The current in an LCR circuit is given by `I=20sin (100pi t+(pi)/(6))A`. The voltage across the the inductance L of `0.1 H` at t = 0 will be

Options

A. (a)31.4 V
B. (b)3.14 V
C. (c)157 V
D. (d)314 V

Answer

To find the voltage across the inductance \( L \) in the given LCR circuit at \( t = 0 \), we will follow these steps:

### Step 1: Identify the current equation
The current in the LCR circuit is given by:
\[ I = 20 \sin(100\pi t + \frac{\pi}{6}) \, \text{A} \]

### Step 2: Determine the parameters from the current equation
From the standard form of the current equation \( I = I_0 \sin(\omega t + \phi) \):
- \( I_0 = 20 \, \text{A} \) (amplitude of the current)
- \( \omega = 100\pi \, \text{rad/s} \) (angular frequency)
- \( \phi = \frac{\pi}{6} \) (phase angle)

### Step 3: Calculate the current at \( t = 0 \)
Substituting \( t = 0 \) into the current equation:
\[ I(0) = 20 \sin(100\pi \cdot 0 + \frac{\pi}{6}) = 20 \sin\left(\frac{\pi}{6}\right) \]
Using the value \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \):
\[ I(0) = 20 \cdot \frac{1}{2} = 10 \, \text{A} \]

### Step 4: Calculate the voltage across the inductor
The voltage across the inductor \( V_L \) can be calculated using the formula:
\[ V_L = I \cdot X_L \]
where \( X_L \) is the inductive reactance given by:
\[ X_L = \omega L \]
Substituting the values:
- \( \omega = 100\pi \)
- \( L = 0.1 \, \text{H} \)

Calculating \( X_L \):
\[ X_L = 100\pi \cdot 0.1 = 10\pi \, \Omega \]

Now substituting \( I(0) \) and \( X_L \) into the voltage equation:
\[ V_L = 10 \cdot (10\pi) = 100\pi \, \text{V} \]

### Step 5: Calculate the numerical value of \( V_L \)
Using \( \pi \approx 3.14 \):
\[ V_L \approx 100 \cdot 3.14 = 314 \, \text{V} \]

### Final Answer
The voltage across the inductance \( L \) at \( t = 0 \) is approximately:
\[ \boxed{314 \, \text{V}} \]