460 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 



which we know to be a harmonic function, as <f> is conjugate to 

 the function log p, both being derived from the function log (x + iy). 

 In the substance of the conductors, there is no magnetic potential. 

 We may find the force by evaluating the expression for the vector- 

 potential, or as above, except that now the line of force does not 

 surround the whole current, but only a portion of it. If the 

 conductor is a solid cylinder of radius R, 



%7TpP = 4-7T . TTp^W = 4-7T/ -^ , 



(9) P-%p. . 



The integral (5) represents H only when //, has the same constant 

 value everywhere, for if it has discontinuities we must add a part 

 corresponding to the apparent current as shown in 228. In the 

 case just treated, however, the apparent current vanishes, for 

 the induction is tangent to the surfaces of the conductors. 



In the general case, if p is constant in the space outside of 

 the conductors there is a magnetic potential, and the equations 

 (2) and (3) become 





dy 'dx 



showing tfiat the function //.fl is conjugate to the vector-potential 

 H, which is accordingly the flux-function for the magnetic induc- 

 tion. The method of functions of a complex variable is accordingly 

 applicable to problems connected with the field of cylindrical 

 conductors. For instance, Fig. 65 represents for external points 

 the lines of force and equipotential lines of the field due to two 

 circular cylinders carrying equal currents in opposite directions. 

 No one of the circles in the figure however represents either of 

 the conductors, whose centers are at the points + a. The surface 

 of a cylindrical conductor is tangent to lines of force only when 

 it is alone in the field, or accompanied by concentric conductors. 

 Within conductors, although there is no magnetic potential, 

 equations (2) and (3) show that R is still the flux-function for 

 the induction. 



If 8 is the area of the cross-section of any conductor, the 

 vector-potential at any point, whether external or internal, is by 



