To determine the value of $R_L$ that will absorb the maximum possible amount of power, we can apply the Maximum Power Transfer Theorem. According to this theorem, the load resistor $R_L$ will absorb maximum power when it is equal to the Thevenin resistance $R_{\text{th}}$ seen from its terminals.

Steps to solve the problem:

1. Identify the Thevenin Equivalent Resistance $R_{\text{th}}$:

   • Remove $R_L$ from the circuit and find the equivalent resistance across its terminals.
   • Since there’s a 3Ω and a 2Ω resistor in series, their combined resistance will be $3 + 2 = 5 \ \Omega$.
   • This combined 5Ω resistance is in parallel with the 5Ω resistor on the right side of the circuit.

   Thus, the Thevenin resistance $R_{\text{th}}$ across the terminals of $R_L$ is: $R_{\text{th}} = \frac{5 \times 5}{5 + 5} = \frac{25}{10} = 2.5 \ \Omega$

2. Set $R_L = R_{\text{th}}$ for Maximum Power:

   • For maximum power transfer, $R_L$ should be equal to $R_{\text{th}}$, which we calculated as 2.5 Ω.

Conclusion:

The value of $R_L$ that will ensure it absorbs the maximum possible amount of power is: $R_L = 2.5 \ \Omega$

Would you like a detailed explanation of each step?

Here are some related questions for further exploration:

1. How would you calculate the power absorbed by $R_L$ once it's set to 2.5 Ω?
2. What if $R_L$ was not adjustable—how would you determine the actual power absorbed by a fixed $R_L$?
3. How does the presence of the 1 A current source affect the Thevenin equivalent?
4. Could you analyze this circuit using Norton's theorem instead? What would change?
5. How can you verify the maximum power transfer condition using a circuit simulator?

Tip: Thevenin and Norton equivalents are powerful tools for simplifying complex circuits when analyzing power transfer to a specific load.