58 Mr. Gr. F. C. Searle on the Impulsive 



§ 14. Now consider the impulsive change of velocity from 

 Hi to u 2 . Before the force F 12 acted the energy in the 

 electromagnetic field was U1+T1, and after the force has 

 acted the energy is U 2 4- T 2 + W. Hence, by § 13, the gain of 

 energy is W 2 — Wi + W, and this must be equal to the work 

 done by the force. Thus, in vector notation, 



-W 2 -W ] +W=^ 12 u 2 dt 



=^F 12 dt, (35) 



since u 2 is constant, the last expression being the scalar 

 product of u 2 and xF^dt. 



Similarly if F 21 be the force required to change the velocity 

 from u 2 to Uj, we have 



W 1 -W 2 + W=u 1 jF 21 ^, .... (36) 



since the energy radiated is the same for the change from 

 u 2 to Ui as for the change from iij. to u 2 . 



From these equations of energy we pass to the equations 

 of momentum. When the sphere is in steady motion, the 

 resultant momentum of the electromagnetic field is in the 

 same direction as the velocity of the sphere. If Mx be 

 the momentum when the velocity is u t and M 2 when the 

 velocity is u 2 , it is known that * 



M, 



Uoi 5111V! og £±_-i_ 2 ]. . . (37) 



2u x l Vll ± °V-W! J v y 



M U^r^+Vj og £±!L._2}. . .. (38) 



zu 2 (. vu 2 G v—u 2 J v 7 



Since ?« 1 = w 1 v, we can write 



».->B(^.)'°^-a 



If we prefer to do so, we may express Mj in terms of m v 

 by means of the formula U /v = 3m t'/4, where m is the 

 electromagnetic mass for infinitely slow motion. 



* These results may be deduced from § 19 of my previous paper, 

 Phil. Mag. Jan. 1907. 



