460 Induction of Currents in Linear Circuits [CH. xiv 



14. Two electrified conductors whose coefficients of electrostatic capacity are y l5 y 2 > r 

 are connected through a coil of resistance R and large inductance L. Verify that the 

 frequency of the electric oscillations thus established is 



J_ /2r + y 1+y . 2 I _ 

 r* L I 



15. An electric circuit contains an impressed electromotive force which alternates 

 in an arbitrary manner and also an inductance. Is it possible, by connecting the 

 extremities of the inductance to the poles of a condenser to arrange so that the current 

 in the circuit shall always be in step with the electromotive force and proportional to it ? 



16. Two coils (resistances R, S ; coefficients of induction Z, M, N) are arranged in 

 parallel in such positions that when a steady current is divided between the two, the 

 resultant magnetic force vanishes at a certain suspended galvanometer needle. Prove 

 that if the currents are suddenly started by completing a circuit including the coils, then 

 the initial magnetic force on the needle will not in general vanish, but that there will be 

 a " throw " of the needle, equal to that which would be produced by the steady (final) 

 current in the first wire flowing through that wire for a time interval 



M-L M-N 

 R S ' 



17. A condenser of capacity C is discharged through two circuits, one of resistance R 

 and self-induction Z, and the other of resistance R' and containing a condenser of capacity 

 C r . Prove that if Q is the charge on the condenser at any time, 



.L^dfqiRRB^dQ Q 

 c + C' +RR )W> + (c + C' + ^)dt + CC' = - 



18. A condenser of capacity C is connected by leads of resistance r, so as to be in 

 parallel with a coil of self-induction Z, the resistance, of the coil and its leads being R. If 

 this arrangement forms part of a circuit in which there is an electromotive force of period 



, shew that it can be replaced by a wire without self-induction if 



and that the resistance of this equivalent wire must be (Rr + L/C}/(R+r). 



19. Two coils, of which the coefficients of self- and mutual-induction are Zi, Z 2 , J/", 

 and the resistances R 1 , R 2 carry steady currents C\, 2 produced by constant electro- 

 motive forces inserted in them. Shew how to calculate the total extra currents produced 

 in the coils by inserting a given resistance in one of them, and thus also increasing its 

 coefficients of induction by given amounts. 



In the primary coil, supposed open, there is an electromotive force which would 

 produce a steady current (7, and in the secondary coil there is no electromotive force. 

 Prove that the current induced in the secondary by closing the primary is the same, as 

 regards its effects on a galvanometer and an electrodynamometer, and also with regard to 

 the heat produced by it, as a steady current of magnitude 



-i 



lasting for a time 



while the current induced in the secondary by suddenly breaking the primary circuit may 

 be represented in the same respects by a steady current of magnitude CM/2L 2 lasting for 

 a time 2Z 2 /7? 2 . 



