444 Mr. A. Russell on the Magnetic 



For the calculation of the mutual inductance standards 

 required for calibrating the galvanometers used in iron- 

 testing, this formula is suitable. In order that the coils be as 

 compact as possible, it is necessary to make h = a and b large, 

 but even in this case the series can be easily summed. 



Let us now consider the case when the cylinder has the 

 same radius as the helix. We may suppose, for instance, that 

 both coils are wound on the same insulating cylinder. The 

 formula (37) now becomes 



M = 47ra 2 ^- 2 "sin 2 6 [R^l-^ 2 sin 2 6)^ 



_R 2 (l--£ 2 2 sin 2 0) 1/2 ]d0. 

 Hence, from (8), 



- R ^S iE - E ^4 (43) 



where 



Ri 2 =4a 2 + (A 1 + A 2 ) 2 , R 2 2 = 4a 2 + (/ ?1 -/g 2 , 

 /t'! 2 = 4« 2 /R 1 2 , and k 2 2 = 4a 2 /R 2 2 . 



The value of M can be readily found from this formula with 

 the help of Dale's Mathematical Tables or by the series f 

 E and F given in § 2. 



If we have, in addition, h 1 = h 2 = h, then 



R 1 = 2(a 2 + A2)V2 j B a = 2a, h 2 = a 2 / (a* + h*) , and Jc 2 2 = l. 

 Hence 



or 



M = ~j-\n s 



If the height 2h equal the diameter 2a, so that the coils are 

 as compact as possible, we may write 



M = 13-589^^ (45) 



9. Formulas for the self-inductance of a helical current when 

 the pitch of the helix is small. 



If we have a wire closely wound on a smooth insulating 

 cylinder, the diameter of the wire being small compared with 

 that of the cylinder, then the mutual inductance between it 

 and a cylindrical current sheet of the same radius as the 

 helix formed by the axis of the wire and having the same 



