An alternating voltage of 100 virtual volt is applied to a circuit of resistance `0.5 Omega` and inductance 0.01 H, the frequency being 50 Hz. What is the current and lag in time between voltage and current ?

Answer

To solve the problem step by step, we will determine the current and the time lag between the voltage and current in the given circuit.

### Step 1: Identify Given Values
- RMS Voltage, \( V_{\text{rms}} = 100 \, \text{V} \)
- Resistance, \( R = 0.5 \, \Omega \)
- Inductance, \( L = 0.01 \, \text{H} \)
- Frequency, \( f = 50 \, \text{Hz} \)

### Step 2: Calculate Angular Frequency (\( \omega \))
The angular frequency \( \omega \) is given by the formula:
\[ \omega = 2 \pi f \]
Substituting the value of \( f \):
\[ \omega = 2 \pi \times 50 = 314 \, \text{rad/s} \]

### Step 3: Calculate Inductive Reactance (\( X_L \))
Inductive reactance \( X_L \) is calculated using the formula:
\[ X_L = \omega L \]
Substituting the values:
\[ X_L = 314 \times 0.01 = 3.14 \, \Omega \]

### Step 4: Calculate Impedance (\( Z \))
The impedance \( Z \) in an R-L circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values:
\[ Z = \sqrt{(0.5)^2 + (3.14)^2} = \sqrt{0.25 + 9.8596} = \sqrt{10.1096} \approx 3.17 \, \Omega \]

### Step 5: Calculate RMS Current (\( I_{\text{rms}} \))
The RMS current can be calculated using the formula:
\[ I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} \]
Substituting the values:
\[ I_{\text{rms}} = \frac{100}{3.17} \approx 31.45 \, \text{A} \]

### Step 6: Calculate Phase Angle (\( \phi \))
The phase angle \( \phi \) is given by:
\[ \tan \phi = \frac{X_L}{R} \]
Substituting the values:
\[ \tan \phi = \frac{3.14}{0.5} = 6.28 \]
Now, calculate \( \phi \):
\[ \phi = \tan^{-1}(6.28) \approx 80.95^\circ \]

### Step 7: Convert Phase Angle to Radians
To convert degrees to radians:
\[ \phi \text{ (in radians)} = \frac{80.95 \times \pi}{180} \approx 1.413 \, \text{radians} \]

### Step 8: Calculate Time Lag (\( \Delta t \))
The time lag \( \Delta t \) is given by:
\[ \Delta t = \frac{\phi}{\omega} \]
Substituting the values:
\[ \Delta t = \frac{1.413}{314} \approx 0.00449 \, \text{s} \approx 4.49 \times 10^{-3} \, \text{s} \]

### Final Results
- The RMS current \( I_{\text{rms}} \approx 31.45 \, \text{A} \)
- The time lag \( \Delta t \approx 4.49 \, \text{ms} \)