168 ADVANCED ELECTRICITY AND MAGNETISM. 



let E volts be the electromotive force or potential difference 

 between these points Then Eq joules is the amount of work 

 done on the charge by the electric field. Of course the charge 

 may be thought of as transferred along a wire. 



The actual physical conditions in an electric field are com- 

 pletely specified when the intensity of the field is everywhere 

 given. Therefore the potential hill can be determined only in 

 that its slope can be everywhere known. It is evident, however, 

 that a hill is not completely determined when its slope is every- 

 where known; its height at some point must be given. There- 

 fore a complete specification of the electric potential in any given 

 case involves an arbitrary and physically meaningless choice of a 

 region of zero potential. When the region of zero potential 

 has been chosen, then the electric potential at any point p is 

 the electromotive force E between that point and the arbi- 

 trarily chosen region of zero potential; and if a charge of q 

 coulombs is carried from p to the region of zero potential, the 

 work done on the charge by the electric field will be Eq joules, 

 or the work done per unit of positive charge will be equal to E. 

 Therefore the electric potential at a point p is equal to the work 

 done by the electric field on one unit of positive charge which is 

 carried from p to the region of zero potential.* 



100. Potential values in the neighborhood of a uniformly 

 charged sphere. As an example of the use of the idea of poten- 

 tial, let us determine the distribution of potential in the neighbor- 

 hood of a uniformly charged sphere 5, Fig. 113; the charged 



J 



AJC 



Fig. 113. 



sphere being at a great distance irom all surrounding objects. 

 Let us choose the region infinitely remote from S as the region 



* A very good discussion of the mathematical theory of potential is given on 

 pages 210-253 of Franklin, MacNutt and Charles' Calculus. 



