MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 3 



return flow of direct current / from the point A to some infinitely 

 distant point and a second return flow of direct current / from this 

 infinitely distant point to the point B. For these component flows 

 the current diverges from A or converges towards B radially and 

 with equal intensity in all directions in the earth; the total current 

 for one of the component flows flowing through any surface in the 

 earth will thus be equal to I/2ir times the solid angle subtended at 

 A or B, respectively, by the boundary of the surface, s'nce the entire 

 solid angle filled by the earth at a point on the surface is 2ir. The 

 total radial flow from A through the lower half of the circular cone 

 having its axis in AB, the elements of the cone making the angle 

 0i with AB, is \I{\— cos 00; similarly, the total radial flow toward 

 B through the lower half of a cone with the angle w — 2 will be 

 $J(l+cos0 2 ). For the combined superposed flows the total current 

 flowing through the semicircle in which the cones intersect is the 

 sum of these two values or \l{2 — cos 0i-f-cos 2 ), from which (1) 

 is immediately obtained, since the total current flowing in the earth 

 from A to B is /. This assumes that the semicircle lies between A 

 and B, but the same formula holds for the entire current sheet of 

 revolution. The lines of flow in the earth are symmetrical about AB 

 and lie in planes through AB, since, in the earth, both component 

 flows are symmetrical about AB. 



For the component flows the equipotential surfaces are hemispherical 

 and, since the resistance of a hemispherical shell of radius r, thickness 

 dr, is pdr/2-irr 2 , the potentials at distances r x or r 2 from A or B, re- 

 ferred to the potential at infinity, are Ip/2irri or —Ip/2irr<i, respec- 

 tively, from which equation (2) follows by addition. 



Fig. 1 accurately reproduces the flow and equipotential lines as 

 given by formulas (1) and (2). At the midpoint of a line of flow its 

 distance from each electrode is ri = r 2 = bI/C and it may be shown 

 that every other point of a line of flow is at a still shorter distance 

 from the nearer electrode. It follows, for example, that less than 

 1/10 of the total current reaches, in its flow through the earth, any 

 point lying at a distance greater than 5 A B from the line AB connect- 

 ing the electrodes. 



If a uniform radial flow of current / in the horizon plane con- 

 verging on the point A is combined with the uniform radial flow 

 in the earth outward from A , we have a closed flow which is sym- 

 metrical about the vertical axis through A. Below the horizon 

 plane the magnetic lines of force will be horizontal circles and the 

 magnetic force at any point distant t\ from A, a being the angle 

 included between r x and the nadir, is H = 2I(l — cos a)/(r jsin a) 



