ELECTRICAL METHODS 583 



applied to structural mapping, the path of effective current flow is cal- 

 culated from the magnetic measurements and variations in the depth of 

 this path or "marker bed" versus electrode spacing or position and is 

 plotted for various points or stations in the area under investigation. 



Theoretical Relationships 



The calculation of the magnetic field at the surface for the case where 

 the earth is homogeneous and non-magnetic can be carried out readily, 

 provided the following assumptions are made: (1) the current pene- 

 trates the earth in all directions radially from the source; (2) the cur- 

 rent enters the sink radially in all directions from the earth; (3) the 

 magnetic effects of the current leaving the source and the current entering 

 the sink may be evaluated separately, and their separate effects combined 

 vectorially; (4) the magnetic effect of the current leaving the source, 

 or entering the sink, may be computed by evaluating the effect (differen- 

 tial) due to a cone of solid angle dzu and intergrating throughout the 

 semi-infinite space occupied by the homogeneous earth; (5) the magnetic 

 field due to the total current in an infinitesimal frustum of a cone is 

 the same as that due to a current which flows along the axis and is equal 

 in magnitude to the total current in the frustrum. 



In Figure 364, the xy plane corresponds to the earth's surface and 

 the origin of coordinates to the current source. / denotes the current 

 leaving the source, x denotes , P X 



the distance between the cur- ^/fsl ? Z~P7 ^A 



rent source and a pomt P on V*'^/^ \ ^sc'^ r ''''' / i 



the X axis. y^/''^ !^ ^Ny ' uL^^\ 



In order to determine the ] ,,--'' j *^nN^ • / -^ 



resultant magnetic field at P, L-il'l l _aS<^'' 



it is convenient to calculate [ nN. 



first the field produced at P by I n^>C***^ 



the current flowing in a solid ^ nP 



angle dzV. The axis / of the Ji.°- 364.— Sketch illustrating geometric. quantities 



o . _ used in evaluating the field produced at a point P by a 



cone of revolution defining the current / entering the earth at the origin of coordinates. 



solid angle makes an angle B 



with the X axis. The perpendicular distance between the axis / of the cone 

 and the point P is denoted by R. The current i passing through the cone 

 is related to the current / leaving the source by the equation 



I — -T^dzv 



Lit 



Let d\ denote the length of a differential element and r denote the dis- 

 tance between dl and the point P. Also, let (dl,r) denote the angle 

 between dl and r. From Ampere's law, the magnitude of the magnetic 



