In a series RL circuit with `L=(175)/(11)mH` and `R=12Omega` and AC source of emf `e=130sqrt(2)V`, 50Hz is applied. Find the circuit impedance and phase difference of EMF and current in circuit.

Options

A. `14.14A,30^(@)`
B. `10sqrt2 A.tan^(-1)5/12`
C. `10 A.tan^(-1)5/12`
D. `20 A,tan^(-1)5/12`

Explanation

To solve the problem step by step, we will find the circuit impedance \( Z \) and the phase difference \( \phi \) between the EMF and the current in the RL circuit.

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

Given:
- \( L = \frac{175}{11} \, \text{mH} = \frac{175}{11} \times 10^{-3} \, \text{H} \)
- \( f = 50 \, \text{Hz} \)

Calculating \( \omega \):
\[ \omega = 2 \pi \times 50 = 100\pi \, \text{rad/s} \]

Now, substituting the values:
\[ X_L = 100\pi \times \left(\frac{175}{11} \times 10^{-3}\right) \]

Calculating \( X_L \):
\[ X_L = \frac{100 \times 175 \times \pi}{11 \times 1000} = \frac{17500\pi}{11000} \approx 5 \, \Omega \]

### Step 2: Calculate the Impedance \( Z \)
The impedance \( Z \) in a series RL circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
where \( R = 12 \, \Omega \).

Substituting the values:
\[ Z = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \Omega \]

### Step 3: Calculate the Phase Difference \( \phi \)
The phase difference \( \phi \) is given by:
\[ \tan \phi = \frac{X_L}{R} \]

Substituting the values:
\[ \tan \phi = \frac{5}{12} \]

Now, calculating \( \phi \):
\[ \phi = \tan^{-1}\left(\frac{5}{12}\right) \]

### Step 4: Calculate the Current \( I \)
The current \( I \) in the circuit can be calculated using Ohm's law:
\[ I = \frac{E}{Z} \]
where \( E = 130\sqrt{2} \, V \).

Substituting the values:
\[ I = \frac{130\sqrt{2}}{13} = 10\sqrt{2} \, A \]

### Final Results
- The impedance \( Z \) is \( 13 \, \Omega \).
- The phase difference \( \phi \) is \( \tan^{-1}\left(\frac{5}{12}\right) \).
- The current \( I \) is \( 10\sqrt{2} \, A \).