386-389] Special Problems 341 



III. The resistance between two spherical electrodes, radii a, 6, at a 

 great distance r apart, in an infinite conducting medium, is, by formula (317), 



4-7T (a b 



388. If two electrodes of any shape are placed in an infinite medium at 

 a distance r apart, which is great compared with their linear distances, we 



may take p^ in formula (317) equal, to a first approximation, to -. This is 



small compared with p n and p^, so that, to a first approximation, we may 

 replace formula (317) by 



It accordingly appears that the resistance of the infinite medium may be 

 regarded as the sum of two resistances a resistance -^ at the crossing of 



the current from the first electrode to the medium, and a resistance -^ at 



the return of the current from the medium to the second electrode. Thus 

 we may legitimately speak of the resistance of a single junction between an 

 electrode and the conducting medium surrounding it. 



For instance, suppose a circular plate of radius a is buried deep in the earth, and acts 

 as electrode to distribute a current through the earth. The value of p u for a disc of 



radius a is ^-, so that the resistance of the junction is 5-. So also if a disc of radius a 



is placed on the earth's surface, the resistance at the junction is , and clearly this 



also is the resistance if the electrode is a semicircle of radius a buried vertically in the 

 earth with its diameter in the surface. 



Flow in a Plane Sheet of Metal. 



389. When the flow takes place in a sheet of metal of uniform thickness 

 and structure, so that the current at every point may be regarded as flowing 

 in a plane parallel to the surface of the sheet, the whole problem becomes 

 two-dimensional. If x t y are rectangular coordinates, the problem reduces to 

 that of finding a solution of 



which shall be such that either V has a given value, or else -r- = 0, at every 



on 



point of the boundary. The methods already given in Chap. vin. for obtain- 

 ing two-dimensional solutions of Laplace's equation are therefore available 

 for the present problem. The method of greatest value is that of Conjugate 

 Functions. 



