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LVIII. On the Coefficients of Self and Mutual Induction of 

 Coaxial Coils. By S. Butterworth, M.Sc, Lecturer in 

 Physics, School of Technology , Manchester*. 



1. A LTHOUGH many formulae have been given for the 



.l\. mutual induction of coaxial circles and solenoids, 

 little seems to have been done on the mutual induction of 

 coils for which the ratio of the inner and outer diameters 

 differs considerably from unity. The present investigation 

 is to supply suitable formulae for such cases. 



The method adopted is to find the mutual induction 

 between two mutually external semi-infinite coaxial coils 

 having zero core diameters and unit winding density (i.e. the 

 number of turns per unit area of channel section is unity), 

 and then by applying the laws of combination of mutual 

 inductances to find the mutual induction between finite 

 hollow coils. 



The results are extended so as to include self-induction. 



The semi-infinite coils of the nature indicated will, for 

 brevity, be referred to as "solid coils. " If we take the 

 radius of the larger coil as the unit of length, then only two 

 variables are involved, viz. : the radius of the smaller coil 

 ranging from zero to unity and the distance of the coil faces. 

 Dimensional considerations will give the correct formula 

 when the radius of the larger coil is not unity. 



2. For any magnetic field possessing circular symmetry 

 about an axis, the magnetic potential £1 satisfies the equation 



V + p*p &»~ ' "' 



in which z represents the distance along, and p the distance 

 from the axis of symmetry. A solution of this equation is 



12= ^ cf>{\)e-^J Q (\p)dk (2) 



* o 



reducing when p = to 



n = { X cj>{\)e->-- d\ (3) 



Jo 



If <£ is the stream function (t. e. the magnetic flux through 

 * Cominuiiicated by the Author. 



