In an LCR series resonant circuit which one of the following cannot be the expression for the Q-factor

Options

A. `(omega_(r))L/(R )`
B. `(1)/(omega CR)`
C. `sqrt((L)/(C )) (1)/(R )`
D. `(R )/(LC)`

Explanation

To determine which expression cannot be the Q-factor in an LCR series resonant circuit, we will analyze the Q-factor and its possible expressions step by step.

### Step-by-Step Solution:

1. **Understanding the Q-factor**:
The Q-factor (Quality factor) of a resonant circuit is a dimensionless parameter that describes how underdamped the circuit is, and it is defined as the ratio of the reactive power to the resistive power in the circuit.

2. **Basic Formula for Q-factor**:
The Q-factor can be expressed in terms of the inductive reactance (XL) and the resistance (R) of the circuit:
\[ Q = \frac{X_L}{R} \]
where \(X_L = \omega L\) (with \(\omega\) being the angular frequency).

3. **Resonance Condition**:
At resonance in an LCR circuit, the inductive reactance equals the capacitive reactance:
\[ X_L = X_C \quad \Rightarrow \quad \omega L = \frac{1}{\omega C} \]
This leads to the resonant frequency:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]

4. **Alternative Expressions for Q-factor**:
Using the resonance condition, we can express the Q-factor in different forms:
- Substituting \(X_L\) into the Q-factor formula:
\[ Q = \frac{\omega L}{R} \]
- Another form can be derived from the relationship between \(X_C\) and \(R\):
\[ Q = \frac{X_C}{R} = \frac{1/\omega C}{R} = \frac{1}{\omega CR} \]

5. **Evaluating Given Options**:
Now, we will check the possible expressions for the Q-factor:
- **Option 1**: \( \frac{\omega L}{R} \) - This is valid.
- **Option 2**: \( \frac{1}{\omega CR} \) - This is also valid.
- **Option 3**: \( \frac{1}{\omega C} R \) - This can be rearranged to show it is valid.
- **Option 4**: \( \frac{R}{LC} \) - This does not fit any derived expression for the Q-factor.

6. **Conclusion**:
The expression that cannot be the Q-factor in an LCR series resonant circuit is:
\[ \text{Option 4: } \frac{R}{LC} \]