Motion of an Electrified Sphere. 63 



pulse generated by the sudden change of velocity no longer 

 cuts the sphere. Dr. Paul Hertz * has obtained a complete 

 solution for the force required in starting the sphere at 

 any time from t = to t = 2a/(v—u), and, by extending 

 his method, I have found the force required to stop the 

 sphere f- 



§ 15. The momentum in the pulse can also be obtained by 

 direct integration. Since in the pulse H= uKVRE, it follows 

 that VEH = vKVEVRE, where R is a unit-vector along the 

 radius. But E is perpendicular to R, and thus VEH = rKE 2 R. 

 Hence VEH is in the same direction as R. Thus, by (40) the 

 momentum in the pulse is 



-&Sfr* 



xdydz, 



the integration extending throughout the volume of the 

 pulse. 



Since the thickness of the pulse is 2a, it follows that, if 

 r be the radius of the pulse and dco an element of solid 

 angle, 



V=f* a {wRda>. 



2ttvJ 



The value of E 2 is given in (3). For brevity we write 

 v — u x cos #! = A l5 v — ii 2 cos 6 2 — Ji 2 , 



• V 2 — 1^2 COS a =S. 



Noting that Q 2 /2Ka = U , we obtain 



The component of P parallel to u, is P., + P x cos a, with 

 the notation of § 14, and thus, since the angle between R and 

 n 2 is # 2 , we have 



* Untersuchungen iiber umtetige Bewegungen ernes Electrons. 



t ' On the Force required to stop a Moving Electrified Sphere," Proc 

 Royal Society, A. vol. lxxix. p. 550. In this paper 1 have given a sketch 

 ol Dr. Hertz s method and have stated his results. 



