MUTUAL IMPEDANCES OF PARALLEL WIRES 515 



composing the wires, or to wires which themselves are fine enough to be 

 regarded as filamentary.^ 



The self and mutual impedances of the wires could be formulated 

 in terms of the self and mutual impedances of their filaments by 

 eliminating the filamentary currents. However, without such elimi- 

 nation it is possible to obtain a type of formulation which is simpler, 

 more compact, and in many ways more enlightening, as will now be 

 shown by aid of the foregoing section: — Considering, in Fig. 1, any 

 filament p of wire A, let Rp denote the resistance of that filament and 

 Ip the current through it; then, by Ohm's law, the product Rpip is 

 equal to the total voltage Wp along p. But Wp is the resultant of the 

 "current voltage" Up due to all of the wire currents, and the "charge 

 voltage" Vp due to all of the charges; however, Vp is equal to Va, the 

 "charge voltage" along wire A as a whole, since the "charge voltages" 

 along all of the various filaments in a wire must be equal. These 

 various facts are expressed by the equation ^ 



i?p/p =Wp^Up+Vp^Up+ Va. (3) 



But, from the definitions of the filamentary self and mutual impedances. 



Up ^ ^ pJ- p ^^ pql q 2^^ p<i>^<i>y \^) 



where Zp = iwLp denotes the inductive part of the self impedance 

 Zpp = Rp + iojLp of filament p, Zpq — icoLpg the mutual impedance^ 

 between p and any other filament qof A, and Zp^ = iooLp^ that between 



* The case where one wire is "thick" and the other filamentary is on the border 

 line, the elementary definition of the mutual impedance being applicable when the 

 "thick" wire is the disturbing wire but not when it is the disturbed wire. 



The generalized definition, to be formulated later herein, must of course be such 

 that the mutual impedance between any two wires will have exactly equal values 

 in the two directions. 



' The distribution of V over the cross-section of the wire being uniform, equation 

 (3) shows that if U is non-uniform I also must be, and conversely. This is exem- 

 plified in skin efTect and proximity effect. 



By averaging the whole set of equations, of which (3) is typical, relating to all of 

 the filaments in wire A, and denoting the total current in this wire by / and its 

 direct current resistance by i?," we find that 



R^I =W = U +V = U + V, 



a bar indicating an average value over the cross-section. The relation 7?°/ = W 

 appears sufficiently useful and interesting to justify its enunciation in the form of a 

 theorem, as follows: When the varying current in a single piece of uniform wire, which 

 may have any cross-sectional shape, has sensibly the same total value I throughout the 

 length of the wire, whose direct current resistance is R", the product R'^I is equal to the 

 cross-sectional average W of the total, or resultant, voltage W along the wire between its 

 two ends. For a wire which is fine enough to be regarded as filamentary, the above 

 equation reduces to RI = W = U -\- V. For a wire carrying direct current, it 

 reduces to R^I = V = V. 



