561-566] Magnetic Energy 491 



while the work done by the electrical forces during displacement is 

 which, by equation (521), is also equal to 



These two quantities would be equal and opposite if the system were a 

 conservative dynamical system acted on by no external forces. In point of 

 fact they are seen to be equal but of the same sign. The inference is that 

 the batteries supply during the motion an amount of energy equal to twice 

 the increase in the energy of the system. Of this supply of energy half 

 appears as an increase in the energy of the system, while the other half is 

 used in the performance of mechanical work. 



This result should be compared with that obtained in 120. 



565. As an example of the use of formula (521), let us examine the 

 force acting on an element of a circuit. Let the 

 components of the mechanical force acting on any 

 element ds of a circuit carrying a current i be de- 

 noted by X, Y, Z. 



To find the value of X, we have to consider a 

 displacement in which the element ds is displaced a 

 distance dx parallel to itself, the remainder of the 



circuit being left unmoved. Let the component of magnetic induction 

 perpendicular to the plane containing ds and dx be denoted by N, then if 

 T denotes the kinetic energy of the whole system, the increase in T caused 

 by displacement will be equal to i times the increase in the number of 

 tubes of induction enclosed by the circuit, and therefore 



dT = iNdsdx. 

 Thus, using equation (521), 



and there are similar equations giving the values of the components T and Z. 



If B is the total induction and if B cos e is the component at right angles 

 to ds, then the resultant force acting on ds is seen to be a force of amount 

 iB cos e ds, acting at right angles to the plane containing B and ds, and in 

 such a direction as to increase the kinetic energy of the system. This is a 

 generalisation of the result already obtained in 497. 



MAGNETIC ENERGY. 



566. We have seen that the energy of the field of force set up by a 

 system of electric currents must be supposed to be kinetic energy. We 

 know also that this field is identical with that set up by a certain system of 



