150 COEFFICIENTS OF INDUCTION. 



In the case of two equal circles at the distance x, we have, if we 

 make y = 0, 



(55) M 



780. SELF-INDUCTION OF A COIL. The preceding formula 

 enables us to calculate the coefficient of self-induction of a coil 

 that is to say, the total flow of force across a surface made up of 

 different windings for unit current. 



Let us consider the central section of the channel, and let P be 

 the outline of this section in a spire of radius a. Neglecting the 

 terms of correction, the flow of force from the windings which 

 correspond to an element dxdy of the section, situate at a distance r, 

 and which traverses the circle of radius a, has the value 



Sa 



['?-] 



If this expression be integrated for the whole extent of the 

 surface w of the section, we shall have for the total flux relative 

 to this circle 



M = 4iran\ I I dxdy / . - 2 . 



Instead of a single winding at a point P, consider the number 

 n}dx'dy' of windings, which correspond to the element of surface 

 dx'dy '. The value M' of the flow of force for this element of 

 surface is 



The total flow for the whole coil, or the coefficient of self- 

 induction in question, will be the integral of this expression ex- 

 tended over the whole surface o>. Considering first a as a constant 

 equal to the mean radius of the coil, we may write 



L = ^an\ HI I dxdydx'dy' (l . - - 

 (56) 



(/. Sa - 2) - ^an\ fill/* r dxdydx'dy' . 



