POTENTIAL OF AN UNLIMITED CURRENT. 425 



We conclude from this that the potential of an unlimited recti- 

 linear current at a point is, within a constant, proportional to the 

 product of the strength by the apparent surface of a plane, unlimited 

 in one direction, and bounded in the other by the current 



In order to determine the sign of this apparent surface, we must 

 remember that, in practice, the unlimited rectilinear current neces- 

 sarily forms part of a closed circuit, and that if the non-rectilinear 

 portion is very distant from the point P, the angle under which the 

 whole circuit is seen, which we may suppose plane, only differs by an 

 inappreciable quantity from the unlimited plane of which it forms 

 part. We shall call that face of the current, which is on the left of 

 the observer placed in the current, and who is looking inwards, the 

 positive face ; the negative face being that on his right ; and we shall 

 take the angle w positive or negative, according as the positive or 

 the negative face is seen from the point P. 



446. THE POTENTIAL OF AN UNLIMITED CURRENT is NOT A 

 SIMPLE FUNCTION OF THE CO-ORDINATES. At a given point, the 

 angle w only gives the value of the potential of an unlimited current 

 to within a constant. It is easy to see what is the significance of 

 this constant. Suppose that a unit positive mass taken at the point 

 P (Fig. 96) describes a circumference about the point O, in the 

 direction of the force, and reverts to its original position. The angle 

 o> has resumed the same value, but the force </> has performed a 



ki 

 work <27T# that is to say, 2irki or 477 , and this mass has traversed 



2 



the plane of the current by the negative face. For n turns of the 



ki 

 mass, the work would be equal to 471-72 , and the potential would 



2 ki 

 have varied by the same quantity - 471-72 . 



ki 

 On the other hand, the expression w is the work which must 



be expended against magnetic forces, in order to bring this mass 

 from infinity to the point P, without traversing the plane of 

 the current. 



If then, by analogy with the properties of magnetic shells, 

 we call the potential at a point, the work necessary to bring a 

 positive magnetic mass equal to unity from an infinite distance, 

 this potential is expressed by 



/ \ TT / N 



(2) V = (0-47772 = (to -47772). 



