Q factor of resonance is given by

Options

A. `(1)/(R ) sqrt((L)/(C ))`
B. `(1)/(R ) sqrt((C )/(L))`
C. `(1)/(L) sqrt((R )/(C ))`
D. `(1)/(C ) sqrt((L)/(R ))`

Answer

To find the Q factor of resonance, we can follow these steps:

### Step 1: Understand the Definition of 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 resonant frequency to the bandwidth of the circuit.

### Step 2: Write the Formula for Q Factor
The Q factor can be expressed mathematically as:
\[ Q = \frac{\omega_r}{\Delta \omega} \]
where:
- \( \omega_r \) is the resonant frequency,
- \( \Delta \omega \) is the bandwidth.

### Step 3: Identify Resonant Frequency
For an LC circuit, the resonant frequency \( \omega_r \) is given by:
\[ \omega_r = \frac{1}{\sqrt{LC}} \]
where:
- \( L \) is the inductance,
- \( C \) is the capacitance.

### Step 4: Determine Bandwidth
The bandwidth \( \Delta \omega \) can be defined in terms of resistance \( R \) and inductance \( L \):
\[ \Delta \omega = \frac{R}{L} \]

### Step 5: Substitute Values into Q Factor Formula
Now, substituting the expressions for \( \omega_r \) and \( \Delta \omega \) into the Q factor formula, we get:
\[ Q = \frac{\frac{1}{\sqrt{LC}}}{\frac{R}{L}} \]

### Step 6: Simplify the Expression
Simplifying this expression:
\[ Q = \frac{L}{R} \cdot \frac{1}{\sqrt{LC}} = \frac{L^{1/2}}{R \cdot C^{1/2}} \]
This can be further simplified to:
\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]

### Final Expression for Q Factor
Thus, the final expression for the Q factor of resonance is:
\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]