Chap. 10] ELECTRICAL METHODS 799 



By analogy with the gravitational potential, the magnetic field for two 

 layers with conductivities a and az is readily obtained (see Fig. 10-114). 

 As the loop is on the ground surface, the magnetic field due to the effect 

 of the upper layer is equal to the potential of a disk with the density 

 Bi = 2Tr^Io(Ti in its own plane. The effect of the interface below is given 

 by the potential of a disk at twice the depth of the interface di with the 

 density hz = 2irVo(o-2 — o^i). The magnetic field due to the latter is 

 therefore, according to eq. (10-616), 



Zq = 2x52[VR=' + 4d? - 2dil 

 to which must be added the field of the disk inside the loop, so that 



Zq = 47r'/o{(riR + i<T2 - <ti)[Vr^ + 4dx - 2di]\, (10-62a) 

 from which the apparent conductivity is 



«r« = (.1 + ^^^ [Vr^ + Ml - 2d^] . (10-626) 



For small values of R, or small depth penetration, the second term in eq. 

 (10-626) approaches zero, and therefore the apparent conductivity ap- 

 proaches the conductivity in the upper layer. On the other hand, if 

 di <$C R (large loop radius), the apparent conductivity approaches <t2 . 



If two interfaces exist (see Fig. 10-114), the magnetic field is composed 

 of three potentials: (1) due to the surface disk with density 5i , (2) due to 

 the disk image of density 62 , and (3) due to the disk image with density 

 S3 = 2x^/0(0-3 — d). Hence, the magnetic field 



Zq = 47r'/o{(riR + {<T2 - ai)[VR' -\-Ml - 2d J 



+ ((73 - <r2)[VR' + Ml - 2di]}. 

 The apparent conductivity in this case is 



aa = a, + ^^^ [V^T4d? - 2dx] 



.+ ^1^' [Vr' -\- Ml - 2d,]. 



It follows from eq. (10-636) that the effect of the n*** interface on the 

 apparent conductivity is given by 



(l0-63a) 



(10-636) 



^a.n = ^5±^- [ Vr^ + 44 - 2dJ. 

 Substituting the ratio r^ = R/2d^ , 



<ra.n = '-^±^^^- [Vr^ + I - 1]. (10-64) 



