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To determine the output voltage \( v_o \) of the operational amplifier (op-amp) circuit in Figure 3, let's break down the analysis. Given that \( V_s = 2 \text{ V} \), we can apply the principles of an inverting op-amp circuit, as well as node voltage analysis, to solve for \( v_o \).

### Step-by-Step Solution:

1. **Identify the configuration and assumptions**:
   - This is an op-amp circuit with feedback, where we assume the op-amp is in an ideal state (i.e., infinite open-loop gain).
   - This assumption implies the virtual short concept: the inverting input (-) and non-inverting input (+) voltages are equal. Since the non-inverting input is grounded, the inverting input voltage is also effectively at 0 V.

2. **Set up node voltage at the inverting input**:
   - Let’s denote the current through each resistor:
     - Current through \( 4 \, k\Omega \) resistor connected to \( 9 \text{ V} \): \( I_1 = \frac{9 \, \text{V} - 0 \, \text{V}}{4 \, k\Omega} \).
     - Current through \( 4 \, k\Omega \) resistor connected to \( V_s = 2 \text{ V} \): \( I_2 = \frac{2 \, \text{V} - 0 \, \text{V}}{4 \, k\Omega} \).
     - Current through \( 8 \, k\Omega \) feedback resistor: \( I_3 = \frac{0 \, \text{V} - v_o}{8 \, k\Omega} \).

3. **Apply Kirchhoff's Current Law (KCL) at the inverting input**:
   - At the inverting input, the sum of currents entering and leaving must equal zero:
     \[
     I_1 + I_2 = I_3
     \]
   - Substituting the currents:
     \[
     \frac{9 \, \text{V}}{4 \, k\Omega} + \frac{2 \, \text{V}}{4 \, k\Omega} = \frac{-v_o}{8 \, k\Omega}
     \]

4. **Solve for \( v_o \)**:
   - Calculate \( I_1 \) and \( I_2 \):
     \[
     I_1 = \frac{9}{4} = 2.25 \, \text{mA}
     \]
     \[
     I_2 = \frac{2}{4} = 0.5 \, \text{mA}
     \]
   - Substitute into the equation:
     \[
     2.25 + 0.5 = \frac{-v_o}{8}
     \]
   - Simplify:
     \[
     2.75 = \frac{-v_o}{8}
     \]
     \[
     v_o = -2.75 \times 8 = -22 \, \text{V}
     \]

### Final Answer:
The output voltage \( v_o \) of the op-amp circuit is \( -22 \text{ V} \), which differs from the expected answer. Double-checking values might reveal assumptions or additional components affecting the circuit's behavior.

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### Related Questions

1. How would the output change if \( V_s \) were increased to 3 V?
2. What if the op-amp had a finite gain instead of being ideal?
3. How does feedback resistance affect the gain of this op-amp configuration?
4. Could we calculate \( v_o \) if the op-amp were in a non-inverting configuration?
5. What role does the \( 2 \, k\Omega \) resistor play in this circuit?

### Tip
In ideal op-amp analysis, always check for virtual short assumptions between inputs in feedback configurations to simplify node analysis.

Determine the output voltage v₀ of the Op-Amp circuit in Figure 3 if Vₛ = 2V.

Learn how to find the output voltage (v₀) of an op-amp circuit with given resistances and input voltage Vₛ = 2V. This solution involves circuit analysis techniques, Kirchhoff's Current Law, and the virtual short concept in ideal op-amps.

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