226 228] ELECTROMAGNETISM. 441 



Consequently we get 



fff~,j ffff (W dF \ f dF 

 (2) &d,T= l/MvU -- -5- - w U --- 



= fff(vN-wM)dr, 



and we obtain for the mechanical forces on the conductor per unit 



volume, 



H = vNwM, 



(3) H=wL-uN, 



Z=uM- vL. 



The mechanical force per unit volume is the vector product of the 

 current density and the magnetic field. 



228. Effect of Heterogeneous Medium. Let us consider 

 what changes are necessitated in our equations by the presence of 

 magnetizable bodies, so that the magnetic inductivity //, is not con- 

 stant throughout space. In the reasoning of 210 we supposed 

 the magnetic force to be both lamellar and solenoidal in all space 

 not traversed by currents. As soon as we have variations in the 

 inductivity, the force is in general no longer solenoidal, but the in- 

 duction is. We cannot, however, apply the reasoning unchanged to 

 the induction, for this, in general, is not lamellar. The reasoning 

 connecting the current strength with the work of carrying a pole 

 around a closed circuit is however unchanged, and if the circuit lie 

 in any other medium than air, the work is the same as if the circuit 

 lay in air, namely zero if the circuit is not linked with the current, 

 kirnl if linked n times positively. For consider a circuit composed 

 of two infinitely near circuits each embracing the current once, 

 corresponding points of the two lying infinitely near each other on 

 opposite sides of a surface separating air from another medium. 

 Then if we carry a pole around the circuit in air in one direction, 

 and back around the circuit in the other medium in the opposite 

 direction, since the double circuit is not linked with the current no 

 work has been done. For otherwise, in going around the double 

 circuit in one direction or the other, we might store up energy, as 

 much as we pleased, by repeating the operation. But this would 

 be in opposition to the principle of conservation of energy, which 

 says that the energy is definitely determined when the positions 

 and strengths of poles and currents are given. Consequently our 



