428 The Magnetic Field produced by Electric Currents [OH. xm 



Thus the force from the element ds may be taken to be a force 

 having components Xds, Yds, Zds, and this again is a force of magnitude 



ss * n , where 6 is the angle between ds and r, the direction of the force 



being at right angles to the plane containing ds and r. This is Ampere's 

 expression for the force from an element of electric current. 



Mechanical action upon a circuit. 



497. Since action and reaction are equal and opposite, a unit pole at P 

 must exert upon the whole circuit a force of which the component in the 

 direction I, m, n will be 



ds\ ds ds 

 where F, G, H are given by equations (420), etc. 



The force exerted on the circuit by any magnetic system can be obtained 

 by addition of such expressions, the magnetic system being regarded as an 



aggregate of simple poles. If fl 1 = ^ ) ^ s the potential of the magnetic 

 system, the force exerted on the circuit in direction I, ra, n is found to be 



where F' = f m r n ^~ 



dz dy 



d l ~ .(423). 



dy i 



The force given by expression (422) can be regarded as made up of a 

 force 



V ds ds 



per unit length. Thus the components of mechanical force acting on the 

 circuit per unit length, which are the coefficients of I, m, n in this last ex- 

 pression, will be 



^ . /an dz an a?/\ 

 & M ^~~ ^ ~^~ ^r > e tc. 



Vdy ds oz csj 

 Let IT be the magnetic intensity at the point, and let l lt m l , n be its 



direction-cosines. Let Z 2 , m 2 , n z be the direction-cosines ~- * * 5~ < of c?s. 



cs ds ds 



Then 



