Electron necessarily Radiate Energy f 413 



that it is limited by certain moving boundaries, which form 

 the surfaces o£ the electrons, and inside which E and H are 

 prescribed to be permanently zero. We know by Poynting's 



theorem that, if the quantity ^— (E 2 + K 2 ) be identified with 



07T 



the energy density, the conservation of energy will hold 

 throughout the whole field except for the space included 

 within the boundaries ; for the Poynting flux is equivalent 

 to the transfer of this quantity from one part of the field to 

 another without net loss or gain. The only places, there- 

 fore, where energy can enter or leave the electromagnetic 

 system are the boundaries, so that if we calculate the 

 total flux of energy which is passing through them at any 

 time, we shall obtain the energy which is being introduced 

 into or is disappearing from the electromagnetic system 

 per unit of time. It must be noted that this is not necessarily 

 zero for a conservative system. Even in the simplest case, 

 that of the uniformly moving electron, considerations of the 

 continuity of the flux *, as well as those based on dynamics 

 and on relativity f, necessitate the postuiation of a definite 

 store of internal energy, of the amount given by (7), within 

 the nuclear boundarv. When the internal energies of the 

 electrons are included in the scheme, conservation in the 

 system will be characterized by the net flux of energy out- 

 wards across the boundaries being equal to the net rate of 

 decrease of the internal ^energies of the electrons. 



Confining attention to the positive electron and to the 

 positive x half of the field, we shall take the boundary to be 

 the spheroid given by the limit of (5) when a is infinitely 

 small. Let P be the Poynting flux outwards through a fixed 

 boundary momentarily coinciding with (5), and let v be the 

 ve]ocity outwards of any point of the boundary surface, then 

 the net flux outwards from the whole moving surface is 

 given by 



j(P-Wv)ndS, 

 where n is a unit vector outwards normal to the surface, and 



W = -^(E 2 + H 2 4 2Ecos6>.X + X 2 ) . . (13) 



07T 



is the energy density of the field at the point. Expressed in 



* Milner, Phil. Mag. Oct. 1920, p. 494. 



t Cf. Lorentz, < Theory of Electrons; § 180. 



