For an RLC 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:

We begin with the standard series RLC circuit, in which the resistor $$R$$, inductor $$L$$ and capacitor $$C$$ are connected in series and are driven by a sinusoidal voltage source of angular frequency $$\omega$$ and peak (amplitude) value $$v_m$$.

For any series RLC circuit the total impedance is written as

$$Z = R + j\left(\omega L - \frac{1}{\omega C}\right),$$

where the symbol $$j$$ represents the imaginary unit $$\sqrt{-1}$$. The magnitude of this impedance is therefore

$$|Z| = \sqrt{R^{2} + \left(\omega L - \frac{1}{\omega C}\right)^{2}}.$$

Resonance in a series circuit occurs at that frequency $$\omega_0$$ at which the net reactance vanishes, i.e.

$$\omega_0 L - \frac{1}{\omega_0 C} = 0.$$

Solving this equation for the resonant angular frequency, we get

$$\omega_0 L = \frac{1}{\omega_0 C} \quad \Longrightarrow \quad \omega_0^2 = \frac{1}{LC} \quad \Longrightarrow \quad \omega_0 = \frac{1}{\sqrt{LC}}.$$

At resonance the impedance magnitude reduces to its purely resistive part, namely $$|Z| = R$$, and hence the current amplitude becomes maximum, signifying the phenomenon of current resonance.

Now we introduce the quality factor, denoted by $$Q$$. For a series resonant circuit, the definition used in most elementary courses is

$$Q = \frac{\text{Energy stored per cycle (maximum)}}{\text{Energy dissipated per cycle}}.$$

An alternative and very convenient expression—derived from that definition and widely quoted—is

$$Q = \frac{\omega_0 L}{R}.$$

We will show why this result holds. In an inductor the peak magnetic energy stored equals

$$U_L^{\text{(max)}} = \frac{1}{2} L I_m^{2},$$

while in a capacitor the peak electric energy stored is

$$U_C^{\text{(max)}} = \frac{1}{2} C V_m^{2}.$$

Because the inductor and capacitor exchange energy sinusoidally with each other, the total maximum energy stored in the reactive elements at resonance can be represented (for one half-cycle) by either of the above forms; customarily we use the inductive one, yielding

$$U_{\text{stored}}^{\text{(max)}} = \frac{1}{2} L I_m^{2}.$$

The average power dissipated in the resistor over a complete cycle is

$$P_{\text{avg}} = I_{\text{rms}}^{2} R,$$

and for a sinusoidal current the relation between peak and rms values is $$I_m = \sqrt{2}\,I_{\text{rms}}$$, so that

$$P_{\text{avg}} = \frac{I_m^{2}}{2}\,R.$$

The energy dissipated per cycle is therefore

$$W_{\text{diss}} = P_{\text{avg}} \times T,$$

where $$T = \tfrac{2\pi}{\omega_0}$$ is the period at resonance. Substituting, we obtain

$$W_{\text{diss}} = \frac{I_m^{2}}{2}\,R \times \frac{2\pi}{\omega_0} = \frac{\pi I_m^{2} R}{\omega_0}.$$

Using the definition $$Q = \dfrac{2\pi \times (\text{Maximum energy stored})}{\text{Energy dissipated per cycle}},$$ we write

$$Q = \frac{2\pi \times \frac{1}{2} L I_m^{2}}{\dfrac{\pi I_m^{2} R}{\omega_0}} = \frac{2\pi \times \frac{1}{2} L I_m^{2} \times \omega_0}{\pi I_m^{2} R} = \frac{\omega_0 L}{R}.$$

The current and its amplitude $$I_m$$ cancel out, leaving a remarkably simple relationship that ties together the three element parameters. This final algebraic form,

$$Q = \frac{\omega_0 L}{R},$$

is exactly the same as Option B in the list given.

None of the other options reproduces the factor $$\omega_0 L$$ in the numerator with $$R$$ in the denominator, so they are incorrect for a series RLC circuit at resonance.

Hence, the correct answer is Option B.