324 CONSTITUTION OF MAGNETS. 



X, Y and Z being the components of the force of the field, A, //, and 

 v the direction cosines of the directions of magnetisation. The 

 expression for the elementary energy is therefore 



and hence the energy of the whole magnet is 

 (20) W= - 



If the field is uniform, the components X, Y and Z are constant. 

 If a, ft and 7 are the cosines of the angles of the force F with the 

 axes, we get 



W= - 



If K be the magnetic moment of the magnet, /, m and n the 

 cosines of the angles which the magnetic axis makes with the axes 

 of the co-ordinates, we have 



(MV 



= K/, 'Rdv = Km , cdv = Kn , 



and the energy becomes 



(21) W= -FK(al+ftm + yn)= - FK cos 8, 



8 being the angle which the magnetic axis makes with the direction 

 of the field. This result may be written directly. 



The energy is a minimum and equal to - FK, and therefore the 

 equilibrium is stable when the angle 8 is zero that is to say, when 

 the magnetic axis is parallel to the direction of the field. The 

 equilibrium is unstable if these two directions are opposite ; the 

 energy is then a maximum and equal to FK. The energy, lastly, 

 is zero if the two directions are at right angles. 



339. ENERGY OF A MAGNETIC SHELL. If the system is a 

 simple magnetic shell S, the magnetic moment of a surface-element 

 of the shell is <I?*/S and the value of its potential energy in the field 

 is 



the energy of the shell is therefore 



W= -</"( AX + ^Y 



