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 angular frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of f:
\[ \omega = 2\pi \left(\frac{200}{\pi}\right) = 400 \, \text{rad/s} \]

### Step 3: Calculate the inductive reactance (X_L)
The inductive reactance (X_L) is calculated using the formula:
\[ X_L = L \cdot \omega \]
Substituting the values of L and ω:
\[ X_L = 1.0 \, \text{H} \cdot 400 \, \text{rad/s} = 400 \, \text{ohms} \]

### Step 4: 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:
\[ Z = \sqrt{(200)^2 + (400)^2} = \sqrt{40000 + 160000} = \sqrt{200000} = 100\sqrt{20} \, \text{ohms} \]

### Step 5: Calculate the phase difference (φ)
The phase difference (φ) between the voltage and current can be found using:
\[ \cos(\phi) = \frac{R}{Z} \]
Substituting the values of R and Z:
\[ \cos(\phi) = \frac{200}{100\sqrt{20}} = \frac{2}{\sqrt{20}} = \frac{2}{\sqrt{4 \cdot 5}} = \frac{2}{2\sqrt{5}} = \frac{1}{\sqrt{5}} \]
Now, to find φ:
\[ \phi = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right) \]

### Step 6: Approximate φ
Using a calculator or trigonometric tables, we find:
\[ \phi \approx 63.43^\circ \]

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