The value of quality factor is:

Options

A. `(omega L)/(R)`
B. `(omega)/(RC)`
C. `sqrt(LC)`
D. L/R

Answer

To find the value of the quality factor (Q) in an LCR resonance circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Quality Factor (Q)**:
The quality factor (Q) of an LCR circuit is defined as the ratio of the resonant frequency to the bandwidth of the circuit. It can also be expressed in terms of circuit components.

2. **Formula for Quality Factor**:
The formula for the quality factor Q in an LCR circuit is given by:
\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]
where R is the resistance, L is the inductance, and C is the capacitance.

3. **Resonant Frequency**:
At resonance, the capacitive reactance (X_C) is equal to the inductive reactance (X_L). The expressions for these reactances are:
\[ X_C = \frac{1}{\omega_0 C} \quad \text{and} \quad X_L = \omega_0 L \]
where \(\omega_0\) is the resonant angular frequency.

4. **Setting Reactances Equal**:
At resonance, we have:
\[ X_C = X_L \implies \frac{1}{\omega_0 C} = \omega_0 L \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{1}{C} = \omega_0^2 L \]

6. **Substituting into the Quality Factor Formula**:
We can substitute \( \frac{1}{C} \) into the quality factor equation:
\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} = \frac{1}{R} \sqrt{L \cdot \frac{1}{\omega_0^2 L}} = \frac{\omega_0 L}{R} \]

7. **Final Expression for Quality Factor**:
Thus, the final expression for the quality factor is:
\[ Q = \frac{\omega_0 L}{R} \]

### Summary:
The quality factor (Q) of an LCR circuit is given by:
\[ Q = \frac{\omega_0 L}{R} \]