Impulsive Motion of Electrified Systems. 127 



does the work U + T + W — U , and this, by (13), is equal to 

 2W. Thus, when a system is set into motion, half the work 

 done by the applied force goes to increase the energy of the 

 field, while the other half is lost by radiation. 



§ 12. It can be shown that the force exerted on any system 

 of charges by the electromagnetic field together with the 

 force 



»=SfJ™^> 



where the integral is taken throughout the space within any 

 surface S which encloses the system, is equivalent to the 

 resultant of the Maxwell stress over S. Here, and throughout 

 the paper, heavy type signifies vectors and YEH denotes the 

 vector product of E and H. The differentiation with regard 

 to the time is to be performed on the values of E and H at 

 points fixed in space. 



When a system, which has been at rest from t = — go to 

 £ = 0, is made to move in any given manner, from £ = 

 onwards, by the application of a set of mechanically applied 

 forces, we can take the surface S so great that, at any time t, 

 the field at every point of S is the old statical field due to 

 the system at rest, and then the resultant of the Maxwell 

 .stress over S vanishes *. In this case, the resultant force 

 exerted by the electromagnetic field upon the system is — Gr 

 and hence, if the system is to move in the given manner, 

 there must be applied to the system by non-electromagnetic 

 means, in addition to any force required to change the 

 momentum of ordinary matter, a resultant force F, where 

 F = Gr. Hence we find that, to produce the given motion, we 

 require a force F, where 



F =5:iOJ vEH ^-^^ • • • (»> 



in addition to any force required for the motion of ordinary 

 matter. 



The force F is measured, on Newton's system, by the rate 

 at which it could generate momentum in ordinary matter. 

 Hence we may speak of F as generating momentum in the 

 electromagnetic field and may say that the expression 



gjJjVl3H^& .... (15) 



is the momentum of the electromagnetic field. 



* It is here supposed that there is no other system within a finite 

 distance of the one under consideration. 



