The magnetic particle corresponding to the mass particle is called a 

 magnetic dipole. We might think of a mass particle as being formed by starting 

 with a body of any shape and imagining its dimensions to shrink toward a 

 point while at the same time supposing its density to increase in such a way 

 that its mass remains constant. The idea of a magnetic dipole may be formed 

 in a similar way. 



P(X,Y,Z) 



Q(u,v,w) ^ ' (u + a, v + b, vv + c) 



Figure 27-11. 



Starting with two poles of strength p and — p, let the coordinates of the 

 negative pole be (u,v,w) and those of the positive pole (u + a, v + b, w + c) . 

 The potential V of such a pair at a point P(x,y,z) distant r and r from the 

 negative and positive poles respectively is given by (fig. 27-11) 



V = — --£■ (37) 



r r' 



By means of Taylor's expansion for a function of three variables, this may be 

 written 



V -p (a— + b— - + c — ) ( — ) + p (a — + p — + c — ) 2 — ) + 



or, if A = \a + \b + kc and v = i — — + j — — + k — — , (38) 



?>u 'dv ^xa 



V = P A • V (— ) + p (A • v )H— ) + (39) 



r r 



The vector m = pA is called the magnetic moment of the pole pair — p and p. 

 If now |A| be made to decrease and p to increase in such a way that m remains 

 constant, all terms after the first in (39) will approach zero and the limiting 



600 



