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.