At resonance, what will be the phase difference between voltage and a current in a series LCR circuit connected to an ac source?

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. 0

Explanation

To determine the phase difference between voltage and current in a series LCR circuit connected to an AC source at resonance, we can follow these steps:

### Step 1: Understand the concept of resonance in an LCR circuit
At resonance, the inductive reactance (XL) and capacitive reactance (XC) in the circuit are equal. This means that the total reactance in the circuit is zero.

### Step 2: Write the condition for resonance
The condition for resonance in a series LCR circuit is given by:
\[ X_L = X_C \]
where:
- \( X_L = \omega L \) (inductive reactance)
- \( X_C = \frac{1}{\omega C} \) (capacitive reactance)
Here, \( \omega \) is the angular frequency of the AC source, \( L \) is the inductance, and \( C \) is the capacitance.

### Step 3: Analyze the phase difference
In a series LCR circuit:
- When \( X_L > X_C \), the circuit is inductive, and the current lags the voltage (phase difference is positive).
- When \( X_C > X_L \), the circuit is capacitive, and the current leads the voltage (phase difference is negative).
- At resonance, since \( X_L = X_C \), the circuit behaves purely resistively.

### Step 4: Conclude the phase difference at resonance
At resonance, the phase difference \( \phi \) between voltage and current is:
\[ \phi = 0^\circ \]
This means that the voltage and current are in phase with each other.

### Final Answer
At resonance, the phase difference between voltage and current in a series LCR circuit connected to an AC source is \( 0^\circ \).

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