﻿Self and Mutual Induction of Coaxial Coils. 589 



(b) Overlapping coils. 



Regard the field inside the outer coil <\s made up of two 

 portions, 



(1) the uniform field calculated by assuming the coil to 



be part of an infinite coil ; 



(2) the field due to the polarity of the coil faces. 



If M x , M 2 are the linkages through the second coil due to 

 these two fields, then M = M! + M 2 ; M 2 is given by (42) 

 and (41) (no regard being paid to the sign of c), and 



M 1 =|ttV^(«2-«i)(^ 3 -&i 3 ) 5 • • (43)* 



k being the length of the overlap. 



When the coils have common centres, A = 0, and (42) 

 becomes 



M 2 = 2{M(Z 1 + / 2 )-M(/ 1 -/ 2 )}. . . . (426) 



If in addition they have the same length (2/), 



M 2 =2{M(2/)-M(0)} . . . . (42c) 



11. For the purposes of calculation, it is convenient to 

 alter the notation as in the following example. 

 Let the coils have the following dimensions : — 



Outer radii a 2 =10 cm., h 2 = 4 cm. 



Inner radii a^= 5 cm., h y = 2 cm. 



Lengths -h= 6 cm., 2 l 2 = 44 cm- 

 Displacement of centres = h = 21 cm. 



Then since the amount of overlapping is 4 cm., we have 

 from (43) 



Mi 9 



2 * = TV X 4 x 5 x (43-2 3 ) = 746-7. 



Again, 



h-^-L^ Cl = 4, c 1 /a 2 =z 1 =0*4, c 1 /a 1 =s 1 ! ^=0'S 

 /6 + Z 1 + Z 2 =c 2 =46, c 2 /a 2 =£ 2 =4'6, c 2 /%— V =9*2 

 7i~Z x 4-Z 2 = c 3 = 40. c z /a 2 = z 3 =&'Q, c 3 /a l = .: z ' = &0 

 7i + Z 1 —/ 2 =c 4 = 2, c 4 /a 2 =s 4 =0*2, c 4 /a 1 =^ 4 , =0 , 4 



&^a,=n=0'4, &i/a 2 =r 2 =02, 63/^=^=0-8, 6 1 / a x ==Sf «' ==0 '*- 

 * Maxwell., ' Electricity and Magnetism,' vol. ii. p. 312. 



