﻿54 Dr. J. G. Leathern : Effect of a Long 



It is readily seen that the scattered wave will have its 

 magnetic force y x parallel to z, and that y x must satisfy the 

 differential equation of wave propagation, which is of the 

 same form as (12). Hence we assume 



7i = 2 A s h 8 {icr) cos sde*P f ; (15) 



so that the total magnetic force in the dielectric is 



7 = y + 7i = ^2[{H^(icr)+AAW}cos^]. . (16) 



From the circuital relation between displacement current 

 and magnetic force, it is readily seen that the component of 

 electric intensity perpendicular to r is a constant multiplied 

 by "by for; the surface condition is that this shall vanish for 

 r = a, since the wire is a perfect conductor. 



Thus for each value of s 



H # s >a)+A s ///(*a)=0, 

 and therefore the scattered wave is given by 



7i=-H^.'(«)^^co.ftf. . . . (17) 



It may be assumed that the radius of the wire is very small 

 compared with the wave-length, so that tea is extremely small. 

 Perhaps the simplest way of arriving at an approximation to 

 g s { K a) for Ka very small is by the usual method of evaluating 

 a coefficient in a Fourier series. Multiply both sides of 

 equation (11) by cos sd and integrate from #=0 to O — tir ; 

 after dividing by 7r, the result is 



1 C 2n 

 9 s i Kr ) = ~ e- iKr C08 -* cos s6 dd 





= -f 2 cossl92 \(-iKrco*0) n dO. . . . (18) 

 ttJo nl y 



Clearly the terms corresponding to values of n less than s 

 give zero on integration, so when Kr is small the most 

 important term is 



~^TJl \ cos* 6 cos sddO (19) 



Thus g s (Ka) is of the order of magnitude of (*a)*, and 

 g s '(Ka) is of the order of magnitude of (tea)*- 1 except in the 

 case of sr=0. So the successive terms in the series for y x 



