20 BELL SYSTEM TECHNICAL JOURNAL 



Thus, the assumption that the return current is confined to the earth's 

 surface does not change the order of magnitude of the effective image 

 distance, but reduces it from about f to about \ of the length of the 

 exposure. For space returns the effective image distance is 



2AB/e i,i = 0A^AB. 



Now take another practical case by assuming that the conductor 

 ab is of negligible length compared with its separation r from the 

 parallel conductor AB and the separation is, in turn, negligible com- 

 pared with the length AB. The formula for dN/dx given in footnote 

 6 shows that the required inductance depends only on the ratio 

 AB/(r\-\-r 2 ) and is thus constant upon an ellipse. Equivalent ex- 

 pressions in logarithmic form are 



Ar = 2aMog(2cos^ lS in^ 2 ^-^- 2 ) (13) 



or approximately 



AB 

 N = 2 ab log , if ab is opposite the midpoint of AB, (15) 



iV=- 2 ab log , if ab is at distance r 2 beyond B. % (16) 



2 L ^2—1 



Thus, from (13) the effective image distance is never greater than 

 twice the geometrical mean distance from ab to A and B. Its maxi- 

 mum value is approximately AB and occurs when ab is opposite 

 the midpoint of AB. Its minimum value is approximately V r 2 AB 

 and occurs when ab is at the distance r 2 from A or B in the prolonga- 

 tion of AB. This makes the inductance one-half of what it is at the 

 symmetrical position. Thus, wherever ab is placed, its mutual in- 

 ductance with AB lies somewhere between 50 per cent, and 100 per 

 cent, of the mutual inductance, due to an effective image distance 

 AB, ab remaining always parallel to AB and the locus of ab being a 

 rectangle with semicircular ends of radius r 2 and centers A and B. 



8. Mutual Impedances of Grounded Circuits Which Depart 

 from the Surface of the Earth 



Consider a system of conductors following any paths in space and 

 insulated from the earth except at two grounding points on the sur- 



8 The shortest distance from ab to AB might have been designated by a single 

 letter in place of using y, r and r 2 in formulas (13), (15) and (16). 



