What is the power factor of an LCR circuit connected to an AC source of angular frequency `(1)/(sqrt(LC))` ?

Explanation

To find the power factor of an LCR circuit connected to an AC source with an angular frequency of \( \frac{1}{\sqrt{LC}} \), we can follow these steps:

### Step 1: Understand the Power Factor
The power factor (PF) in an AC circuit is defined as the cosine of the phase angle \( \phi \) between the voltage and the current. It is given by the formula:
\[ \text{Power Factor} = \cos \phi = \frac{R}{Z} \]
where \( R \) is the resistance and \( Z \) is the impedance of the circuit.

### Step 2: Determine the Impedance \( Z \)
The impedance \( Z \) in an LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \( X_L \) is the inductive reactance and \( X_C \) is the capacitive reactance.

### Step 3: Calculate \( X_L \) and \( X_C \)
The inductive reactance \( X_L \) and capacitive reactance \( X_C \) are defined as:
\[ X_L = \omega L \quad \text{and} \quad X_C = \frac{1}{\omega C} \]
Here, \( \omega \) is the angular frequency.

### Step 4: Substitute the Given Frequency
We are given that the angular frequency \( \omega = \frac{1}{\sqrt{LC}} \). Now, substituting this into the equations for \( X_L \) and \( X_C \):
\[ X_L = \frac{1}{\sqrt{LC}} L = \frac{L}{\sqrt{LC}} = \sqrt{\frac{L}{C}} \]
\[ X_C = \frac{1}{\frac{1}{\sqrt{LC}} C} = \sqrt{\frac{1}{LC}} C = \sqrt{\frac{C}{L}} \]

### Step 5: Calculate \( X_L - X_C \)
Now we can find \( X_L - X_C \):
\[ X_L - X_C = \sqrt{\frac{L}{C}} - \sqrt{\frac{C}{L}} \]
To simplify this, we notice that when \( L \) and \( C \) are such that \( \omega = \frac{1}{\sqrt{LC}} \), we find that:
\[ X_L - X_C = 0 \]

### Step 6: Substitute into the Impedance Formula
Substituting \( X_L - X_C = 0 \) into the impedance formula gives:
\[ Z = \sqrt{R^2 + 0^2} = R \]

### Step 7: Calculate the Power Factor
Now substituting \( Z \) back into the power factor formula:
\[ \text{Power Factor} = \frac{R}{Z} = \frac{R}{R} = 1 \]

### Conclusion
Thus, the power factor of the LCR circuit connected to an AC source at the given frequency is:
\[ \text{Power Factor} = 1 \]