A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 553 



On Magnetic Equipotential Surfaces. 



(51) If we explore the field with a uniformly magnetized bar, so long that 

 one of its poles is in a very weak part of the magnetic field, then the mag- 

 netic forces will perform work on the other pole as it moves about the field. 



If we start from a given point, and move this pole from it to any other 

 point, the work performed will be independent of the path of the pole between 

 the two points ; provided that no electric current passes between the different 

 paths pursued by the pole. 



Hence, when there are no electric currents but only magnets in the field, 

 we may draw a series of surfaces such that the work done in passing from one 

 to another shall be constant whatever be the path pursued between them. Such 

 surfaces are called Equipotential Surfaces, and in ordinary cases are perpendicular 

 to the Lines of magnetic force. 



If these surfaces are so drawn that, when a unit pole passes from any one 

 to the next in order, unity of work is done, then the work done in any motion 

 of a magnetic pole will be measured by the strength of the pole multiplied by 

 the number of surfaces which it has passed through in the positive direction. 



(52) If there are circuits carrying electric currents in the field, then there 

 will still be equipotential surfaces in the parts of the field external to the con- 

 ductors carrying the currents, but the work done on a unit pole in passing 

 from one to another will depend on the number of times which the path of 

 the pole circulates round any of these currents. Hence the potential in each 

 surface will have a series of values in arithmetical progression, differing by the 

 work done in passing completely round one of the currents in the field. 



The equipotential surfaces will not be continuous closed surfaces, but some 

 of them will be limited sheets, terminating in the electric circuit as their common 

 edge or boundary. The number of these will be equal to the amount of work 

 done on a unit pole in going round the current, and this by the ordinary 

 measurement =4iry, where y is the value of the current. 



These surfaces, therefore, are connected with the electric current as soap- 

 bubbles are connected with a ring in M. Plateau's experiments. Every current 

 y has i-ny surfaces attached to it. These surfaces have the current for their 

 common edge, and meet it at equal angles. The form of the surfaces in other 

 parts depends on the presence of other currents and magnets, as well as on 

 the shape of the circuit to which they belong. 



VOL. i. 70 



