In series LCR circuit at resonance

Options

A. Power factor is zero
B. Power developed across the resistor is maximum
C. Power developed across the inductor is zero
D. Both (2) & (3)

Answer

To solve the question regarding the series LCR circuit at resonance, we will follow these steps:

### Step 1: Understand Resonance in LCR Circuit
At resonance in a series LCR circuit, the inductive reactance (\(X_L\)) is equal to the capacitive reactance (\(X_C\)). This means:
\[ X_L = X_C \]

### Step 2: Express Inductive and Capacitive Reactance
The inductive reactance is given by:
\[ X_L = \omega L \]
And the capacitive reactance is given by:
\[ X_C = \frac{1}{\omega C} \]

### Step 3: Derive Resonant Frequency
Setting \(X_L\) equal to \(X_C\), we have:
\[ \omega L = \frac{1}{\omega C} \]
Multiplying both sides by \(\omega\) gives:
\[ \omega^2 = \frac{1}{LC} \]
Thus, the resonant frequency (\(\omega_0\)) is:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]

### Step 4: Calculate Impedance at Resonance
The impedance (\(Z\)) of the LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
At resonance, since \(X_L = X_C\), we have:
\[ Z = \sqrt{R^2 + 0} = R \]

### Step 5: Determine Power Factor
The power factor (\(cos \phi\)) in an LCR circuit is defined as:
\[ \cos \phi = \frac{R}{Z} \]
Substituting \(Z = R\):
\[ \cos \phi = \frac{R}{R} = 1 \]
This indicates that the power factor is maximum at resonance.

### Step 6: Analyze Power Developed Across Components
1. **Power across the resistor**: The power developed across the resistor is maximum because:
\[ P = V \cdot I \cdot \cos \phi \]
At resonance, \(\cos \phi = 1\), so the power is maximized.

2. **Power across the inductor**: The power developed across the inductor is zero because:
\[ P_L = V_L \cdot I_L \cdot \cos \phi \]
In an inductor, the voltage leads the current by \(90^\circ\) (or \(\frac{\pi}{2}\) radians), making \(\cos \phi = 0\). Therefore, \(P_L = 0\).

### Conclusion
From the analysis:
- The power factor is 1 (not zero).
- The power developed across the resistor is maximum.
- The power developed across the inductor is zero.

Thus, the correct options are:
- Power developed across the resistor is maximum.
- Power developed across the inductor is zero.

The final answer is that both the second and third options are correct.