An inductance L having a resistance R is connected to an alternating source of angular frequency `omega`.The quality factor Q of the inductance is

Options

A. `(R)/(omegaL)`
B. `((omegaL)/(R))^(2)`
C. `((R)/(omegaL))^(1/2)`
D. `(omegaL)/(R)`

Explanation

To find the quality factor \( Q \) of an inductance \( L \) having resistance \( R \) connected to an alternating source of angular frequency \( \omega \), we can follow these steps:

### Step 1: Understand the Definition of Quality Factor
The quality factor \( Q \) is defined as the ratio of the voltage across the inductor \( V_L \) to the voltage across the resistance \( V_R \).

### Step 2: Express the Voltages
For an inductor, the voltage across it can be expressed as:
\[ V_L = I \cdot X_L \]
where \( I \) is the current through the circuit and \( X_L \) is the inductive reactance given by:
\[ X_L = \omega L \]

For the resistance \( R \), the voltage across it is:
\[ V_R = I \cdot R \]

### Step 3: Write the Expression for Quality Factor
Now, substituting the expressions for \( V_L \) and \( V_R \) into the definition of \( Q \):
\[ Q = \frac{V_L}{V_R} = \frac{I \cdot X_L}{I \cdot R} \]

### Step 4: Simplify the Expression
The current \( I \) cancels out from the numerator and denominator:
\[ Q = \frac{X_L}{R} \]

### Step 5: Substitute the Inductive Reactance
Now, substituting \( X_L = \omega L \) into the equation:
\[ Q = \frac{\omega L}{R} \]

### Final Answer
Thus, the quality factor \( Q \) of the inductance is:
\[ Q = \frac{\omega L}{R} \]

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