In a series LCR circuit, the voltages across an inductor, a capacitor and a resistor are 30 V, 30 V, 60 V respectively. What is the phase difference between the applied voltage and the current in the circuit ?

Options

A. `60^@`
B. `30^@`
C. `90^@`
D. `0^@`

Answer

To solve the problem, we need to find the phase difference (Φ) between the applied voltage and the current in a series LCR circuit, given the voltages across the inductor (V_L), capacitor (V_C), and resistor (V_R).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Voltage across the inductor (V_L) = 30 V
- Voltage across the capacitor (V_C) = 30 V
- Voltage across the resistor (V_R) = 60 V

2. **Understand the Relationship in LCR Circuits:**
In a series LCR circuit, the phase difference (Φ) between the total voltage and the current can be found using the formula:
\[ \tan(\Phi) = \frac{X_L - X_C}{R} \]
where:
- \(X_L\) = inductive reactance
- \(X_C\) = capacitive reactance
- \(R\) = resistance

3. **Relate Voltages to Reactances and Resistance:**
The voltages across the components can be related to the current (I) as follows:
- \(V_L = I \cdot X_L\)
- \(V_C = I \cdot X_C\)
- \(V_R = I \cdot R\)

From the given voltages:
- \(X_L = \frac{V_L}{I} = \frac{30}{I}\)
- \(X_C = \frac{V_C}{I} = \frac{30}{I}\)
- \(R = \frac{V_R}{I} = \frac{60}{I}\)

4. **Substitute into the Tan Formula:**
Substitute the expressions for \(X_L\), \(X_C\), and \(R\) into the tan formula:
\[ \tan(\Phi) = \frac{\frac{30}{I} - \frac{30}{I}}{\frac{60}{I}} \]
Simplifying this gives:
\[ \tan(\Phi) = \frac{0}{\frac{60}{I}} = 0 \]

5. **Find the Phase Difference:**
Since \(\tan(\Phi) = 0\), it implies that:
\[ \Phi = 0^\circ \]

### Final Answer:
The phase difference between the applied voltage and the current in the circuit is \(0^\circ\).

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