Question

I would love some assistance on this question.

Answer 1

@TuringTest would you care to take a look?

Answer 2

I have,

\(\Large{V_3=\frac{\Epsilon R^{\prime}}{(R^{\prime}+2.07}}\)

\(\Large{R^{\prime}=\frac{5.33~R}{5.33+R_3}}\)

Answer 3

Lovely...

Answer 4

never seen a problem like this, give me a sec. I assume they mean maximum power dissipation in R3?

Answer 5

Well, the Epsilon that isn't displaying is actually a fancy E, that is for EMF.

Answer 6

And those are \(\large{R~^{'}}\)

Answer 7

What is the absolute minimum value of R3 or of any resistor for that matter?

At this minimum, all current will flow through R3 and NO current will flow through R2.

Think of the extreme case.

Answer 8

i was doing the same thing using maxima and minima.

Answer 9

tho i used P=I^2 R

Answer 10

$$
P_3=\cfrac{V_3^2}{R_3}
$$
Which is maximum when \(R_3=0\), i.e. that branch is short-circuited. In reality this would not be useful. So we tend to maximize output power by "matching" \(R_3\) by making it equal to \(R_2||R_1\), but that is not what the problem asks.

When we choose \(R_3=R_2||R_1\), 1/2 the power is consumed by the circuit and 1/2 the power is consumed by \(R_3\). We would, of course, prefer that all the power be consumed by the load (i.e. \(R_3\)), which in theory means \(R_3=0\), but this is just not possible.

Answer 11

I can say with a certainty that \(\large{R_3\ne0~\Omega}\)

Answer 12

I think you are correct. I was surprised that \(R_3\ne0\).

Using the chart above:
$$
\Large{
I_{R_3}=\cfrac{V}{\frac{R_1R_2}{R_1+R_2}+R_3}\\
=\cfrac{V(R_1+R_2)}{R_1R_2+(R_1+R_2)R_3}\\
P_{R_3}=I_{R_3}^2R_3\\
P_{R_3}=\left(\cfrac{V(R_1+R_2)}{R_1R_2+(R_1+R_2)R_3}\right )^2R_3\\
\text{After substituting values,}\\
\cfrac{d}{dR_3}P_{R_3}=\cfrac{ 1.49096-R_3}{(R_3+1.49096)^3}\\
\text{So,}P_{R_3}\text{ is maximized when}\\
P_{R_3}=0\\
\implies R_3=1.50\Omega
}
$$
That the maximum occurs here does appear to be the case:
https://www.wolframalpha.com/input/?i=plot+%28x*1^2%282.07%2B5.33%29^2++%29%2F%282.07*5.33%2B%282.07%2B5.33%29*x%29^2%2C+0%3C%3Dx%3C%3D10