APPENDIX I 



THE ELECTRIC FIELD IN A CONDUCTING MEDIUM 



When two metallic electrodes are placed in a body of water and a potential difference created 

 between them, an electric field will be set up in all parts of the water, and a current will flow from 

 one electrode to the other. The exact magnitude and direction of the electric field and current den- 

 sity at any point depend on the shape and location of the electrodes, and upon the boundary conditions 

 of the conducting medium, the body of water. The solution of this problem in general is not easily 

 amenable to mathematical analysis, but special electrode shapes and boundary conditions can be 

 chosen which make possible a straightforward mathematical analysis. In such an analysis one must 

 employ mathematically idealized conditions which as closely as possible approximate the actual 

 conditions to be described. 



In this instance the actual conditions to be described consist of two electrodes deeply submerged 

 in a large body of water such as the ocean; the size of the electrodes is small (about 1/ 100th) com- 

 pared to the distance between thenn, and the distance between the electrodes is small (about 1/ 100th) 

 compared with the distance to any boundary (top, bottom, sides) of the body of water. These are 

 conditions not difficult to satisfy in the open ocean. 



The ideal conditions which closely approximate these actual boundary conditions may be taken as 

 two nnetallic spheres of radius a as the electrodes, separated by a distance of 2d in a body of water 

 of uniform conductivity er and of infinite expanse in all directions. The solution of this ideal problem 

 can then be taken as the solution of the actual problem with an error of less than 1 percent up to 

 distances froni the electrodes equal to about 10 times the electrode spacing. 



We shall proceed, then, to solve the idealized problem. There are four things we wish to learn 

 about this system: The electric field at any point in the medium, the current density at any point, the 

 total current between electrodes, and the net resistance between the electrodes. 



The approach to be used here will give the potential at any point in the medium. From this the 

 electric field can be found by the relationship E = -VV, where E is the electric field (a vector 

 quantity, having magnitude and direction), V is the potential (a scalar quantity), and V is the vector 

 differential operator which in Cartesian coordinates is given by V = Ta "j"a Tii • ^^'^ i. J> ^> 



are the unit vectors in the X-, Y-, and Z-directions respectively. ^ ^ dy * ^ 



The current density is found by the relationship J = crE, and the total current, I, is found by 

 integrating the normal connponent of the current density over any surface completely enclosing one of 

 the electrodes. The most convenient surface for this is the infinite plane which bisects and is normal 

 to the line joining the two electrodes. 



If one of the two electrodes is at a potential Vq and the other -Vq, then the total potential 

 difference is 2Vq. The net resistance between the two electrodes is then given by R = — 2_ . 



Let us take as the Z-axis of our coordinate system the line through the centers of the two 

 electrodes, and the origin at the midpoint of the line joining the electrodes. Let the X-axis be di- 

 rected horizontally, and the Y-axis vertically downward, as shown in the figure on the following page. 



It is obvious that the field will be symmetrical about the Z-axis. This suggests that a spherical 

 polar coordinate system might be the most convenient to use. In this system the coordinates of some 

 point in space are given by r, 6, and (f , where r is the radial distance from the origin, 6 is the angle 

 between the radial line and the Z-axis, and Cf is the angle between the projection of the radial line 

 into the XY-plane and the X-axis. Since the field is symmetrical, it will depend only on the 

 coordinates r and 6. 



16 



