Find the quality factor os an oscillating circuit connected in series to a source of alternating emf if at resonance the voltage across the capacitor is `n` time that of the source.

Explanation

Given `V=V_(m) cos omegat`
`I=I_(m) cos ( omegat- varphi)`
where `I_(m)=(V_(m))/( sqrt(R^(2)+(omegaL-(1)/( omegaC))^(2)))`
Then `V_(C)=(intIdt)/( C)=(I_(m) sin ( omegat- varphi))/( omegaC)`
`=(V_(m))/( sqrt((1- omega^(2) lC)^(2)+( omegaRC)^(2)))sin ( omegat- varphi)`
As resonance the voltage amplitude across the capacitor
`=(V_(m))/( RC(1)/(sqrt(LC)))=sqrt((L)/(CR^(2)))V_(m)=nV_(m)`
so ` (L)/( CR^(2))=n^(2)`
Now `Q=sqrt((L)/9CR^(2)-(1)/(4))=sqrt(n^(2)-(1)/(4))`