467-470] Energy of a Magnetic Field 403 



permanent magnetisation at any point, the components of the total magnet- 

 isation at any point will be 



A = II . + KCL "I 



B = Im + Kj3 I (401), 



C=In 



and the components of induction are 

 a = a + 4-Tr J. = 



b = 4>7rlm + n0 (402). 



c = 47T/W -f yu/y ' 



For such a substance, it is clear that equations (395) and (396) will not 

 in general be satisfied. 



ENERGY OF A MAGNETIC FIELD. 



470. To obtain the energy of a magnetic field in which both permanent 

 and induced magnetism may be present, we return to the general equation 

 obtained in 451, 



Q .................. (403). 



On substituting for a, 6, c from equations (402), this becomes 



47T Iff I (la + m/3 + nj) dxdydz + 1TL (a 2 + /3 2 + 7 2 ) dxdydz = (404). 



Whether or not induced magnetism is present, it is proved in 448, that the 

 energy of the field is 



where the integral is taken through all space. This is equal to ^ times the 

 first term in equation (404). Thus 



a? + p + ^)dxdydz ............... (405). 



This could have been foreseen from analogy with the formula 

 W=- jjJK(X*+ T 2 + ^ 2 ) dxdydz, 

 which gives the energy of an electrostatic field. 



From formula (405) we see that the energy of a magnetic field may be 

 supposed spread throughout the medium, at a rate ^ per unit volume. 



262 



