MAGNETISM 



or the difference is the .same at all pairs of points one on each side 

 of the shell. 



Potential of pole m due to a shell. Since the potential 

 at any point of the unit pole is </>w, the potential of pole m is (/>mw. 

 We have seen that pole m sends out 4>7rm lines of force or unit tubes 

 through the whole sol id angle 4?r surrounding it ; therefore through 

 solid angle w it sends out 0?/iw, or its potential is </> x number of lines 

 of force it sends through the closed curve forming the rim of the 

 shell. Conversely the potential of the shell with respect to the 



pole is (j) x number of lines from the 

 pole passing through the shell. It is 

 easy to see that if lines of force which 

 enter the shell on the positive side are 

 reckoned positive, and if those which 

 enter on the negative side are reckoned 

 negative, the number will be tlu- same. 

 whatever the form of the shell, so long 

 as it has the same bounding curve. For 

 FIG. 171. let ABCDE and A IT, ( Fig. 1 71 ) repre- 



sent two forms of the shell. A line ( 'H 



which passes through the first shell at C in one direction and at It 

 in the other direction will not count in the sum ; while a line 1)F 



which passes only once through the first shell miM pass through 

 the second shell also, and will count equally for each. 



Potential^of a shell in any field/ If we regard the field 

 as made by a distribution of polarity, the potential of a shell will 

 be equal to </>X ^ lines sent through, summed up for e\cr\ element 

 of polarity, or 0xN, where N is the total number of lino sent 

 through. 



Force on a shell in any direction. Let a shell l>c 

 displaced parallel to itself through small distance < ./ in the gi\en 

 direction. Let the rate at which N. the total number of lino of 



force through the shell, increases in the given direction be 



Then the number paing through in the displaced position N 



d^ 

 N -f y-.cM' and the potential has been increased bv 



This is the work which must be done in the displacement. There 

 is therefore a force resisting the displacement equal to ~ or the 

 force tends to decrease N, and in the positive direction of oV is 



