CHAPTER IV. 



SYSTEMS OF CONDUCTORS. 



98. IN the present Chapter we discuss the general theory of an electro- 

 static field in which there are any number of conductors. The charge on 

 each conductor will of course influence the distribution of charges on the 

 other conductors by induction, and the problem is to investigate the distri- 

 butions of electricity which are to be expected after allowing for this mutual 

 induction. 



We have seen that in an electrostatic field the potential cannot be a 

 maximum or a minimum except at points where electric charges occur. It 

 follows that the highest potential in the field must occur on a conductor, or 

 else at infinity, the latter case occurring only when the potential of every 

 conductor is negative. Excluding this case for the moment, there must be 

 one conductor of which the potential is higher than that anywhere else in 

 the field. Since lines of force run only from higher to lower potential ( 36), 

 it follows that no lines of force can enter this conductor, there being no 

 higher potential from which they can come, so that lines of force must leave 

 it at every point of its surface. In other words, its electrification must be 

 positive at every point. 



So also, except when the potential of every conductor is positive, there 

 must be one conductor of which the potential is lower than that anywhere 

 else in the field, and the electrification at every point of this conductor must 

 be negative. 



If the total charge on a conductor is nil, the total strength of the tubes 

 of force which enter it must be exactly equal to the total strength of the 

 tubes which leave it. There must therefore be both tubes which enter and 

 tubes which leave its surface, so that its potential must be intermediate 

 between the highest and lowest potentials in the field. For if its potential 

 were the highest in the field, no tubes could enter it, and vice versa. On 

 any such conductor the regions of positive electrification are separated from 

 regions of negative electrification by " lines of no electrification," these lines 

 being loci along which a = 0. In general the resultant intensity at any 



