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BELL SYSTEM TECHNICAL JOURNAL 



positive direction around the loop is counter-clockwise; the straight 

 grounded conductor A B of Fig. 6 is of unit length while in Fig. 7 

 grounded terminal B alone appears, terminal A being at an infinite 



Fig. 7 — Contour lines of the mutual Neumann integral between a counter-clockwise 

 small loop on the surface of the earth, per unit area, and a straight grounded filament 



of infinite length 



distance to the left. These curves show the vertical component of 

 the magnetic field due to unit current in AB. The formulas employed 

 for AB = l and infinity, respectively, are 6 



2y(r 1 +r 2 ) 



<PN 



dxdy rir 2 [(ri-f-/-2) 2 - 



d*N y 1 la 



T~T = —( i \ = "~ tan £02- 



dxdy r 2 (x 2 + r 2 ) r 2 



= ri + r 2 l l-Qi 

 1] r x r 2 \{r x + r, 



-Oi-r 2 ) ! 



r 2 ) 2 -l' 



(8) 



(9) 



Large loop mutual inductances may be calculated either by inte- 

 grating the value of —d 2 N/dxdy over the loop or by integrating the 

 value of dN/ds around the boundary. If the boundary may be ap- 

 proximated to by a broken straight line, the curves of Figs. 2 and 3 

 may be employed. 



6 These formulas may be derived by differentiating (8) of the paper loc. cit., 

 dN/dx = 2 ta.nh -1[ab/(A>+sB)~\, with respect to y. 



