Finding the Optimal Resistance for Maximum Dissipation

[1MileCrash]

Homework Statement

attached

Homework Equations

The Attempt at a Solution

First I decided to find the equivalent resistance.

Since r2 and r3 are parallel, this can be combined into a resistor of resistance r2r3/(r2 + r3). Since r2 is known, I went ahead and wrote that as 14r3/(14+r3).

Now, this resistor is in series with r1, so I can find the equivalent resistance to be

2 + [14r3/(14+r3)]

Now, I can call the current due to the battery

i = V/R = V/(2 + [14r3/(14+r3)])

Allowing me to use the dissipation formula for r3

P = r3[V/(2 + [14r3/(14+r3)])]^2

Now I figured it would be a matter of differentiating this function and setting it equal to zero, but the derivative is unreasonably complex.

Any ideas?

 

[gneill]

Take another look at the problem statement; You want to maximize the power dissipated in R3, not in the whole network.

 

[1MileCrash]

I don't understand what you mean. I don't know the current through the battery and represented the equivalent resistance as a function of r3, allowing me to show the current of the battery as a function of r3 (with V constant) allowing me to create a dissipation due to r3 function (P=i^2(r3)).



I really suck at this electrical engineering crap. What i goes into P=i^2(R)? And how do I find it?

 

[gneill]

You want the power dissipated by R3 as a function of R3. that way you will be in a position to maximize that function.

That means you need to find either the voltage across R3 or the current through R3 so that you can write the power as I²R3 or V²/R3.

 

[1MileCrash]

Ok, so the current through r3 isn't the current supplied by the battery?

Also, you said I either need the current through r3 or the potential difference across r3, but don't I need one to know the other?

 

[gneill]

Ok, so the current through r3 isn't the current supplied by the battery?

Nope, because the battery also supplies current to R2 which does not flow through R3.

Also, you said I either need the current through r3 or the potential difference across r3, but don't I need one to know the other?

If you know one you can find the other via Ohm's law, but either one solved in terms of the circuit component values will do. You want to write KVL and/or KCL equations to solve for either the potential across R3 or the current through R3.

 

[1MileCrash]

I'm going to do some more reading. Thanks for your guidance!