Given the circuit in the image, we need to find the value of $R$ given that the gain of the operational amplifier (op-amp) is -12.

Step-by-step Analysis

1. Circuit Configuration:

   • The circuit is an inverting op-amp configuration with multiple resistors.
   • Resistors: $10 \, k\Omega$ resistors are connected as feedback resistors along with the unknown resistor $R$ that branches at point B.

2. Inverting Op-amp Gain Formula: For an inverting op-amp, the voltage gain ($A_v$) is given by: $A_v = -\frac{R_f}{R_{in}}$ Where:

   • $R_f$ is the total feedback resistance.
   • $R_{in}$ is the input resistor (here, $10 \, k\Omega$).

3. Total Feedback Resistance:

   • In the circuit, the resistors form a combination in the feedback loop.
   • The resistors at the point B (two $10 \, k\Omega$ resistors in series with $R$ in parallel): $R_f = 10 \, k\Omega + \left( \frac{10 \, k\Omega \times R}{10 \, k\Omega + R} \right)$ Here, the two $10 \, k\Omega$ resistors are in series, and their combination is in parallel with $R$.

4. Given Gain:

   • The gain is provided as -12.
   • Plugging into the gain formula: $-12 = -\frac{R_f}{10 \, k\Omega}$ Simplifying: $R_f = 12 \times 10 \, k\Omega = 120 \, k\Omega$

5. Finding $R$:

   • Substitute $R_f = 120 \, k\Omega$ into the expression for total feedback resistance: $120 \, k\Omega = 10 \, k\Omega + \left( \frac{10 \, k\Omega \times R}{10 \, k\Omega + R} \right)$
   • Subtract $10 \, k\Omega$ from both sides: $110 \, k\Omega = \frac{10 \, k\Omega \times R}{10 \, k\Omega + R}$
   • Cross-multiply to solve for $R$: $110 \, k\Omega \times (10 \, k\Omega + R) = 10 \, k\Omega \times R$ $1,100,000 \, k\Omega^2 + 110 \, k\Omega \times R = 10 \, k\Omega \times R$
   • Isolate $R$: $1,100,000 \, k\Omega^2 = 10 \, k\Omega \times R - 110 \, k\Omega \times R$ $1,100,000 \, k\Omega^2 = -100 \, k\Omega \times R$
   • Divide by $-100 \, k\Omega$ to get $R$: $R = -\frac{1,100,000 \, k\Omega^2}{100 \, k\Omega} = 11 \, k\Omega$

Final Answer

The value of $R$ is $11 \, k\Omega$.

Would you like further explanation or have any questions?

Related Questions

1. How would the gain change if $R$ were doubled?
2. What is the effect of changing the input resistor on the overall gain?
3. How would you design a non-inverting amplifier with a gain of 12?
4. What happens to the op-amp gain if one of the resistors in the feedback loop is open-circuited?
5. How does the ideal op-amp assumption affect the accuracy of real-world circuit performance?

Tip: For accurate gain calculations in op-amp circuits, always check the configuration (inverting or non-inverting) and the arrangement of resistors in the feedback loop.