ELECTRICAL METHODS 471 



where the operator V^ is defined by the relation 



^ " dA-^ dy^ d^^ 



It is clear from the derivation of this equation that whatever the shape 

 of the conductor through which a steady current flows or whatever the 

 particular conditions of the problem, the potential V must satisfy Equation 

 4. This is true everywhere except at points where there is either a source 

 or a sink of current.* Equation 4 alone does not determine the solution 

 of any particular problem because it expresses only one of the conditions 

 which V must satisfy. 



Furthermore, Laplace's equation (4) holds only for steady currents 

 in isotropic, homogeneous media. 



In any particular problem there are "boundary conditions" which must 

 be satisfied. The boundary conditions require : ( 1 ) at any boundary 

 separating two media of different resistivities, Fi = Fo where Fi and Fa 

 are the potentials on opposite sides of the boundary, and (2) the normal 

 component of the current entering the boundary through one side is equal 

 to the normal component of the current leaving through the opposite side. 

 That is, 



• - 1 dVi_l dV2 

 ^« — -^ — -^ 



pi on p2 on 



Any solution of Laplace's equation which also satisfies the boundary 

 conditions constitutes the unique solution of the given problem.** 



Solutions of Laplace's Equation 



Solutions of Equation 4 for several simple cases of current flow in 

 media of uniform resistivity p will be discussed in the following para- 

 graphs. For exploration purposes, certain portions of the earth's crust are 

 approximately homogeneous ; other portions approximate the mathematical 

 ideal of two or three homogeneous layers of uniform resistivities ; still other 

 portions comprise, approximately, two homogeneous, semi-infinite media 

 separated by a bounding plane (fault), etc. The flow of current through 

 such structures may be described by a solution of Laplace's equation which 

 satisfies the boundary conditions at all the boundaries. 



Case I. — Consider a small source of current surrounded by an infinite 

 isotropic homogeneous conductor of resistivity p. From considerations of 

 symmetry, it is clear that the potential F will be a function only of the 

 distance r from the current source. 



* Points where current flows out of a conducting medium or into a conducting 

 medium from an external source. 



** For a proof that V is determined uniquely by Laplace's equation and the boun- 

 dary conditions see J. H. Jeans, The Mathematical Theory of Electricity and Mag- 

 netism (Cambridge University Press). 



