and 



and Inductance of a Concentric Main. 

 3 



547 



L = 2fi' log - + 



'./X 



V sj2 8 \ / 2m 2 a 2 SmWj 



2fi sinli m(c — b) */ 2 — sin m(c — b) sj2 



mb sj2 coshm(c — b) y2 + cosm(c— 6) y2 



(104) 



It is to be noticed that R becomes infinite when b = c, but 

 L becomes 



2fi' log (b/a) + (2fi/ma)(l/ V2-3/8 </2m 2 a 2 -3/$m*a»). 



The first term in (103) gives tbe resistance of the inner 

 conductor and the second that of the outer conductor. In 

 (104) the first term gives the linkages of the magnetic flux 

 in the dielectric with the current in the inner conductor. 

 The second term gives the linkages of the magnetic flux in 

 the substance of the inner conductor, and the third the 

 linkages of the flux in the outer conductor. 



1. With very high /requeue// currents. 



AY hen the frequency is very high, provided that c be not 

 nearly equal to b, we may write 



R = 



ptn pm 



77 ay/ 2 2irb^/: 



=^>G + 9- 



(10.5) 



Similarly when ma is very great we may write 



*-V.**| + £v^n>. (106) 



AVe see, therefore, that as / increases R continually in- 

 creases, but L approaches the value 2// log (b/a) asymptoti- 

 cally. Whatever the frequency, we see that the value of L 

 lies between the value given by (96) and 2// log (b/a). It 

 is easy to see that it has the latter value when the currents 

 are confined to an infinitely thin skin on the outer and inner 

 surfaces of the inner and outer conductors respectively. 



Formulae (105) and (106) agree with those given by Sir 

 Joseph Thomson (see 'Recent Researches/ p. 295). They 

 are also given in Heaviside's ' Electrical Papers,' vol. ii. 

 p. 193. 



2P2 



