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The problem asks us to design a circuit that has a specific input resistance and a variable gain using a potentiometer. Let's break down the steps to find the voltage gain when the potentiometer is set exactly at its middle value.

### Key Information:
- **Input Resistance (R_in):** 100 kΩ
- **Gain Range:** -1 V/V to -100 V/V
- **Potentiometer R4 value:** 100 kΩ

### Circuit Overview:
- The circuit is a typical inverting amplifier, with the potentiometer \( R_4 \) in the feedback loop to control the gain.
- The voltage gain \( A_v \) of an inverting amplifier is given by:

\[
A_v = -\frac{R_f}{R_{\text{in}}}
\]

Where:
- \( R_f \) is the feedback resistance (which includes \( R_2 \), \( R_4 \), and \( R_3 \) depending on the configuration).
- \( R_{\text{in}} \) is the input resistance (here, \( R_1 \)).

### Step 1: Calculate Gain at Middle Position of Potentiometer
When the potentiometer \( R_4 \) is set to its middle value, it will be effectively split into two resistors of 50 kΩ each. So, the total feedback resistance \( R_f \) will be:

\[
R_f = R_2 + 50\ \text{k}\Omega
\]

Now, applying the gain formula for the inverting amplifier:

\[
A_v = -\frac{R_2 + 50\ \text{k}\Omega}{R_1}
\]

### Step 2: Use Given Information
We are told that the input resistance \( R_{\text{in}} = 100\ \text{k}\Omega \), so:

\[
R_1 = 100\ \text{k}\Omega
\]

Substitute this into the gain equation:

\[
A_v = -\frac{R_2 + 50\ \text{k}\Omega}{100\ \text{k}\Omega}
\]

At the middle position, \( R_4 \) adds 50 kΩ, so if we assume \( R_2 = 50\ \text{k}\Omega \), then the gain becomes:

\[
A_v = -\frac{50\ \text{k}\Omega + 50\ \text{k}\Omega}{100\ \text{k}\Omega} = -1
\]

### Final Answer:
The voltage gain when the potentiometer is set to its middle value is **-1 V/V**.

---

Would you like further clarification on any of the steps or how to handle the variable gain? Here are some related questions for practice:

1. How would you calculate the gain if the potentiometer was set to 25% of its value?
2. What happens to the input impedance when the gain is increased?
3. Can you derive a general formula for the gain based on the potentiometer setting?
4. How does the configuration change if the circuit is made non-inverting?
5. What practical effects do the resistor tolerances have on the actual gain of the circuit?

**Tip:** Always check the resistance of the potentiometer at different positions to understand its effect on the gain.

Design the circuit shown in Fig. P2.35 to have an input resistance of 100 kΩ and a gain that can be varied from -1 V/V to -100 V/V using the 100-kΩ potentiometer R4. What voltage gain results when the potentiometer is set exactly at its middle value?

Design a circuit with an input resistance of 100 kΩ and a variable gain range from -1 V/V to -100 V/V using a 100-kΩ potentiometer. Learn how to calculate the voltage gain when the potentiometer is set at its middle value. Suitable for electrical engineering students.

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