An AC source is connected to the LC series circuit with `V = 10 sin (314t)`. Find the current in the circuit as function of time ? (`L = 40 mH, C = 100 mu F`)

Options

A. `5.2 cos 314t`
B. `0.52cos 314t`
C. `0.52sin 314t`
D. `10 cos 314t`

Answer

To find the current in the LC series circuit connected to an AC source given by \( V = 10 \sin(314t) \), we can follow these steps:

### Step 1: Identify the given values
- Voltage source: \( V(t) = 10 \sin(314t) \)
- Inductance: \( L = 40 \, \text{mH} = 40 \times 10^{-3} \, \text{H} \)
- Capacitance: \( C = 100 \, \mu\text{F} = 100 \times 10^{-6} \, \text{F} \)
- Angular frequency: \( \omega = 314 \, \text{rad/s} \)

### Step 2: Calculate the reactance of the inductor and capacitor
- Inductive reactance \( X_L \) is given by:
\[ X_L = \omega L = 314 \times 40 \times 10^{-3} = 12.56 \, \Omega \]

- Capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{\omega C} = \frac{1}{314 \times 100 \times 10^{-6}} \approx 31.83 \, \Omega \]

### Step 3: Calculate the impedance of the circuit
Since there is no resistance in the circuit (\( R = 0 \)), the impedance \( Z \) is given by:
\[ Z = |X_L - X_C| = |12.56 - 31.83| = 19.27 \, \Omega \]

### Step 4: Determine the phase angle
The phase angle \( \phi \) can be calculated using:
\[ \tan(\phi) = \frac{X_L - X_C}{R} = \frac{12.56 - 31.83}{0} \text{ (undefined)} \]
Since \( X_C > X_L \), the current will lead the voltage by \( \phi = -90^\circ \) or \( \phi = -\frac{\pi}{2} \).

### Step 5: Calculate the peak current
The peak current \( I_0 \) can be calculated using Ohm's law:
\[ I_0 = \frac{V_0}{Z} = \frac{10}{19.27} \approx 0.519 \, \text{A} \]

### Step 6: Write the current as a function of time
Since the current leads the voltage by \( \frac{\pi}{2} \), we can express the current as:
\[ I(t) = I_0 \sin(\omega t + \phi) = 0.519 \sin(314t - \frac{\pi}{2}) \]
Using the identity \( \sin(x - \frac{\pi}{2}) = \cos(x) \), we can rewrite this as:
\[ I(t) = 0.519 \cos(314t) \]

### Final Answer
Thus, the current in the circuit as a function of time is:
\[ I(t) \approx 0.52 \cos(314t) \, \text{A} \]

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