492 BELL SYSTEM TECHNICAL JOURNAL 



over the area of the disk. This gives: 



I pa = -AppakTup. (16A) 



The analogous expression for a hemispherical contact area of radius 

 fft, obtained from (7), is: 



I pa = —lirrbpakTup. (17A) 



If a comparison is made on the basis of equal radii, (17 A) is larger than 

 (16A) by a factor of t/2. On the more reasonable basis of equal contact 

 areas, (16A) is larger than (17 A) by a factor of -i/ir. 



An approximate solution which includes surface recombination can be 

 obtained as follows. A solution of Laplace's equation which satisfies 

 (8A)^and (lOA) js: 



y^^J^Te-UrD'^dt. (18A) 



That (18A) satisfies (8A) may be verified by direct substitution; 



4^^> 



= ?L° /* j,{rt) sin ptdt = for r > p. (19A) 

 = (2yoA)(p' - rT'" for r < p. (20A) 



2yo f 



TT Jo 



Expression (18A) satisfies (9A) approximately if Xc is large. Using 

 (20A) and neglecting X in comparison with X^ we have: 



y= pa- (2yoAXc)(p2 - f2)-i/2 for z = 0, r < p. (21A) 



Except for r almost equal to p, the second term on the right of (21 A) is 

 very small. It is not possible to obtain an explicit expression for y for 

 r < p. For 2 = 0, r = p, 



, ^"'•fjy' dl^y.FM. (22 A) 



The integral, F(\p), can be evaluated from a more general integral in 

 Watson's Bessel Functions, p. 433. We have: 



/<■(/;) = 2 r Jjh c) sin xdx ^ ^^^ ^ ^^^^^ _^ ^.^ ^ y^^^^ (22B) 



TT Jo X -\- k 



The factor multiplying yo is unity for Xp = 0, and decreases as Xp increases. 

 Since y is approximately equal to Pa, we have, approximately, 



yo = Pa/FM. (23A) 



