Chap. 10] ELECTRICAL METHODS 797 



Zp is the combined in-phase component and Zq the quadrature component. 

 According to Stefanescu,^^ the 180° component is negUgible for low fre- 

 quencies, andZp — 2'kIq/'R (field in air, see eq. [10-506]). The quadrature 

 component is Zq = ^Tr/ok^R, where /o is the loop current, R its radius, and 

 k an induction factor similar in significance to the factor q previously 

 defined by eq. (10-57). Since k^ = 47ro-co (to = frequency, a = conduc- 

 tivity), the quadrature component is 



Z„ = 2.V...R = '-1^13 = '2.4.10-'/R/.^ (jO_g^^) 



P P 



where /o is in amp., R in cm, / in c.p.s., and p in ohm-cm. 



As a rule, the ground is not homogeneous, so that when the loop radius 

 is increased, beds of different conductivities affect the magnetic field. As 

 in the resistivity method, formula (10-60a) is still applicable, provided 

 that p is understood to represent now an apparent resistivity, pa , which 

 in accordance with (10-60a) follows from the observed parameter, Zq/Z 

 (quadrature field in gauss/amp. primary current) : 



Pa = 12.4.10~VR.^. (10-606) 



In resistivity methods the effect of layers of different conductivity is 

 calculated by reflecting the source on the formation boundaries. The 

 same procedure is applicable here, with the significant difference that, for 

 each interface, only one image is required. ^^ The analysis is simplified 

 considerably by considering low frequencies only, in which case the 180° 

 field is always zero and the quadrature field is given by Zq = 

 -lim./=o dZq/df. 



If a circular loop energized by low-frequency current is suspended at 

 an elevation d above an interface (for example, the surface of the ground), 

 the magnetic field at any point P{z) in its axis is equal to the gravity potential 

 of a disk at a depth 2d (see Fig. 10-114) of thickness 1 and density 2ir^Ioai , 

 if <Ti is the conductivity of the medium above which the loop is suspended. 

 The gravity potential above a disk at a distance h from its center is given 

 by U = 27rG5 rf/i(r — /i), where G is gravitational constant and 5 is density. 

 li dh ^ 1, Gh = h', h = 2d - z, and r' = R' + {2d - zf, the magnetic 

 field at the point P is 



Zq = 2x6i[VR' + a^d - zY - {2d - z)], (10-61a) 



so that when P is moved up to the center of the loop {z = 0), 



Zq = 27r5i[VR2 + 4^2 - 2d]. (10-616) 



"Beitr. angew. Geophys., 5(2), 188 (1935). 

 98 Ibid. 



