Field of Cylindrical Currents. 435 



Particular cases of the formula are interesting. For 

 instance, when 7i { is infinite we get 



=Hk> 



and when r is zero, 



Both of these results are well known. It is not difficult to 

 show that the latter result is true for a helical current i, the 

 radius of the helix being a, the axial length 2h u and the 

 number of turns Ni. 



When z is zero, that is at points on the plane bisecting 

 the axis of the cylindrical current sheet at right angles, we 

 have 



Sg-Ntt n 1 r» 3 .fy+2a') 



ft 3 + « 2 + '") 12 L ' 2'A 1 2 + a 2 + r 2 "*" 8 ' (A, 2 + a 2 + r 2 ) 3 



5 H(r 2 + 3« 2 ) -i 



+ i¥-(A 1 2 +a 2 +^ + -J- • ( 28 > 



When r 2 is small compared with A x 2 + a 2 + ? ,2 ? the series con- 

 verges with great rapidity. This result, however, is not 

 true for a helical current. 



7. Formulae for the mutual inductance between two coaxial 

 cylindrical-current sheets. 



Let us suppose that there is a cylinder of radius b and 

 axial length 2h 2 , coaxial and concentric with the cylinder 

 considered above. We shall suppose that b is less than a and 

 that we have N 2 filaments equally spaced on the inner cylinder 

 so that N 2 /(27i 2 ) gives the turns per unit length. We have 

 now to calculate the linkages M of the flux, due to unit 

 current in the outer cylinder, with the filaments of the inner 

 cylinder. 



If cCSi denote the linkages due to the flux through the 

 element of area rd6 dr from z = h 2 to z= — 7i 2 , we have 



-A 2 



and hence, by (26), 



r+ho 

 dM=l ZrdQdrT$ 2 (dz/2h 2 ), 



J —h. 



M2J0J0 



-{(*i- A 3 ) 2 +A 2 } 1/2 ]#^. 



2 H2 



