A series RL circuit is connected to an AC power supply of 120 V . The frequency of source is 50 Hz. Calculate the phase difference between the current and voltage in the circuit if values of R and L are 10 `Omega` and 0.01 H, respectively. Also find the current in the circuit .

Answer

To solve the problem step by step, we will calculate the phase difference between the current and voltage in a series RL circuit and then find the current in the circuit.

### Step 1: Calculate the angular frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
where \( f \) is the frequency.

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

Calculating ω:
\[ \omega = 2 \times \pi \times 50 = 2 \times 3.14 \times 50 \approx 314 \, \text{rad/s} \]

### Step 2: Calculate the inductive reactance (X_L)
The inductive reactance (X_L) is given by the formula:
\[ X_L = L \omega \]
where \( L \) is the inductance.

Given:
- \( L = 0.01 \, \text{H} \)

Calculating \( X_L \):
\[ X_L = 0.01 \times 314 \approx 3.14 \, \Omega \]

### Step 3: Calculate the impedance (Z)
The impedance (Z) in a series RL circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]

Given:
- \( R = 10 \, \Omega \)

Calculating Z:
\[ Z = \sqrt{10^2 + 3.14^2} = \sqrt{100 + 9.8596} = \sqrt{109.8596} \approx 10.48 \, \Omega \]

### Step 4: Calculate the phase difference (θ)
The phase difference (θ) between the current and voltage in an RL circuit can be calculated using:
\[ \tan(\theta) = \frac{X_L}{R} \]

Calculating θ:
\[ \tan(\theta) = \frac{3.14}{10} \]
\[ \theta = \tan^{-1}(0.314) \approx 17.43^\circ \]

### Step 5: Calculate the current (I)
The current (I) in the circuit can be calculated using Ohm's law:
\[ I = \frac{V}{Z} \]

Given:
- \( V = 120 \, \text{V} \)

Calculating I:
\[ I = \frac{120}{10.48} \approx 11.45 \, \text{A} \]

### Final Answers
- The phase difference between the current and voltage is approximately \( 17.43^\circ \).
- The current in the circuit is approximately \( 11.45 \, \text{A} \).