MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 19 



is small compared with the length AB is, by (25) of the appendix, 

 neglecting the first and higher powers of 1/s, 1 



^AB 



N = 2AB log ^-r— = 2AB log ° 73 , 6 AB . (11) 



to Aa b Aa 



This expression has the form (2 / log s/r) of the commonly employed 

 mutual inductance formula for two long parallel conductors, each of 

 length /, separated by distance r, the common return being a perfectly 

 conducting earth in which the image of each conductor is at the 

 distance 5 from the other physical conductor. For our direct-current 

 case, therefore, the effective distance to the images is about f of 

 the length of either grounded conductor. Since this distance is by 

 assumption large compared with the distance between the conductors, 

 the images are approximately at this same depth below the actual 

 surface of the earth, and the hypothetical perfectly conducting earth 

 would be at one-half this depth, or %AB. The effective image dis- 

 tance is necessarily directly proportional to the dimensions of the 

 grounded circuits and independent of the earth's resistivity because 

 the shape and relative distribution of the lines of flow are independent 

 of the resistivity and of the length of the grounded circuits. Inspec- 

 tion of Fig. 1 shows that somewhat over \ of the return flow attains a 

 distance \AB from AB, while the remainder of the current remains 

 closer to the grounded conductor. 



It may be inquired what would be the effect of confining both return 

 currents to a thin uniform conducting layer on the earth's surface, 

 so that they become horizon return flows. For the closed flows 

 {JC-Jt) and (-»—•*) in general and the particular flows (tt—Jt) and 

 {r— A) t where & and r are close parallel straight conductors, 



N{jr- JV)(*-a) = NJT*- Njta - NjZr+Njto, 



N{s?-jr){r-A) = Nj?r-2Ab+2Aa 



*AB 

 = 2ABlo g ^=2ABlo g °-^*. (12) 



7 If the term \/s — Aa/AB of the expansion is retained the equivalent ground plane 

 has the depth (AB-\- Aa)/e and thus becomes deeper as ab is moved away from AB. 

 But the equivalent ground plane may be kept fixed at the distance AB/e from AB 

 provided it is tipped at the angle sin -1 2/e=47° so that ab moves away from the 

 ground plane as it moves away from AB. If it were worth while, still closer ap- 

 proximations might be secured by using a perfectly conducting cylindrical earth of 

 suitable cross-section. 



