MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 5 



result also applies to any closed flow which does not extend above 

 the horizon plane and may be resolved into any number of com- 

 ponent flows, each of which is radially symmetrical about a vertical 

 axis. 



3. Mutual Resistance of Grounded Circuits 



By definition, if e.m.f.'s E and c in grounded conductors AB and 

 ab produce the currents / and ;' = () in the conductors, the mutual 

 impedance between the two conductors is e/I. In the present case 

 we are dealing with direct current and thus the mutual impedance 

 is a mutual resistance Q, and by (2) its value is 2 



v 2tt \Aa Ab Ba^ BbJ 



^2tJ J 

 2ttJ 



-2dUdu + d Vdv + d Wdw 

 R 3 



COS (0i — e) COS (02 — e) 

 ri 2 r 2 2 



ds. 



(3) 



The third form of (3) shows that the mutual resistance falls off 

 as the inverse third power of the distance between grounded circuits 

 when this distance has become large compared with the length of 

 these circuits between grounding points. 



The first form of (3) show r s that the mutual resistance between 

 grounded circuits does not depend upon the location of the conductors 

 but only upon the location of the terminal grounding points A, B, a, b. 



The mutual resistance for the case p = 27r is obtained from Fig. 1 

 by taking the value of V/I at the point corresponding to a reduced 

 by its value at the point corresponding to b; if b is anywhere on the 

 center line, for which V/I = 0, the diagram gives directly the value of 

 the mutual resistance. Employing ordinary units the diagram gives 

 the mutual resistance directly in ohms if AB = 1 mile and the earth 

 has a resistivity of about one million ohms per centimeter cube (more 

 exactly 1.011 XlO 6 ) which is its actual order of magnitude. 



2 In addition to the earlier notation there are employed in the different expressions 

 for formula (3), and also in formula (5) below, the following: R is the distance be- 

 tween two elements dS and ds of any two paths extending from A to B and a to b; 

 the rectangular projections of these elements along and perpendicular to R are dU, 

 dV, dW and du, dv, dw, the two sets being parallel and with the same positive direc- 

 tions; 0i, 02, e are the angles which r u r 2 and ds make with AB, when the path ab 

 lies in a plane with A and B. 



