A `12 ohm` resistor and a `0.21` henry inductor are connected in series to an `AC` source operating at 20volts,` 50` cycle/second. The phase angle between the current and the source voltage is

Options

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

Explanation

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

### Step 1: Identify the given values
- Resistance (R) = 12 ohms
- Inductance (L) = 0.21 henry
- Frequency (f) = 50 Hz
- Voltage (V) = 20 volts (not directly needed for phase angle calculation)

### Step 2: Calculate the inductive reactance (X_L)
The inductive reactance (X_L) can be calculated using the formula:
\[ X_L = \omega L \]
where
\[ \omega = 2\pi f \]
Substituting the values:
\[ \omega = 2 \pi \times 50 = 100\pi \, \text{rad/s} \]
Now calculate \( X_L \):
\[ X_L = 100\pi \times 0.21 \approx 66.0 \, \text{ohms} \]

### Step 3: Calculate the impedance (Z)
The total impedance (Z) in a series R-L circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values:
\[ Z = \sqrt{12^2 + 66^2} = \sqrt{144 + 4356} = \sqrt{4500} \approx 67.08 \, \text{ohms} \]

### Step 4: Calculate the phase angle (φ)
The phase angle (φ) can be calculated using the tangent function:
\[ \tan(\phi) = \frac{X_L}{R} \]
Substituting the values:
\[ \tan(\phi) = \frac{66.0}{12} \approx 5.5 \]

Now, to find φ, we take the arctangent:
\[ \phi = \tan^{-1}(5.5) \]

### Step 5: Calculate φ
Using a calculator:
\[ \phi \approx 80.54^\circ \]

### Conclusion
The phase angle between the current and the source voltage is approximately \( 80.54^\circ \).

---