A 0.07 H inductor and a 12`Omega` resistor are connected in series to a 220 V, 50 Hz ac source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. [Take `pi` as `(22)/(7)`]

Options

A. `8.8A and tan^(-1)((11)/(6))`
B. `88A and tan^(-1)((11)/(6))`
C. `0.88A and tan^(-1)((11)/(6))`
D. `8.8 A and tan^(-1)((6)/(11))`

Answer

To solve the problem, we need to find two things: the approximate current in the circuit and the phase angle between the current and the source voltage. Let's break it down step by step.

### Step 1: Calculate the Inductive Reactance (X_L)
The inductive reactance (X_L) is given by the formula:
\[ X_L = \omega L \]
where \( \omega = 2\pi f \) and \( f \) is the frequency.

Given:
- \( L = 0.07 \, \text{H} \)
- \( f = 50 \, \text{Hz} \)

First, calculate \( \omega \):
\[ \omega = 2 \times \frac{22}{7} \times 50 = \frac{2200}{7} \approx 314.29 \, \text{rad/s} \]

Now, substitute \( \omega \) into the formula for \( X_L \):
\[ X_L = \omega L = 314.29 \times 0.07 \approx 22 \, \Omega \]

### Step 2: Calculate the Impedance (Z)
The total impedance (Z) in a series circuit with resistance (R) and inductive reactance (X_L) is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Given:
- \( R = 12 \, \Omega \)

Now, substitute the values:
\[ Z = \sqrt{12^2 + 22^2} = \sqrt{144 + 484} = \sqrt{628} \approx 25.05 \, \Omega \]

### Step 3: Calculate the Current (I)
Using Ohm's Law for AC circuits, the current (I) can be calculated as:
\[ I = \frac{V}{Z} \]
Given:
- \( V = 220 \, \text{V} \)

Now, substitute the values:
\[ I = \frac{220}{25.05} \approx 8.8 \, \text{A} \]

### Step 4: Calculate the Phase Angle (θ)
The phase angle (θ) can be calculated using:
\[ \tan \theta = \frac{X_L}{R} \]
Substituting the values:
\[ \tan \theta = \frac{22}{12} = \frac{11}{6} \]

Now, calculate θ using the inverse tangent function:
\[ \theta = \tan^{-1} \left( \frac{11}{6} \right) \]

### Summary of Results
- The approximate current in the circuit is \( I \approx 8.8 \, \text{A} \).
- The phase angle \( \theta \) can be calculated as \( \tan^{-1} \left( \frac{11}{6} \right) \).