608 EXPLORATION GEOPHYSICS 



where 



tan^ = -^ (12) 



wLg 



On replacing /3 by its equivalent in terms of ^, we obtain 



and 



sin (wt — /?) = sin (wt — — + 6) = — cos (wt + 6) 



J- _ wMIprn COS {wt + B) .^^ 



On comparing Equation 13 and the expression for the primary current, 

 it is apparent that the time phase angle between the secondary and primary 

 currents is tt + ^. 



At any point such as Q on the plane mm (Figure 380), the magnetic 

 flux density due to the secondary current is given by the relation 



where fa is a function of the geometry of the system, the location of the 

 point Q, and the permeabiHty of the medium surrounding the secondary 

 coil. The magnetic flux density due to the primary current is given by 

 the relation 



Hp = fplpm cos wt ( 15 ) 



where fp is a function of the geometry of the system, etc. 



At a particular instant of time, therefore, the fields at Q may be 

 represented by vectors Hp and Hs as shown in Figure 380. These vectors 

 are displaced in space by an angle a and in time phase by an angle ir + 6. 

 Hence, in general, they will combine to give a rotating vector for the 

 resultant field, and the tip of the rotating vector will trace an ellipse.* 



In certain special cases, however, 6 is approximately equal to zero so 

 that the time phase angle between the secondary and primary fields is 

 approximately 180°. For such cases, the major axis of the ellipse is much 

 larger than the minor axis and the resultant field at any point may be 

 represented without appreciable error by a vector which is constant in 

 direction rather than by a rotating vector. 



Consider, for example, a mineralized, highly conductive subsurface 

 zone and assume that with respect to inductive phenomena this mineralized 

 zone is equivalent to a coil. The resistance Rg of the mineralized zone will 

 be small, while the inductance Lg will not, and it may be shown that 

 tan 6 = Rg/ivLg approaches zero ; that is, 6 is approximately equal to zero. 



* The mathematical analysis proving that two fields which differ in time phase and 

 "space phase" combine to produce an elliptically polarized field is given on p. 614. 



