368 Permanent Magnetism [CH. xi 



If we take the line drawn from the first magnet to the second as pole in 

 spherical polar coordinates, and denote the azimuths of the axes of the two 

 magnets by T/T, -*//, then the polar coordinates of the directions of the axes of 

 the two magnets will be 0, -fy and ', ty' respectively, and we shall have 

 cos e = cos cos 0' + sin sin & cos (ty x/r'). 



On substituting this value for cos e in equation (354), we obtain 



/ 



W = -~ {sin sin 0' cos (-\Ir il/) 2 cos cos 6'} (355). 



r 3 l 



422. Knowing the mutual potential energy W, we can derive a know- 

 ledge of all the mechanical forces by differentiation. For instance the 

 repulsion between the two magnets, i.e. the force tending to increase r, is 

 _d_W 



^ {sin sin 0' cos (ty A^') 2 cos cos 0'}. 



Thus, whatever the position of the magnets, the force between them 

 varies as the inverse fourth power of the distance. 



If the magnets are parallel to one another, 0' and -^ = -vjr', so that the 

 repulsion 



(sin 2 - 2 cos 2 0). 



C* ? 



the force is an attractive force -~- . When = -^ , so that the magnets are 



Thus when = 0, i.e. when the magnets lie along the line joining them, 



its ar< 



at right angles to the line joining them, the force is a repulsive force -~- . 



In passing from the one position to the other the force changes from one of 

 attraction to one of repulsion when sin 2 2 cos 2 = 0, i.e. when = tan" 1 \/2. 



The couples can be found in the same way. If % is any angle, the couple 



dW 

 tending to increase the angle % is -= , or 



~ i*" ir ' sin e sin & cos W - ^') ~ 2 cos cos &}> 



so that all the couples vary inversely as the cube of the distance. 



For instance, taking % to be the same as -^r, we find that the couple tend- 

 ing to rotate the first magnet about the line joining it to the second, in the 

 direction of i|r increasing 



9 W IJLuf ./!./!//, , ,x 



= - = ^ sin 6 sin sin (^ - ^'), 



so that this couple vanishes if either of the magnets is along the line joining 

 them, or if they are in the same plane, results which are obvious enough 

 geometrically. 



