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To solve this problem, we need to ensure that the current \( i(t) \) is in-phase with the voltage \( v_s(t) \), which implies that the total impedance \( Z \) seen by the source must be purely resistive (no imaginary component).

### Steps to Solve the Problem

1. **Impedance Analysis**:
   - The circuit consists of a series resistor \( 50 \, \Omega \) and a parallel combination of a \( 25 \, \Omega \) resistor, a \( 4 \, \mu \text{F} \) capacitor, and an inductor \( L \).

2. **Impedance Conditions for \( i(t) \) to be In-Phase**:
   - For \( i(t) \) to be in-phase with \( v_s(t) \), the imaginary part of the impedance of the parallel branch must cancel out with the inductor.

3. **Impedance of the Capacitor and Inductor**:
   - The capacitive reactance \( X_C = \frac{1}{\omega C} \), where \( C = 4 \, \mu \text{F} \) and \( \omega = 10^4 \, \text{rad/s} \).
   - The inductive reactance \( X_L = \omega L \).

4. **Equating Imaginary Parts to Zero**:
   - Set up the equation so that the imaginary parts in the parallel combination cancel out, allowing us to solve for \( L \).

Let's calculate step-by-step:

### Calculations

1. **Capacitive Reactance**:
   \[
   X_C = \frac{1}{\omega C} = \frac{1}{10^4 \times 4 \times 10^{-6}} = 25 \, \Omega
   \]

2. **Inductive Reactance Requirement**:
   - Since the capacitor and inductor are in parallel, we want their reactances to cancel each other out, so \( X_L = X_C = 25 \, \Omega \).
   
3. **Calculate \( L \)**:
   \[
   X_L = \omega L \Rightarrow L = \frac{X_L}{\omega} = \frac{25}{10^4} = 2.5 \times 10^{-3} \, \text{H} = 2.5 \, \text{mH}
   \]

### Answer

The required value of \( L \) is \( 2.5 \, \text{mH} \).

Would you like more details on the calculations or any specific step?

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### Related Questions

1. What is the role of the resistor in series with the parallel branch in determining the phase relationship?
2. How would the solution change if the frequency \( \omega \) were different?
3. What happens to the current \( i(t) \) if the inductance \( L \) is not chosen correctly?
4. How do the properties of parallel resonance influence this circuit's behavior?
5. What would be the effect on \( i(t) \) if an additional resistor was added in series with the capacitor?

### Tip
In AC circuits, to achieve in-phase conditions between voltage and current, ensure that the total impedance has no imaginary component.

In the circuit of Fig. P7.44, what should the value of L be at ω = 10^4 rad/s so that i(t) is in-phase with v_s(t)?

Learn how to find the required inductance L in an AC circuit to make the current in-phase with the voltage. This solution uses impedance and reactance concepts, including capacitive and inductive reactance, to ensure a purely resistive impedance. Suitable for Electrical Engineering students.

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