A 12 `Omega` resistor and a 0.21 H inductor are connected in series to an a.c. source operating at V, 50 cycle second. The phase angle between current and source vottage is

Options

A. `30^(@)`
B. `40^(@)`
C. `80^(@)`
D. `90^(@)`

Answer

To find the phase angle between the current and the source voltage in an AC circuit with a resistor and an inductor in series, we can follow these steps:

### Step 1: Identify the given values
- Resistance (R) = 12 Ω
- Inductance (L) = 0.21 H
- Frequency (f) = 50 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 frequency:
\[ \omega = 2\pi \times 50 = 100\pi \text{ rad/s} \]

### Step 3: Calculate the inductive reactance (X_L)
The inductive reactance (X_L) is calculated using the formula:
\[ X_L = \omega L \]
Substituting the values:
\[ X_L = (100\pi) \times 0.21 = 21\pi \text{ Ω} \]

### Step 4: Calculate the impedance (Z)
The impedance (Z) of the circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values:
\[ Z = \sqrt{(12)^2 + (21\pi)^2} \]

Calculating \( (21\pi)^2 \):
\[ (21\pi)^2 \approx 441 \times 9.87 \approx 436.68 \]
So,
\[ Z = \sqrt{144 + 436.68} = \sqrt{580.68} \approx 24.1 \text{ Ω} \]

### Step 5: Calculate the phase angle (φ)
The phase angle (φ) can be calculated using the formula:
\[ \tan(\phi) = \frac{X_L}{R} \]
Substituting the values:
\[ \tan(\phi) = \frac{21\pi}{12} \]
Now, calculating φ:
\[ \phi = \tan^{-1}\left(\frac{21\pi}{12}\right) \]
Using a calculator:
\[ \phi \approx \tan^{-1}(5.49) \approx 80^\circ \]

### Final Answer
The phase angle between the current and the source voltage is approximately \( 80^\circ \).

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