SOME MAGNETIC FIELDS 



The shell, therefore, tends to move into the position in which 

 as many lines of force as possible pass through in the negative 

 direction. This can easily be remembered if it is considered that 

 it will tend to set with its negative face towards a positive pole. 



The resultant force on the shell evidently acts in the direction 

 in which a given motion produces the maximum decrease in N. 



Representation of the force by elementary forces 

 each acting on an element of the rim of the shell. 

 The value of N can alter only by lines moving in or out across the 

 boundary curve of the shell as it is displaced. This is seen at once 

 if we imagine the lines of force to be materialised into wires. 

 The number can only be altered by the rim cutting through the 

 wires. We may expect then that the force on the whole shell can 

 be represented by forces acting at the rim where the cutting takes 

 place. 



Let us suppose that the force on an element $s of the boundary 

 in direction &r is 0?*, where n is the number of lines cut by Ss per 

 unit displacement in the direction &x. Then the total force is 



0Zra = -j-, since 2n must be equal to -^. Then we may 



regard the total force </> -=- as the resultant of forces such as 



(uC 



n on the elements of the boundary. The direction in which the 

 force on each element acts is that in which a given motion will make 

 n the greatest. 



If H, Fig. 172, represents the field at an element Ss, it is evident 

 that n is the greatest when &r is perpendicular to H and to Ss. Then 



H 



* 



F 



FIG. 172. 



the direction of the resultant force F on the element is along the 

 normal to the plane through Ss and H. The number of lines of 

 force cut per unit motion of Ss along F is H sin 9Ss, where 6 is the 

 angle between H and Ss. Then F = 0H sin BSs. 



We shall see when we study the magnetic field of a current that 

 the current equivalent to a shell circulates counter-clockwise round 

 the rim when seen from the side towards which the positive layer 

 faces. Hence in the figure we may assign to ^considered as an 

 element of the current, the direction inwards through the paper wheii 

 the shell lies above Ss with the positive layer to the right. The force 



