A series circuit consisting of a capacitor and a coil with active resistance is connected to a source of harmonic voltage whose frequency can be aried, keeping the voltage amplitdue are `n` times less than the resonance amplitude. Find `:`
`(a)` the resonance frequency,
`(b)` the quality factor of the circuit.

Explanation

At resonance
`omega_(0)L=(omega_(0)C)^(-1)` or `omega_(0)=(1)/(sqrt(LC))`
`(I_(m))_(res)=(V_(m))/(R)`.
Now `(V_(m))/( nR)=(V_(m))/sqrt(R^(2)+(omega_(1)L-(1)/(omega_(1)C))^(2))=(V_(m))/( sqrt(R^(2)+(omega_(2)L-(1)/( omega_(2)C))^(2)))`
Then `omega_(1)L-(1)/( omega_(1)C)=sqrt(n^(2)-1)R`
`omega_(2)L-(1)/( omega_(2)C)=sqrt(n^(2)-1)R` ( assuming `omega_(2) gt omega_(2))`
or `omega_(1)-(omega_(0)^(2))/( omega_(1))=- omega_(2)+(omega_(0)^(2))/( omega_(2))=- sqrt (n^(2)-1 )(R)/(L)`
or` omega_(1)+ omega_(2)=( omega_(0)^(2))/( omega_(1)omega_(2))(omega_(1)+ omega_(2))implies omega_(0)=sqrt( omega_(1) omega_(2))`
and ` omega_(2) - omega_(1)= sqrt( n^(2)-1) (R)/(L)`
`beta=(R)/(2L) = ( omega_(2)-omega_(1))/( 2 sqrt( n^(2)-1))`
and `Q=sqrt( (omega_(0)^(2))/(4 beta^(2))-(1)/(4))=sqrt(((n^(2)-1)omega_(1)omega_(2))/( (omega_(2)-omega_(1))^(2))=(1)/(4))`