In a series LCR circuit connected to an ac source of variable frequency and voltage `v =v_(m) sin omega t, ` draw a plot showing the variation of current (I) with angular frequency `(omega)` for two different values of resistance `R_(1) and R_(2) (R_(1) gt R_(2))`. Write the condition under which the phenomenon of resonance occurs. For which value of the resistance out of the two curves, a sharper resonance is produced ? Define Q-factor of the circuit and give its significance.

Explanation

Figure shows the variation of `i_(m)` with `omega` in a `LCR` series circuit for two values of resistance `R_(1)` and `R_(2)(R_(1) gt R_(2))`
The condition for resonance in the `LCR` circuit is.
`omega_(0)=1/sqrt(LC)`
We see that the current amplitude is maximum at the resonant frequency `omega`.Since `i_(m)=V_(m)//R` at resonance. the current amplitude of case `R_(2)` is shaper to that for case `R_(1)`.
Quality factor or simply the `Q`-factor of a resonant `LCR` circuit is defined as the ratio of voltage drop across the resistance at resonance.
`Q=V_(L)/V_(R)=(omegaL)/R`
Thus, finally, `Q=1/Rsqrt(L/C)`
The `Q` factor determines the sharpness at resonance as for higher value of `Q` factor the tuning of the circuit and it sensivity to accept resonating frequency signals will be much higher.