74 STATIC ELECTRICITY 



the circuit BACDB, we shall get a supply of energy each time 

 we go round. This could only be accounted for by a gradual 

 discharge of the system due to the motion, and experiment shows 

 that no such discharge occurs. Hence the work along AB equals 

 the work along CD. As a particular case the work from A to B 

 is the same by all paths, such as AEB or AFB. If then we 

 choose some convenient surface as having zero-level, corresponding 

 to sea-level with gravity, the level of any other point, measured 

 by the work done in carrying the unit positive charge from zero- 

 level to that point, is definite, whatever be the path pursued. 

 Level as thus estimated is termed Potential, and we have the 

 following definition : 



The potential at a point is the work done in carrying a small 

 body with unit positive charge from zero-level to the point. 



The potential at a point is usually denoted by V. 



It is evident that the work done in carrying unit charge from 

 a point A at potential V A to a point B at potential V B is V B V A . 

 For if O is a point in the zero-level, V A is the work done in going 

 from O to A, V B that in going from O to B, and we may go to H 

 through A. Hence the work along AB is V V A . 



Alteration of surface chosen as zero -level or zero- 

 potential. If we alter our starting-point or zero-level to one at 

 potential U referred to the old starting-point, the potential at 

 every other point is evidently decreased by U. 



Equipotential surfaces. The surfaces which we have 

 hitherto termed level, surfaces everywhere cutting the lines of 

 force at right angles, are also equipotential surfaces. Drawing a 

 series of such surfaces corresponding to potentials 0, 1, 2, 3, &c., we 

 may map out the variations of level in the system. 



A conducting surface is an equipotential surface, and any closed 

 conductor containing no electrification within it is all at one 

 potential, for there is no component of the intensity along the 

 surface, and if there is no charge within there is no intensity 

 within, and no work is done in taking unit charge from one point 

 to another. 



The intensity at a point in any direction expressed 

 in terms of the rate of change of potential. Let A be the 

 given point and B a point very near it such that AB is the direction 

 in which the component of the intensity is to be expressed. Then 

 the work done in carrying the unit charge from A to B is V B V A ; 

 but if X is the component of the intensity along AB, the component 

 along BA is X, and this is the force against which the unit charge 

 is carried. 



Then - X.AB = V - V A 



or, using the notation of the differential calculus and putting 

 AB = dx and V A - V B = dV, 



- X.cLr = d\ 



