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We are tasked with finding the value of the inductor \(L\) in an RL circuit based on the provided current equation \(i(t) = 10 \cos(5t - \pi/4)\) A and the voltage source \(v(t) = 200 \cos(5t)\) V.

### Step 1: Impedance of the Circuit
In an RL circuit, the impedance \(Z_{RL}\) is given by:
\[
Z_{RL} = R + j\omega L
\]
where \(R\) is the resistance, \(L\) is the inductance, and \(\omega = 5 \, \text{rad/s}\) (since the frequency of the voltage is \(5t\)).

### Step 2: Phasor Form
To analyze the circuit in phasor form:
- Voltage: \(V(t) = 200 \cos(5t)\) corresponds to phasor \(V = 200 \angle 0^\circ\).
- Current: \(i(t) = 10 \cos(5t - \pi/4)\) corresponds to phasor \(I = 10 \angle -45^\circ\).

### Step 3: Applying Ohm's Law in Phasor Domain
Using Ohm's law for phasors, the total impedance \(Z_{RL}\) is given by:
\[
Z_{RL} = \frac{V}{I}
\]
Substituting the known phasors:
\[
Z_{RL} = \frac{200 \angle 0^\circ}{10 \angle -45^\circ} = 20 \angle 45^\circ \, \Omega
\]
This means that the impedance has a magnitude of 20 ohms and a phase angle of \(45^\circ\).

### Step 4: Expressing Impedance in Rectangular Form
The impedance \(Z_{RL} = 20 \angle 45^\circ\) can be expressed in rectangular form as:
\[
Z_{RL} = 20 (\cos 45^\circ + j \sin 45^\circ) = 20 \left( \frac{1}{\sqrt{2}} + j \frac{1}{\sqrt{2}} \right)
\]
Simplifying:
\[
Z_{RL} = 20 \times \frac{1}{\sqrt{2}} (1 + j) = 14.14 + j14.14 \, \Omega
\]

### Step 5: Identifying the Resistance and Inductance
From the impedance, we identify:
- The real part, \(R = 14.14 \, \Omega\), which represents the resistance.
- The imaginary part, \(\omega L = 14.14 \, \Omega\).

### Step 6: Solving for Inductance \(L\)
Since \(\omega = 5 \, \text{rad/s}\), we can solve for \(L\):
\[
\omega L = 14.14 \implies L = \frac{14.14}{5} = 2.828 \, \text{H}
\]

### Final Answer:
The inductance \(L\) is \(2.828 \, \text{H}\).

Let me know if you'd like further details or have questions!

Here are 5 related questions to expand your understanding:
1. How would the value of \(L\) change if the frequency of the voltage source increased?
2. Can you derive the time-domain voltage across the inductor using the current \(i(t)\)?
3. What would happen to the current phase angle if \(R\) increased?
4. How would the impedance of the circuit change if the frequency doubled?
5. What is the relationship between voltage and current in a purely inductive circuit?

**Tip**: When analyzing AC circuits, converting to the phasor domain simplifies solving for impedance and current relationships.

The current in the RL circuit shown below is i(t) = 10 cos(5t - π/4) A. Find the value of the inductor?

Learn how to find the inductance value in an RL circuit with a given current equation i(t) = 10 cos(5t - π/4) A and voltage source 200 cos(5t) V. This problem covers key concepts of AC circuit analysis and phasor mathematics, suitable for undergraduate electrical engineering students.

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