Resistance and Inductance of a Helical Coil. 89 



and the self-induction is changed by the concentration of 

 current to an amount 



-£ co , a(1 _ 5s ^j{i-i(^)vj- (^y} 



In these results it is not possible to proceed to the limit of 

 zero frequency, for it has been assumed that k"a 2 is large. 

 Although, therefore, the expression for R does not become 

 cr/Vr 2 when p is made zero, no error is involved on this 

 account. Near p = Q another form of expression must be 

 used. 



The limitations of these results are determined by the con- 

 siderations that kr shall be less than 2, and that r d ja d and 

 ka~* shall be negligible. As an average practical case, we 

 take a copper wire of radius 2 millimetres, and determine 

 the requisite conditions for a three figure accuracy. In the 

 first place, r 3 /« 3 cannot be ignored, unless a is greater than 

 12r approximately. This condition is usually fulfilled. Even 

 if it is not, the order of accuracy may extend to two figures 

 for a much greater value of rja, so that this limitation is not 

 of great moment. 



Secondly, kr is less than 2, so that in the present case, if 

 fbe the frequency, and er=16% C.G.s. units, approximately 



! /;/'' < 2 or rf* < 10 





where r is in centimetres. The upper limit in frequency is 

 therefore about 2500 per second. 



Thirdly, (Ica)~ A may be neglected if ka>10, so that, to 

 determine the lower limit, there is an inequality 



a/£ > 50, 



or if a=12r in the most unfavourable case,/ is about 400 

 per second. The formula?, therefore, range between fre- 

 quencies of 400 and 2500 for a wire of 2 millimetres radius, 

 wound on an appropriate cylinder. In general the conditions 

 for a three figure accuracy are 



a > 127^, af± > 50, r/i <: 10, . . . (44) 



the radii being in centimetres. 



