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The value of quality factor is:...

The value of quality factor is:

A

`(omega L)/(R)`

B

`(omega)/(RC)`

C

`sqrt(LC)`

D

L/R

Text Solution

AI Generated Solution

The correct Answer is:
To find the value of the quality factor \( Q \), we can follow these steps: ### Step-by-Step Solution: 1. **Understand the Definition of Quality Factor**: The quality factor \( Q \) is defined as: \[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \] where \( R \) is the resistance, \( L \) is the inductance, and \( C \) is the capacitance. 2. **Resonance Condition**: At resonance in an RLC circuit, the inductive reactance \( X_L \) is equal to the capacitive reactance \( X_C \): \[ X_L = X_C \] This can be expressed as: \[ \omega L = \frac{1}{\omega C} \] where \( \omega \) is the angular frequency. 3. **Rearranging the Resonance Condition**: From the equation \( \omega L = \frac{1}{\omega C} \), we can multiply both sides by \( \omega \): \[ \omega^2 = \frac{1}{LC} \] This implies: \[ \frac{1}{C} = \omega^2 L \] 4. **Substituting into Quality Factor Formula**: Now, substitute \( \frac{1}{C} \) back into the quality factor formula: \[ Q = \frac{1}{R} \sqrt{L \cdot \frac{1}{\omega^2 L}} = \frac{1}{R} \sqrt{\frac{L}{\frac{1}{\omega^2 L}}} \] Simplifying this gives: \[ Q = \frac{1}{R} \sqrt{\frac{L^2}{\frac{1}{\omega^2}}} = \frac{1}{R} \cdot \frac{L \omega}{1} \] 5. **Final Expression for Quality Factor**: Thus, we arrive at the final expression for the quality factor: \[ Q = \frac{\omega L}{R} \] ### Conclusion: The value of the quality factor \( Q \) is given by: \[ Q = \frac{\omega L}{R} \]
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