192 EXPLORATION GEOPHYSICS 



The component in the direction x is 



s 



nix 



F.= \^dx = 0* 



Hence, the resultant field for an infinite linear distribution of magnetic 

 poles has the direction r and a magnitude 2m/r. 



The potential due to the linear distribution of charge is given by the 



'^"^^^°" F = 2mlogr (79)** 



Field Dice to Two Parallel Linear Distributions of Equal 

 Strength and Opposite Polarity (Figure 83) 



Fig. 83. — Sketch illustrating geometric re- 

 lations between two parallel linear distribu- 

 tions and an external point P. 



If the separation a of the two parallel lines is small compared to r, 

 then the potential F = Fi + Vo = 2m log ri — 2ni log To 



r-i + a cos 



To 



= — 2m log -^ = — 2ni log 



( r-[ + a cos d \ 

 n ) 



= — 2m log 



(. a cos 9 \ 

 ri / 



, _ , mx dx 



To evaluate the integral l^^a.r— IT^JljrjJT^ 



eo o 



set 



This yields 



x = r tan i 



F.= 



v/2 



r tan r sec'9 dd 





7r/2 



sine d 6 = 



r ser d 

 -7r/2 -7r/2 



** It will be noticed that this expression for V makes V infinite at r equal to infinity. 

 However, we are chiefly interested in the magnetic field, i.e., the derivative of the poten- 

 tial at a finite value of r, and this derivative always has a finite value. 



