418 The Magnetic Field produced by Electric Currents [CH. xm 



MAGNETIC POTENTIAL OF A FIELD DUE TO CURRENTS. 



486. Let us fix upon a definite equivalent shell to represent a current of 

 strength i. Let us bring a unit pole from in- 

 finity to any point A, by a path which cuts 

 the equivalent shell in points P, Q, ...Z. For 

 simplicity, let us at first suppose that at each 

 of these points the path passes from the 

 positive to the negative side of the shell, and 

 let the points on the two sides of the shell be 

 denoted, as before, by P + , P.; Q + , Q_; and 

 so on. 



Then, if H denotes the magnetic potential due to the equivalent shell, 

 the work done in bringing the unit pole from infinity to P + will be fl p+ . In 

 the limit P+ and P. are coincident, so that the work in taking the unit pole 

 on from P + to P_ is infinitesimal. In taking it from P_ to Q + work is done of 

 amount O Q+ O P _, from Q + to Q_, the work is infinitesimal, and so on, until 

 ultimately we arrive at A. Thus the total work done in bringing the unit 

 pole to A is 



n p+ + (0 Q+ - n p j + (n* + - n Q _) + . . . + (n A - n z _), 



or, rearranging, is 



&A + (flp + - HP.) + (<V - n Q _) + .... 



Now each of the terms ft p H P , n e+ H Q , etc. is equal by equation 

 (410) to 4nri, so that if n is the number of these terms, the whole expression 

 is equal to 



Replacing 1 A by io, where o> is the solid angle subtended by the shell at 

 A, we find for the potential at A due to the electric current 



(CD + 4<7rn)i .............................. (4H)- 



If the path cuts the equivalent shell n times in the direction from + to , 

 and ra times in the opposite direction, the quantity n must be replaced by 

 n m. 



Expression (411) shews that the potential at a point is not a single- valued 

 function of the coordinates of the point. The forces, which are obtained by 

 differentiation of this potential, are, however, single-valued. 



Current in infinite straight wire. 



487. As an illustration of the results obtained, let us consider the 

 magnetic field produced by a current flowing in a straight wire which is of 

 such great length that it may be regarded as infinite, the return current 

 being entirely at infinity. 



