Impulsive Motion of Electrified Systems. 125 



motion. For the pulse due to the part of the system which 

 lies between / and the corresponding plane FF' has already 

 passed over /', while the pulse due to the remainder of the 

 system has not yet reached /. Jf Q x is the charge between 

 / and the plane FF' and Q 2 is the remainder of the total 

 charge, and if E x and E 2 are the electric forces due to Q x and 

 Q 2 , then the electric force at / has the three components E 1? 

 E 2 , and E, the latter being given by (11). The electric force 

 Eo has the value Q 2 /Kr 2 and is ultimately radial to 0, but the 

 force Ej is ultimately radial to 0' (fig. 2) and has the value 



^ 1_ ~KR 2 (l--n 2 sm 2 <£)f ' 



where E = 0'P and <f> is the angle between O'P and OX. 



The magnetic force at /has the three components H 1? H 23 . 

 and H, where H is given by (12) and 



H 1 =KwEisin^, H 2 =0. 



Both Hi and H are in circles in planes normal to OX. 



§ 9. The electric and magnetic forces in the pulse ultimately 

 vary as 1/r, while the thickness of the pulse, for given angular 

 coordinates of P, ultimately becomes constant, and hence the 

 electromagnetic energy in the pulse tends to a constant value, 

 Wj as vt, the radius of the pulse, becomes infinite. Since only 

 the squares of E and H are concerned, it follows that the 

 energy in the pulse is ultimately the same, whether the system 

 is suddenly set into motion at speed u or is suddenly reduced 

 to rest from speed u, the direction of u relative to the system 

 being the same in both cases. The energy W, which is 

 radiated away in the pulse, cannot be recovered and may 

 therefore be spoken of as (i lost." 



§ 10. We now proceed to show that the total energy of the 

 system, for steady motion at speed u, is simply equal to- 

 W + Uo, where U is the electrostatic energy of the system at 

 rest. The total energy at the speed uis U + T, the sum of the 

 electric and magnetic energies, where 



U^Wd,d,d, } T=^J, 



dydz. 



If the moving system be brought to rest suddenly, the 

 charges do not move under the influence of the electric forces, 

 which act on them after the system has been brought to rest, 

 and therefore the electromagnetic field does no work upon 

 the system after its motion is destroyed*. Hence the total 



* See M, Abraham, Thcorie dev Eleldrkitat, vol, ii. p. 230. 



