For an \(LCR\) circuit driven with voltage of amplitude \(V_m\) and frequency \(\omega_0=\frac{1}{\sqrt{LC}}\) the current exhibits resonance. The quality factor, \(Q\) is given by:
1. \(\frac{\omega_0 L}{R}\)
2. \(\frac{\omega_0R}{L}\)
3. \(\frac{R}{\omega_0C}\)
4. \(\frac{CR}{\omega_0}\)

A circuit connected to an AC source of emf \(\varepsilon=\varepsilon_0\sin(100t)\) with \(t\) in seconds, gives a phase difference of \(\frac{\pi}{4}\) between the emf \(\varepsilon\) and current \(i\). Which of the following circuits will exhibit this?

An inductance coil has a reactance of \(100~\Omega\). When an AC signal of frequency \(1000\) Hz is applied to the coil, the applied voltage leads the current by \(45^\circ\). The self-inductance of the coil is:
1. \( 1.1 \times 10^{-2} \mathrm{~H} \)
2. \(1.1 \times 10^{-1} \mathrm{~H} \)
3. \(5.5 \times 10^{-5} \mathrm{~H} \)
4. \(6.7 \times 10^{-7} \mathrm{~H} \)

An AC circuit has \(R=100~\Omega\), \(C=2~\mu\text{F}\), \(L=80~\text{mH}\), connected in series. The quality factor of the circuit  is:
1. \(0.5\)
2. \(20\)
3. \(2\)
4. \(400\)

A resonance circuit having inductance and resistance \(2\times 10^{-4}\) H and \(6.28~\Omega\) respectively oscillates at \(10\) MHz frequency. The value of quality factor of this resonator is: [\(\pi=3.14\)]
1. \(1000\)
2. \(2000\)
3. \(3000\)
4. \(4000\)

A series \(LCR\) circuit is designed to resonate at an angular frequency \(\omega_0=10^5~\mathrm{rad/s}\). The circuit draws \(16\) W power from \(120\) V source at resonance. The value of resistance '\(R\)' in the circuit is:
1. \(300~\Omega\)
2. \(600~\Omega\)
3. \(900~\Omega\)
4. \(100~\Omega\)

A transmitting station releases waves of wavelength \(960~\text{m}\). A capacitor of \(2.56~\mu F\) is used in the resonant circuit. The self-inductance of the coil necessary for resonance is:
1. \(10 \times 10^{-8} \mathrm{~H}\)
2. \(5 \times 10^{-8} \mathrm{~H}\)
3. \(10 \times 10^{-5} \mathrm{~H}\)
4. \(5 \times 10^{-4} \mathrm{~H}\)