To determine $i_2(t)$, let’s go through the solution in steps. We are dealing with an AC circuit containing both resistive and capacitive components.

Given Information

1. $v_{s1}(t) = 5 \cos(2 \pi f t)$, where $f = 1 \, \text{kHz}$.
2. $V_{s2} = 15 \, \text{V}$.
3. The capacitor value is $0.22 \, \mu\text{F}$.
4. Resistors are $1 \, \text{k}\Omega$ and $2 \, \text{k}\Omega$.

Step-by-Step Solution

1. Convert Frequency to Angular Frequency: Since $f = 1 \, \text{kHz} = 1000 \, \text{Hz}$, $\omega = 2 \pi f = 2 \pi \times 1000 = 2000 \pi \, \text{rad/s}.$

2. Impedance of the Capacitor: For a capacitor, the impedance $Z_C$ is given by: $Z_C = \frac{1}{j \omega C}.$ Substituting the values of $\omega$ and $C = 0.22 \, \mu\text{F} = 0.22 \times 10^{-6} \, \text{F}$: $Z_C = \frac{1}{j \cdot 2000 \pi \cdot 0.22 \times 10^{-6}}.$ Calculating this gives: $Z_C \approx -j723.4 \, \Omega.$

3. Total Impedance Seen by $v_{s1}(t)$: The total impedance $Z_{total}$ from $v_{s1}(t)$ to ground includes $Z_C$, the $1 \, \text{k}\Omega$ resistor, and the parallel combination of $2 \, \text{k}\Omega$ and $V_{s2}$:

   • Combine the capacitor’s impedance $Z_C$ with the $1 \, \text{k}\Omega$ resistor in series.
   • Then calculate the current divider for the $2 \, \text{k}\Omega$ resistor.

4. Calculate $i_2(t)$: Use Ohm’s law and AC analysis techniques to find $i_2(t)$.

Would you like a more detailed walkthrough of each step or additional clarifications?

Related Questions

1. How would the solution change if the capacitor value were different?
2. What if the frequency $f$ was increased to 10 kHz? How would $i_2(t)$ be affected?
3. How can we analyze this circuit in the frequency domain?
4. How would the response change if we replaced the AC source with a DC source?
5. How does the current $i_2(t)$ vary with changes in $V_{s2}$?

Tip

In AC circuits, always convert all elements to their impedance form for easier analysis, especially when working with complex circuits involving capacitors and inductors.