280 BELL SYSTEM TECHNICAL JOURNAL 



high. The lightning stroke will then arc directly to the cable from the 

 point where it enters the ground, often at the base of a tree. Furrows 

 exceeding 100 feet in length have been found along the ground path of such 

 an arc. 



For a crest current / in the lightning stroke, the arcing distance in meters 

 is given by^ 



r = k{Jpyi^ (1) 



where / is in kiloamperes, p is the earth resistivity in meter-ohms and k 

 is a constant depending on the surface breakdown gradient of the soil. 

 Low resistivity soil, up to p = 100 meter-ohms, has an average breakdown 

 gradient of about 2500 volts/cm, and the corresponding value of k is about 

 ,08. For high resistivity soil, p = 1000 meter-ohms and up, the average 

 breakdown gradient is about 5000 volts/cm and k = .047. Thus, for an 

 earth resistivity of 2000 meter-ohms, and / = 100 ka, r = 21 meters or 

 70 feet. 



The number of strokes arcing to a cable of length ^ may conveniently 

 be expressed as 



N = Ifsn (2) 



where n is the number of strokes to ground per unit area, f is an equivalent 

 arcing distance, and Ifs an equivalent area near the cable within which the 

 cable is assumed to attract all lightning strokes. In obtaining r, the 

 number of strokes arcing to the cable from various distances r as given by 

 (1), depending on the current in the strokes, must be evaluated. This 

 number and the equivalent arcing distance will thus depend on the crest 

 current distribution of lightning strokes. For the distribution curve shown 

 by Curve 1 in Fig. 1, the effective distance in meters is^ 



r^ .36p'^^whenp < 100 meter-ohms 

 ^ . 22p'/2 when p > 1000 meter-ohms ^^ ' 



Thus, for soil having an average resistivity near the surface (to a depth of at 

 least 10 meters) of 2000 meter-ohms, r = 10 meters = ZZ feet. 



A cable will thus collect direct lightning strokes for an eflfective distance 

 f to either side of it, and when the incidence of strokes to ground per unit 

 area is known, the number of strokes to a cable of a given length may read- 

 ily be calculated. The incidence of strokes to ground has been estimated, 

 on the average, as about 2.4 per square mile for each 10 thunderstorm days, 

 and the corresponding expectancy- of strokes to a buried cable, per 100 

 miles of length, is about 2.1 for 10 thunderstorm days when the earth re- 

 sistivity is 1000 meter-ohms and 3.0 when it is 2000 meter-ohms. 



The distribution of the crest currents in direct strokes to a cable may be 



