MUTUAL IMPEDANCE OF GROUNDED WIRES 



411 



and we thus obtain Ex, Ey, E^ in the compact form 



(18) 



_ „ „, Ids/ ah 



(£.,£„,£.) = 2"-, (-.-^ 



cvq_ 



d'-p 



(VQ 



rV-P 



dz" ' dxd \'dz ' dxd: 



where P and Q are given by (15) and (16). In deriving this form we 

 use the fact that Q satisfies the wave equation 



X'^ ay- az- 



At the surface of the earth {z = 0) the electric force takes the simple 

 form 



(19) (£., E.) = ^ 



d\~ \ r 



1 + ir 



cT 



d.Vfh' \ r 



where we have used the expressions for the derivatives ^ of the Bessel 

 functions, lo'iz) = Ii(z), Kq{z) = — Ki{z), and also the identity^ 

 h{z)K,{z) + h{,z)K,{z) = 1/s. 



The mutual impedance dZi^ between two infinitesimal elements dS 

 and ds is now written down as the ratio of the resulting electric force in 

 one element to the current in the other, with sign reversed: 



(20) dZ^i 



dSds 

 IttX 



dSds 

 27rX 



cos e -T—; - — COS e r. C~ 



d \~ \ r / r 



sm e 



d- 



dxdy \ r 



3 sin $ sin </> — cos e cos e ., , ^ ^ 



where $ and are the angles which the elements dS and ds make with 

 r, and e == $ — is the angle they make with each other. 



Integration over the two wires 5 and 5 gives a general formula for 

 the mutual impedance of grounded wires lying on the surface of the 

 earth: 



(^') ^'-2^//l7ls(-.)+^t.-(. + ..).-"]}^. 



Sds 



= //[ 



J ^ / 1 



27rX ' dSds 



+ fco ^1 ^, [1 - (1 + 7r)e-] } ] dSds. 



* G. N. Watson, op. cit., page 79, formula (7) of § 3.71. 



' G. N. Watson, op. cit., page 80, formula (20) of § 3.71, with v = <d. 



