10 ON POTENTIAL. 



A to B. The work of the electrical forces which corresponds to this 

 displacement, is independent of the path following in passing from 

 A to B. This is a necessary consequence of the hypothesis (19) that 

 the forces are central ; for if it were otherwise, it is obvious that, by 

 moving an electrical mass on suitable paths between the points 

 A and B, we might produce an indefinite quantity of work, without a 

 corresponding expenditure. The work in question only depends, 

 then, on the co-ordinates of the points A and B ; it is equal to the 

 difference of the values V A and V B which the same function V has at 

 these two points, and representing this work by W, we may write 



(0 W>V A -V B . 



The function V plays a paramount part in the study of electrical 

 phenomena ; it has been called potential. As this function is only 

 denned by an integral, its value is only determined to within a 

 constant, and the variations are measured by the electrical work. 



From equation (i), the excess of the potential at a point A over the 

 potential at B, is equal to the work done by the electrical actions on unit 

 mass in passing from A to B ; or conversely, it is the work which 

 must be expended against electrical force to move this mass from B to A. 



If unit mass moves along a line of force, the work for an infinitely 

 small displacement ds is fds t and the total work from A to B is 

 expressed by 



(2) W' 



23. EQUIPOTENTIAL SURFACES. ELECTROMOTIVE FORCE. A 

 level surface, or equipotential surface, is a surface perpendicular at 

 every point to the direction of the force ; that is to say, a surface 

 which is perpendicular to all the lines of force which it meets. In 

 the case of central forces, a surface satisfying this condition can 

 always be drawn through any given point. If an electrical mass 

 moves along such a surface, the elementary work is constantly zero, 

 for the force is always perpendicular to the displacement. The 

 potential has the same value for all points of the same electrical level. 



Let us consider two equipotential surfaces, ! and S 2 , whose 

 potentials are respectively V 1 and V 2 . The work corresponding to 

 the displacement of unit mass from a point of the former to a point 

 of the latter, has the value V x - V 2 ; it is independent of the path 

 traversed, and even of the position of the point of departure from, 

 and of the point of arrival at, the two surfaces. 



