ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 551 



Surface Impedance of a Solid Wire 



On page 547 we defined the surface impedances of a coaxial pair as 

 the ratios of the longitudinal electromotive intensities on the adjacent 

 surfaces of the cylinders to the total currents flowing in the respective 

 conductors. In that place, however, we were unable to give explicit 

 formulae for the impedances so defined because we did not yet have a 

 precise value for E^. Now that this omission has been supplied, we 

 are prepared to compute Zh and ZJ'. 



We consider the case of a solid inner cylinder surrounded by any 

 coaxial return, and seek to determine the constants A and B in (60). 

 Since the E.M.I, must be finite along the axis of the wire we must 

 make B = 0, because the X-function becomes infinite when p = 0. 

 On the surface of the wire the magnetomotive intensity is Ijl-Kh if 

 I is the total current in the wire. By equation (56) this intensity 

 equals AIi{(Th) ; hence, 



and the final expression for the electromotive intensity within the 

 wire is 



Thus, we have the following expression for the surface impedance of 



the solid wire: 



E^ib) r]Io((Tb) , , ,,-v 



Zh = — J— = ^ 7 7- / .X , ohms/cm. (65) 



1 2ir01i{aO) 



As the argument increases, the modified Bessel functions of the 

 first kind (the /-functions) become more and more nearly proportional 

 to the exponential functions of the same argument. Thus, if the 

 absolute value of ab exceeds 50, the Bessel functions in the preceding 

 equation cancel out and the following simple formula holds within 

 1 per cent: 



^^ = A = ?A \/- (1 + ^)' ohms/cm. (66) 



ZTTO Zb \ TTg 



This surface impedance consists of a resistance representing the 

 amount of energy dissipated in heat, and a reactance due to the mag- 

 netic flux in the wire itself. Separating (66) into these two parts, 

 we have, approximately, 



1 Kf 



Rh = o)Lb = :7r\/ — ■ 



20 \ TTg 



