An `LCR` circuit is connected to a source of alternating current. At resonance, the applied voltage and the current flowing through the circuit will have a phase difference of

Options

A. zero
B. `pi` / 4
C. `pi` / 2
D. `pi`

Answer

To solve the problem, we need to analyze the behavior of an LCR circuit at resonance. Here are the steps to arrive at the solution:

### Step 1: Understand the LCR Circuit
An LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series to an alternating current (AC) source. The circuit's behavior is characterized by the relationship between the inductive reactance (XL), capacitive reactance (XC), and resistance (R).

**Hint:** Recall that at resonance, the inductive reactance and capacitive reactance are equal.

### Step 2: Resonance Condition
At resonance, the condition is given by:
\[ X_L = X_C \]
This means that the voltage across the inductor (VL) and the voltage across the capacitor (VC) will be equal in magnitude but opposite in phase.

**Hint:** Remember that \( X_L = \omega L \) and \( X_C = \frac{1}{\omega C} \).

### Step 3: Phasor Diagram Analysis
In the phasor diagram:
- The voltage across the resistor (VR) is in phase with the current (I).
- The voltage across the inductor (VL) leads the current by 90 degrees.
- The voltage across the capacitor (VC) lags the current by 90 degrees.

**Hint:** Visualize the phasor diagram to see how the voltages relate to the current.

### Step 4: Voltage Relationships at Resonance
Since \( V_L = V_C \) at resonance, we can express the total voltage (V) in the circuit as:
\[ V = V_R \]
This means that the effective voltage across the circuit is purely resistive.

**Hint:** Consider how the voltages add up in a series circuit.

### Step 5: Phase Difference Calculation
At resonance, since \( V_R \) is in phase with the current (I), the phase difference (φ) between the total applied voltage and the current is:
\[ \phi = 0 \]
This indicates that the voltage and current reach their maximum and minimum values at the same time.

**Hint:** Recall that a phase difference of zero means the waveforms are perfectly aligned.

### Conclusion
Thus, at resonance in an LCR circuit, the applied voltage and the current flowing through the circuit will have a phase difference of **0 degrees**.

**Final Answer:** The phase difference is **0 degrees**.