POTENTIAL ENERGY OF MAGNETS. 323 



On the other hand, the functions V and tt are identical for all 

 points external to the magnetised media points for which the induc- 

 tion and the magnetic force are themselves identical. Hence the 

 function 



= f($_ 



may be considered as the potential of magnetic induction of a lamellar 

 magnet. 



338. POTENTIAL ENERGY OF MAGNETS. The general expression 

 for the energy of a permanent magnet in a magnetic field produced 

 by an invariable system, where m is the magnetic mass situate at the 

 point where the potential of the field is V, is 



or again, as a function of the surface density and of the volume 

 density of the magnetism, 



= / 



This energy is the work which must be expended to bring the 

 magnet in question from an infinite distance to the position which 

 it occupies, or conversely the work done in moving it to an infinite 

 distance. 



In order to express the energy as a function of the intensity 

 of magnetisation, we must replace the densities by their known 

 values ; but it is simpler to consider the problem directly. A volume 

 element dv, the magnetic moment of which is Idv, is equivalent to a 

 small magnet of mass m, and length ds y parallel to the direction of 

 magnetisation. If V and V are the potentials of the field at the 

 points at which are the masses -m and +;;z, the energy of this 

 element of volume is 



dW = m(V - V) = mds^-^-=ldv . 

 ds & 



If 8 be the angle which the direction of magnetisation makes 

 with the direction of the field, and dn the perpendicular distances 

 of the two equipotential surfaces V and V at the point in question, 

 we have 



dV dV 



y- = cos 8= -Fcos 8= - 



Y 2 



