182 BELL SYSTEM TECHNICAL JOURNAL 



Case II — Current Uniformly Distributed in a Solid 

 Cylindrical Disturbing Wire 



Consider next the case where the total disturbing current is uni- 

 formly distributed over the cross section of a solid cylindrical wire. 

 Such a distribution exists exactly with direct current only, but is 

 closely approximated at very low frequencies. Since the magnetic 

 field outside of a conductor carrying a uniformly distributed current 

 is the same as would exist if the total current were concentrated in the 

 center filament, the total induced voltage in a filamentary wire located 

 in this field is again equal to — jwMI, where M is the same as in the 

 case of two filaments similarly located in space. 



Case III — Current Symmetrically Distributed in a Solid 

 Cylindrical Disturbing Wire 



The a.-c. distribution in a solid cylindrical wire is not uniform. 

 However, when the wire is at a considerable distance from its return, 

 the current distribution is practically symmetrical about the axis of 

 the wire although its density varies from a minimum value at the center 

 to a maximum value at the surface. Such a distribution is caused by 

 the fact that the counter-electromotive force induced in a filament 

 near the center of the wire due to the current in all of the other fila- 

 ments is greater than that induced in a filament at the surface. This 

 is the well-known skin eff^ect. 



In this case the total current, /, may be considered as distributed in 

 infinitely thin concentric rings in any one of which the current is the 

 same in phase and magnitude at all points. Since the field outside of 

 one such ring is the same as would exist if all of the ring current were 

 concentrated in a filament at the center, the total field due to the sum 

 of the currents in all the concentric rings is the same as would exist 

 if the total current were concentrated at the center of the wire. Thus, 

 the total voltage induced in a filamentary wire by the field set up 

 by a symmetrically distributed current in the disturbing wire is 

 again expressed by —jcoAfl, where M is again a pure number as in 

 the case of two filaments. 



Case IV — ^Current Unsymmetrically Distributed in a Solid 

 Cylindrical Disturbing Wire 



If a solid wire and its return are placed close together, as in a cable 

 pair, the a.-c. distribution is neither uniform nor symmetrical about 

 the axis of the wire. In this case the magnetic field set up by the 

 current in the return wire contributes to the counter-electromotive 



