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XXVI. A Formula for Calculating approximately the Self- 

 induction of a Coil. By Professor John Perry, D.Sc, 

 F.R.S* 



AN easy empirical formula for enabling the coefficient of 

 self-induction (the inductance) of any coil to be calcu- 

 lated with approximate correctness is very much wanted. I 

 have been led to take up the matter through having to design 

 an instrument with various cylindric coils whose inductances 

 should bear certain relations to one another, and it was neces- 

 sary for me to see at a glance with some exactness the effect 

 of altering their dimensions. It is to be observed that for 

 practical purposes extreme accuracy of calculation is of no 

 importance, because the mechanical inaccuracies in construct- 

 ing coils are considerable. It will be seen that my method of 

 arriving at what may almost be called a rational formula has 

 been suggested by Prof. James Thomson's ingenious method 

 of considering the flow of water over a rectangular gauge- 

 notch. 



I consider only hollow cylindric coils. Thus fig. 1 shows 

 the section of a coil whose dimensions are c centimetres mea- 

 sured radially, b parallel to the axis, mean radius of coil a, 

 o o' being the axis. The supposition I make is that all the 



coils considered are so long (b so great) that, - being the same 



in all, the field near the ends is the same in all. Without 

 much straining of the meaning of words, we may say that the 

 inductance of a coil is the total induction when unit current 

 passes in the coil ; and if there is only one convolution, it is 

 proportional to the reciprocal of what may be called the mag- 

 netic resistance. My supposition comes to this : that if the 

 dimension b is great enough, increase of the magnetic resist- 

 ance produced by increasing b is proportional to the increase 

 of b. Let MNM' (fig. 2) show the limiting length beyond 

 which increase of resistance will be proportional to increase of 

 length. Evidently M W must be a function of c and a, and 

 if written in the shape 2sc it may be expected that s will be 



constant for small values of -. The magnetic resistance due, 



a 



then, to the end parts, A B, of the coil (fig. 1) will be the 

 same as that of the coil, fig. 2. Now the magnetic resistances 

 of all similar coils are inversely proportional to their dimen- 

 sions, and for all coils similar to that shown in fig. 2 for any 



* Communicated by the Physical Society : read June 20, 1890. 



'R2 



