202 



EXPLORATION GEOPHYSICS 



due to the side reduces to 



^ = /, log 3!ol±l£i 



■XoV 



yx^+ (^xi — xoY 



(88) 



Fig. 93. — ^Surface irregularity. (A cHfF.) 



where yi = yo + the width of the layer. 



The vertical anomaly due to the side of the thick vein is plotted in Figure 91 for 

 an assumed negative value of h. (To simplify calculations, auxiliary curves are plotted 

 in Figure 92.) 



Effects Due to Surface Irregularities. (A Cliff.) — Consider the 

 vertical anomaly in the neighborhood of a vertical cliff. (Figure 93.) The 



vertical component of the field along a 

 line A A' is the resultant of the effects 

 due to surfaces A, C, and B. The effect 

 of surface A will be approximately 

 constant, as the instrument is usually 

 relatively close to the ground and the 

 effects due to the edge are not apparent 

 until the distance from the edge be- 

 comes comparable with the height of the instrument above ground. Sur- 

 faces C and B may be considered as strips and their effects calculated from 

 the formulas already given. 



C. Magnetic Anomalies Produced by a Uniformly Magnetized 

 Sphere. — The effect of induction in the earth's magnetic field for a 

 sphere may be analyzed as follows : t 

 Consider that the sphere has two vol- 

 ume densities of magnetic charge, +t 

 and — T, which coincide when there is 

 no external field. (Figure 94. )| In the 

 presence of a field, the sphere of posi- 

 tive charge is displaced relative to the 

 sphere of negative charge in the direc- 

 tion of the field by a distance 00'. As a consequence of this displacement the 

 sphere becomes uniformly magnetized with an intensity of magnetization. 



Fig. 94. — A uniformly magnetized 

 sphere is equivalent to two spheres 

 having equal and opposite densities of 

 magnetic charge. 



/ = T • 00' 



(89) 



It may be proved that the field at outside points due to a spherical 

 distribution of magnetic charge is the same as that due to a single charge 

 concentrated at the center of the sphere. Hence, the magnetic field at an 

 external point P is the same as that produced by a dipole of magnetic 

 moment : 



M = 4/3 (ttR^ r • 00') = 4/3 (ttR^I) 

 where R is the radius of the sphere. 



(90) 



t Compare Starling, loc. cit., p. 142. 

 t Starling, loc. cit., pp. 268, 269. 



