Hi,
Here's a way to get the max power in a resistor connected in series with another resistor connected to a voltage source.
We call the upper resistor R1 and the lower resistor R2. We want to see what value R2 should be to get max power
in R2.
Start with:
v=Vs*R2/(R1+R2)
i=Vs/(R1+R2)
p=v*i
Then the derivative with respect to R2:
dp/dR2=-(Vs^2*(R2-R1))/(R2+R1)^3
The set dp/dR2 equal to zero:
-(Vs^2*(R2-R1))/(R2+R1)^3=0
Solving, we get:
R2=R1
So the max power occurs in R2 when R2 is equal to R1.
Just to try this in a circuit, we'll say that Vs=20v and R1=10 ohms, and now change R2 to several values and see what happens:
R2=10, P=10, V=10, I=1
R2=9, P=9.9723, V=9.4737, I=1.0526
R2=11, P=9.9773, V=10.4762, I=0.95238
From the above we can see that the power always goes lower when R2 is either a lower or higher value than R1. The power is greatest
when R2 equals R1 exactly.
The attached drawing shows the power as R2 changes from 0 to 50 ohms with R1=10 ohms and Vs=10 volts.