Obtain an equation for the sharpness of resonance in an $L-C-R$ series $AC$ circuit. What is the quality factor $Q$? Explain bandwidth.

(N/A) $1$. Sharpness of Resonance: The sharpness of resonance is characterized by the ratio of the resonant frequency $\omega_0$ to the bandwidth $2\Delta\omega$. It is given by the expression $\frac{\omega_0}{2\Delta\omega} = \frac{\omega_0 L}{R}$. $A$ higher value indicates a sharper resonance.
$2$. Quality Factor $(Q)$: The quality factor $Q$ is defined as the ratio of the resonant frequency to the bandwidth. It measures the sharpness of the resonance curve. $Q = \frac{\omega_0 L}{R} = \frac{1}{R} \sqrt{\frac{L}{C}}$.
$3$. Bandwidth: The bandwidth is defined as the frequency range $\Delta\omega_1 + \Delta\omega_2 = 2\Delta\omega$ over which the current in the circuit is at least $\frac{1}{\sqrt{2}}$ times its maximum value. These frequencies are known as half-power frequencies.