How to calculate maximum power when value of load resistance is unknown?

Once again same circuit and same voltage source / current source and resistances. However now I need to calculate the maximum power that is consumed by the RL. How do I do that, and most importantly when I don't have the value on the load resistor?

Here is the shematic of the previous problem I solved (Thévenin equivalent):

simulate this circuit – Schematic created using CircuitLab

And the new question is this one:

Note: I1 should be 2 mA.

And here is the link to the question I submitted before this one:

How do I calculate load resistance?

• I1 is not 2 mΩ (well not in this universe anyway) Apr 1, 2020 at 14:19
• Yes, you are correct. It's 2 mA. Apr 1, 2020 at 14:25

The power in the load resistor is: $$P_L= I^2R_L$$

$$I=\frac{V}{R_{th}+R_L}.$$

Therefore:

$$P_L=(\frac{V}{R_{th}+R_L})^2R_L.$$

To obtain the maximum power, differentiate and assign zero to obtain maximum:

$$\frac{d P_L}{dR_L}=0 \text{ therefore } R_L=R_{th} ~~~~~~\text{(NB.)}$$

This gives $$P_{max} = \frac{V^2}{4\cdot R_{th}}.$$

• I think you've misread the question. Apr 1, 2020 at 14:20
• No, I have not, I just skipped the middle part that he should practice. Apr 1, 2020 at 14:25
• Once again, how can I calculate all of this when the RL is unknown? Apr 1, 2020 at 14:29
• Follow the above step-by-step. Apr 1, 2020 at 14:31
• I'm sorry, but I can't figure out how I can solve the first step when I have 2 unknowns ( PL and RL) to begin with? Apr 1, 2020 at 14:51