A series L-C-R circuit is operated at resonance . Then

Options

A. voltage across R is minimum
B. impedance is minimum
C. impedance is maximum
D. current amplitude is minimum

Answer

To solve the question regarding the behavior of a series L-C-R circuit at resonance, we will analyze the properties of the circuit step by step.

### Step-by-Step Solution:

1. **Understanding Resonance in L-C-R Circuit**:
At resonance, the inductive reactance (X_L) equals the capacitive reactance (X_C). This condition can be expressed as:
\[ X_L = X_C \]

2. **Impedance Calculation**:
The impedance (Z) of a series L-C-R circuit is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Since at resonance \(X_L = X_C\), the equation simplifies to:
\[ Z = \sqrt{R^2 + 0^2} = R \]
This indicates that the impedance is at its minimum value, which is equal to the resistance (R).

3. **Current Calculation**:
The root mean square (RMS) current (I_rms) in the circuit can be calculated using Ohm's law for AC circuits:
\[ I_{rms} = \frac{V_{rms}}{Z} \]
Since we found that \(Z = R\), we can rewrite the equation as:
\[ I_{rms} = \frac{V_{rms}}{R} \]
Given that Z is at its minimum (equal to R), the current I_rms is at its maximum.

4. **Voltage Across Resistor**:
The voltage across the resistor (V_R) in the circuit can be calculated using Ohm's law:
\[ V_R = I_{rms} \cdot R \]
Since the current is at its maximum when the impedance is minimum, the voltage across the resistor will also be significant but not minimum.

5. **Conclusion**:
From the analysis above, we find that:
- The impedance (Z) is minimum at resonance.
- The current (I_rms) is maximum at resonance.
- The voltage across the resistor is not minimum but remains significant.
- Therefore, the correct statement is that the impedance is minimum.

### Final Answer:
The correct answer is: **Impedance is minimum**.