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 inductive reactance (X_L)
The inductive reactance (X_L) can be calculated using the formula:
\[ X_L = 2\pi f L \]
Substituting the values:
\[ X_L = 2\pi (50) (0.21) \]
\[ X_L = 2\pi (10.5) \]
\[ X_L = 21\pi \, \Omega \]

### Step 3: Calculate the phase angle (φ)
The phase angle (φ) between the current and the voltage in an R-L circuit is given by:
\[ \phi = \tan^{-1}\left(\frac{X_L}{R}\right) \]
Substituting the values of X_L and R:
\[ \phi = \tan^{-1}\left(\frac{21\pi}{12}\right) \]

### Step 4: Simplify the expression
To simplify:
\[ \phi = \tan^{-1}\left(\frac{21}{12} \cdot \pi\right) \]
This can be approximated using a calculator or trigonometric tables.

### Step 5: Calculate the numerical value
Using a calculator:
\[ \phi \approx \tan^{-1}(5.4978) \]
This gives:
\[ \phi \approx 79.67^\circ \]
Rounding off, we find:
\[ \phi \approx 80^\circ \]

### Conclusion
The phase angle between the current and the source voltage is approximately **80 degrees**.

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