426 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 



218. Forces on Conductor carrying Current. The 



magnetic energy of a pole m at P in the field due to a current is 



(i) 



where fl m is the potential due to the pole. For any number of 

 of poles, in like manner H m being the potential due to them all, 



(2) W = 



which is the flux of force through the current circuit in the 

 negative direction, due to all magnets. The potential energy 

 tends to decrease, consequently a current in a magnetic field 

 tends to move so as to make the surface integral a maximum, that 

 is, to embrace the largest possible number of tubes of force linked 

 with it in the positive direction. This statement of the mechanical 

 action of magnetic forces on a current is due to Faraday. 



219. Mechanical Force acting on Element of Circuit. 



We may consider the forces acting on the whole circuit as the 

 resultant of the forces acting on each element ds lt with the same 

 degree of arbitrariness as in the case of the field due to the current. 

 By the principle of reaction the force on ds 1 due to the presence of 

 a unit pole P must be equal and opposite to the force dL, dM, dN 

 on the unit pole, due to the current element ds^ Consequently if 

 dB,, dH, dZ, are the components of the mechanical force acting on 



ds, 



I 



da = -^ \dy-L (*| z) dz l (y-^ y)}, 



(3) dH =- {dz l (X -x)- dx^ (z, - z)}, 

 dZ = [dss l (2/1 y) dy-L (x x)}. 



T^ 



But 



1 #! x T 1 yi y ,., 1 z l z 



_ -\r 



w> 



are the components of the fiel,d at ds^ due to the unit pole at P. 



