518 BELL SYSTEM TECHNICAL JOURNAL 



general not only a reactance component which is not quite proportional 

 to the frequency, but also a resistance component which does not vary 

 in any simple way with the frequency. On the basis of formula (8) 

 these facts are to be accounted for by the consideration that in general 

 the various filamentary currents in a wire are not only not in phase but 

 have no simple phase relations. Thus if (8) is written in the form 



Zab — icoJ^Y^L p4,J pJ^ = Rab + IwLab, (11) 



then Rab is not zero, and Rab and Lab vary with co although Lp^ does 

 not. 



That the self impedance Zaa of the wire A is not a pure reactance 

 can be accounted for similarly, with the additional reason that the 

 self impedance of each filament is complex, because of its resistance. 

 Thus if (7) is written in the form 



Zaa = HiRp + io^Lp)// + icoE E LpJpJ, = Ra + io:LA, (12) 



p p Qy^p 



then Ra is not zero, and Ra and La vary with co although Rp, Lp, Lpg 

 do not. 



It may be noted that in the idealized case of perfect conductivity 

 the mutual and self impedances of the wires would be pure reactances 

 and directly proportional to the frequency; for in this case the fila- 

 mentary currents in any wire would all be in phase and their distribu- 

 tion would be independent of the frequency. (The current dis- 

 tribution would be the same as the charge distribution and hence 

 purely superficial.) 



Case of Negligible Proximity Effect 



The case here considered is that in which the wires A and B are of 

 circular or of annular cross-section (but external to each other) and 

 are far enough apart so that the proximity effect > is negligible and so 

 that therefore the current distribution over the cross-section of each 

 wire is sensibly axially symmetrical. 



For this particular case the mutual impedance Zab = Zba is not 

 complex but is pure reactance, being equal to the mutual impedance 

 Za'b' = Zb'A' between two filamentary wires A' and B' having the 

 same interaxial spacing as the given " thick" wires ^ and B. Although 

 this statement is clearly true when only one of the wires is "thick," it 

 really needs a proof in the general case where both are "thick." The 

 following simple proof depends on the fact that everywhere (except 

 near its ends) outside of a long straight wire, of circular or of annular 

 section, carrying an axially symmetrical current the magnetic field 

 produced by that current is the same as though the current were con- 



