Figure $(a)$ shows a plot of voltage across the capacitor as a function of the driving frequency for a sinusoidally driven electromagnetic $LCR$ circuit. Figure $(b)$ shows the phase angle $\phi$ (phase difference between voltage and current) vs $\omega / \omega_0$ graph for the same circuit,for three different quality factors corresponding to graphs $1, 2, 3$ of figure $(a)$. Each graph in figure $(a)$ can be matched by one of the graphs $a, b, c$ in figure $(b)$. Which of the following statements is correct?

  • A
    Graph $(3)$ corresponds to graph $(a)$
  • B
    Graph $(1)$ corresponds to graph $(c)$
  • C
    The circuit of graph $1$ has a high quality factor
  • D
    The circuit of graph $3$ has a high quality factor

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$A$ $10 \Omega$ resistance,$5 \ mH$ coil,and $10 \ \mu F$ capacitor are joined in series. When an alternating current source of suitable frequency is joined to this combination,the circuit resonates. If the resistance is halved,the resonant frequency is

In a series $LCR$ circuit with $C = 2 \mu F$,$L = 1 \ mH$,and $R = 10 \ \Omega$,when the current in the circuit is maximum,what is the ratio of the energy stored in the capacitor to the energy stored in the inductor?

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Suppose the frequency of the source in the previous example can be varied.
$(a)$ What is the frequency of the source at which resonance occurs?
$(b)$ Calculate the impedance,the current,and the power dissipated at the resonant condition.

In a series $R-L-C$ $AC$ circuit,for a particular value of $R, L$ and $C$,the power supplied by the source is $P$ at resonance. If the value of inductance is halved,then the power from the source again at resonance is $P'$. Then:

The power factor of an $LCR$ circuit at resonance is:

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