A coil of resistance 200 ohms and self inductance 1.0 henry has been connected in series to an a.c. source of frequency `200/pi Hz`. The phase difference between voltage and current is

Options

A. `30^(@)`
B. `63^(@)`
C. `45^(@)`
D. `75^(@)`

Explanation

To find the phase difference between the voltage and current in a coil connected to an AC source, we can follow these steps:

### Step 1: Identify the given values
- Resistance (R) = 200 ohms
- Self-inductance (L) = 1.0 henry
- Frequency (f) = \( \frac{200}{\pi} \) Hz

### Step 2: Calculate the inductive reactance (X_L)
Inductive reactance (X_L) is given by the formula:
\[ X_L = 2 \pi f L \]
Substituting the values:
\[ f = \frac{200}{\pi} \quad \text{and} \quad L = 1.0 \text{ H} \]
\[ X_L = 2 \pi \left(\frac{200}{\pi}\right) \cdot 1.0 = 2 \cdot 200 = 400 \text{ ohms} \]

### Step 3: Calculate the impedance (Z)
The impedance (Z) in a series R-L circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values of R and X_L:
\[ Z = \sqrt{(200)^2 + (400)^2} = \sqrt{40000 + 160000} = \sqrt{200000} = 447.21 \text{ ohms} \]

### Step 4: Calculate the phase difference (φ)
The phase difference (φ) between the voltage and current in an R-L circuit can be calculated using the formula:
\[ \tan(\phi) = \frac{X_L}{R} \]
Substituting the values:
\[ \tan(\phi) = \frac{400}{200} = 2 \]
Now, to find φ, we take the arctan:
\[ \phi = \tan^{-1}(2) \]

### Step 5: Final Calculation
Using a calculator, we find:
\[ \phi \approx 63.43^\circ \]

### Final Answer
The phase difference between the voltage and current is approximately \( 63.43^\circ \).

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