MAGNETIC METHODS 



193 



where m is the pole strength per unit length. Since is small when a 



is small compared with ri, the power series expansion of the logarithm 



gives F = — 2m 1- terms of the order | -^ 



That is, 



ri 



fe) 



etc. 



, - ^ a cos ^ , , 



V = — Zm (approx.) 



(80) 



General Expressions for the Magnetic Potential and Field Pro- 

 duced by Surface Distributions of Charge. — The magnetic potential 

 produced by a uniformly magnetized body at any external point P is given 

 by the equationt 



V 



=0^ 



cos a 



dS 



where r is the distance between P and the element of surface dS and a is 

 the angle between the magnetization / and the outward normal at dS. 

 For the cases to be considered, the magnetization is normal to the surface, 

 i.e., a = ; hence, 



"idS 

 r 



V 



=((^ 



(81) 



The magnetic field at P is obtained by taking the derivative of V. 

 Differentiation of Equation 81 with respect to r yields :* 



---f=-|M^)=i^ 



(82) 



Equations 78 to 82 are the fundamental equations utilized in deriving 

 the magnetic effects produced by two-dimensional geologic bodies. 



Magnetic Anomalies Due to a 

 Buried Contact of Two Very Thick 

 Horizontal Layers Having Dif- 

 ferent Susceptibilities. — Assume 



that the materials A and B (Figure 

 84) are separated by a vertical plane, 

 i.e., have a vertical contact, and that d 

 is the thickness of layer C, ki sus- 

 ceptibility of layer A, ^2 susceptibility 



of layer B, h susceptibility of layer C. ^'''- S^.-Buried contact of two thick layers. 



On the upper surface of C there is a uniform distribution of magne- 

 tism of surface density h — kzZ units of pole per unit area.** This is south 

 polarity or magnetism if C is paramagnetic and north polarity if C is 



t J. H. Jeans, Electricity and Magnetism, 5th Edition, p. 374. 



* In carrying out the differentiation with respect to r, the quantity IdS is held 

 constant. 



** Note that Z is the vertical component of the earth's field. 



