There is a `5 Omega` resistance in an `AC`, circuit. Inductance of `0.1 H` is connected with it in series. If equation of `AC` emf is `5 sin 50 t` then the phase difference between current and e.m.f. is

Options

A. `pi/2`
B. `pi/6`
C. `pi/4`
D. 0

Answer

To find the phase difference between the current and the EMF in the given AC circuit, we follow these steps:

### Step 1: Identify the given values
- Resistance (R) = 5 Ω
- Inductance (L) = 0.1 H
- AC EMF equation: \( V(t) = 5 \sin(50t) \)

### Step 2: Determine the angular frequency (ω)
From the EMF equation, we can identify the angular frequency:
- \( \omega = 50 \, \text{rad/s} \)

### Step 3: Calculate the inductive reactance (XL)
The inductive reactance (XL) is given by the formula:
\[ X_L = L \cdot \omega \]
Substituting the values:
\[ X_L = 0.1 \, \text{H} \cdot 50 \, \text{rad/s} = 5 \, \Omega \]

### Step 4: Calculate the total impedance (Z)
The total impedance (Z) in a series circuit with resistance and inductance is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values:
\[ Z = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \, \Omega \]

### Step 5: Calculate the phase difference (φ)
The phase difference (φ) between the current and the EMF can be calculated using:
\[ \cos(\phi) = \frac{R}{Z} \]
Substituting the values:
\[ \cos(\phi) = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 6: Find φ
To find φ, we take the inverse cosine:
\[ \phi = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) \]
This gives:
\[ \phi = 45^\circ = \frac{\pi}{4} \, \text{radians} \]

### Conclusion
The phase difference between the current and the EMF is:
\[ \phi = \frac{\pi}{4} \, \text{radians} \]