714 



ELECTRICAL METHODS 



[Chap. 10 



that are produced by reflection on the formation boundary and the earth's 

 surface. The source / is reflected at the formation boundary and produces 

 the image kl. The value of k is given by eq. (10-32c) and is equal to 

 (P2 — Pi)/(p2 + Pi), since it applies to the formation boundary (p2Pi). The 

 image /i = kl is now reflected at the earth's surface and produces the 

 image /i = k'kl above. In this case k' as applied to the boundary 

 between the upper medium and air is 1, since po = » . Hence, the image 

 /i = kl. The potential at any surface point will therefore result from 

 a summation of an infinite series of images. Thus, h = kl at & depth 2h; 

 1 1 = kl at a height 2h; h = k^I at a depth 4/i; ![ = k^I at a height 4^; 

 so that 7n = k^I at a depth 2nA, and /q = k^I at a height 2nA. Since 

 ■^n = ^n , ^n + ^n = 2k^I , and the potential at any point on the surface is 



^=f:[^ 



°y 2fc°7"| 

 n-l Tn J 



(10-34) 



The potential at a point A{xiyi} due to a source / at and its series of 

 images is therefore 



V^o = 



2irL- 



1 



+ 2E 



fc° 



Vx? + y\ «=i Vx\ + y\-\- {2nhf\ 



and the potential due to an equal sink at 0' at the same point is 



(10-35a) 



V AO' — :r— 



':l 



+ 2E 



/c" 



2t L Va - XrY + y\ -^ V{1 - x,f + y\ + {2nhf\ ' 



(10-356) 



where I is the distance between the electrodes so that the resulting po- 

 tential is 



'.=¥i 



1 



+ 2Z 



A;° 



'^^Wx\ + y\ y/{l-x,f + y\ n=i Vx? + 1/? + (2iiA)' 



n=«) ,n -, 



-22: ' 



n=i V(Z - x,f -\-y\ + (2nhy 

 Similarly for another point B{x2y^, 



(10-36a) 



2irW, 



x\ + yl 



V(Z - x,f + y\ 



+ 2E 



A;° 



-2E 



i ^xl + 1/2 + {2nhf 

 k'' 



\ (10-366) 



^1 V{1 - xzf + 1/2 + (2n/i) 



1 



