An oscillating circuit consisting of a coil and a capacitor connected in series if fed an alternating emf, with coil inductance being chosen to provide the maximum current in the circuuit. Find the quality factor os the system, provided an `n-`fold increase of inductance results in an `n-` fold decrease of current in the circuit.

Explanation

For maximum current amplitude
`I_(m)=(V_(m))/( sqrt(R^(2)+( omegaL-(1)/( omegaC))^(2)))`
`L=(1)/( omega^(2) C) `and then `I_(m0)=(V_(m))/(R)`
Now `(I_(m0))/( eta)=(V_(m))/( sqrt(R^(2)+((n-1)^(2))/(omega^(2)C^(2))))`
So` eta=sqrt(1+((n-1)^(2))/((omegaRC)^(2)))`
`omega RC=(n-1)/( sqrt(eta^(2)-1))`
Now `Q=sqrt(((L)/( CR^(2)))^(2)-(1)/(4))=sqrt(((1)/(omegaRC))^(2)-(1)/(4))=sqrt((eta^(2)-1)/(( n-1)^(2))-(1)/(4))`