) .2irrdr, 



6 =2 



which reduces to u.-^— O on inserting the values of iv and u 



from (4) and (7). If this flux is produced by the equatorial 

 section of the nucleus moving with velocity v past energy 



* This view agrees with the attribution of the energy to the Poincare 

 internal stress, for the activity of the stress in the moving nucleus will 

 give an energy current of — Tv, which will just counteract the convective 

 •current -\-Tv, where T is the stress or the (equal) energy density. 



