CHAPTER VIII. 



THE IDEA OF POTENTIAL AND ITS USE IN THE STUDY OF THE 

 ELECTRIC FIELD. 



99. Definition of velocity potential of a moving fluid. In the 



study of a moving fluid it is sometimes* helpful to imagine a sort 

 of a "hill" whose "slope" at each point is equal to the fluid 

 velocity at that point, and the "height" of this imagined hill 

 at a point is called the velocity potential of the moving fluid at 

 that point. Let Ah be the change of height of the potential hill 

 corresponding to the distance Ax measured along the base of 



Ah 

 the hill. Then is the slope of the hill, and according to 



the above definition this slope is equal to the velocity of the fluid 



Ah 



in centimeters per second. Therefore is expressed in centi- 

 meters per second, and if Ax is expressed in centimeters it is 

 evident that Ah must be expressed in centimeters squared per 

 second. It is evident, therefore, that the velocity potential of a 

 moving fluid is not a geometrical hill. Indeed, when a fluid is 

 moving in three dimensions the entire region occupied by the 

 fluid is, as it were, the "base" of the potential hill. A clear 

 idea of velocity potential in this case may be reached by imagining 

 the temperature in a region to vary from point to point in such a 

 way that the temperature gradient or temperature slope at each 

 point may represent the fluid velocity at that point in magnitude 

 and in direction. Then the temperature at each point represents 

 the height of the potential hill at that point. 



Definition of electric potential. In the study of electric field 

 distribution it is sometimes helpful to imagine a sort of hill whose 



* The simplification of the mathematical treatment of fluid motion by the use 

 of the idea of velocity potential is exemplified on page 246 of Franklin, MacNutt 

 and Charles's Calculus, published by the authors, South Bethlehem, Pa., 1913. 



165 



