MUTUAL IMPEDANCE OF GROUNDED WIRES 417 



We may obtain a fifth relation from the fact that the current / 

 flows through the earth from one grounding; point to the other. To 

 utiHze this fact let us compute the total current flowing out through 

 five faces of a rectangular prism in the earth, the sixth face being a 

 rectangle in the surface of the earth surrounding the grounding point 

 (a, 0, 0), the prism extending from .r = a — ^ to .r = a + ^, from 

 y = — 7] to y = Vi and from ;: = — j" to 2 = 0. The components of 

 the electric force being given by (31)- (33), and X being the conductivity 

 of the earth, we obtain for this current the expression 



(41) - 4X f rV /in aM sing, sin ,. ^^^^^^^ 



Jo Jo 1^" 



after simplifying by means of the divergence condition (37). This 

 current flowing out through the prism is / if the face in the surface of 

 the earth includes only the one grounding point (a, 0, 0), but is zero 

 if it includes both grounding points; that is, the above integral (41) 

 equals / if ^ < 2a. but equals zero if 2a < $, for any positive value of r?. 

 It is readily verified that the Fourier integral 



(42) ^ r r^'"^^A/sin?Msin..^^^^^ 



has the desired properties. Accordingly, we must have 



2/ 



(43) F^= — ^sina/i. 



TT-A 



To obtain the one additional relation which is needed, we make use 

 of the fact that the current / flows through the straight wire from one 

 grounding point to the other. Let us integrate the magnetic force 

 around a rectangle in a plane perpendicular to the wire, that is, 

 perpendicular to the x axis, the rectangle extending from y — — r] 

 to y = 7} and from s = — f to s = f , the path of integration being 

 taken in the clockwise direction looking along the positive direction of 

 the X axis, and then equate this integral to 47r times the total current 

 threading the rectangle. The components of the magnetic force 

 which we need, Hy and Hz, are found from the fact that curl E= —iuH, 

 that is, 



(44) io:Hy 



(45) iccH, 



