Analysis of the Circuit

To determine the value of resistor "r" in the given circuit, such that the maximum power is transferred to it, we need to analyze the circuit and apply the maximum power transfer theorem.

Maximum Power Transfer Theorem

According to the maximum power transfer theorem, the maximum power is transferred from a source to a load when the resistance of the load is equal to the internal resistance of the source. In other words, the load resistance should be equal to the Thevenin equivalent resistance of the circuit seen from the load terminals.

Steps to Determine the Value of r

To find the value of resistor "r" for maximum power transfer, we can follow these steps:

1. Calculate the Thevenin equivalent voltage (Vth) and Thevenin equivalent resistance (Rth) of the circuit.

2. Substitute the load resistance with the Thevenin equivalent resistance (Rth) to find the maximum power.

3. Calculate the power dissipated in the load (Pload) using the formula P = V^2/R, where V is the voltage across the load and R is the load resistance.

4. Differentiate the power equation with respect to the load resistance (R) and equate it to zero to find the maximum power point.

5. Solve the equation to determine the value of the load resistance (R) at the maximum power point.

6. Substitute the value of R back into the original circuit to find the value of resistor "r".

Conclusion

By following the above steps, we can determine the value of resistor "r" in the given circuit such that the maximum power is transferred to it. It is important to note that the maximum power transfer theorem is applicable only when the load resistance is equal to the Thevenin equivalent resistance of the circuit seen from the load terminals.