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Ive one of the objectives for ap physics. For this unit. Is they want want you to be able to find the external resistor that will give you the power for this type of a circuit.

So if you have an emf a battery with an emf and an internal resistance r. And theres a this external resistor here. How much should the r be if these are no lets say thats 1 ohm.

And thats 2 volts. If thats 2 volts. And thats what ohm.

What should our be to maximize your power. Okay. So this is this how this works.

The power across this resistor is going to be i squared. Times r. Just like that and but the i through here this i going through here is going to be e over the total resistance and so thats going to be e over the total resistance r.

Plus capital r. So thats going to be i will square that times r ok well it turns out that if we were to plot power. Versus.

This where r is our variable. And what wed find out is that if we plotted this power versus are assuming that the little r. And the e were just constant.

Then what we get is that it when r was 0. The power would be 0. Because if this were just a wire.

How would there be any power through it so thats just going to be 0. But certainly if r goes to infinity. If we have a huge resistance then therell be no current and if theres no current again.

Well get no power. Because you need you need some current to get power and so it would again be 0. So what it does though is it goes up and then it drops down and it approaches as you go as you go out to to infinity is our capital r.

Goes to infinity that goes to 0. But what they really want is this point right here where you get the maximum power pmax and so were going to find out what ill call that r. Naught.

R. Naught. Is where you get the maximum power.

And lets find out what that is all were going to do is were going to take the derivative of this function with respect to r the derivative. Remember is the slope. So you see how the slope starts out positive and then it goes to zero.

See how the tangent line is zero and then it goes negative well right at this maximum. We are here we have at this maximum. We have the slope equal to zero.

So ill just set the derivative equal to zero. And thats how i will know that its a at its max all right so lets do that take the derivative of this function. Let me write this function again.

Ill bring it over here. Im just going to write this at the top so its power is equal to maybe ill pretty it up a little bit. Its equal to e squared.

Times r. All over and that bottom term maybe. Ill square right away.

So its e squared r. And im going to go. R.

Squared. Plus. Two little r.

Big r. Plus capital. R.

Squared. This video is going to be a lot of math. See if you can follow along so i just did this i squared this term.

I squared that term and here. I am okay now so im going to what im going to do is im going to take the derivative of p. With respect to r and ill set that equal to zero.

And that should tell me if i solve from capital r. That will tell me when capital r. Is when capital r.

Is whatever value it is thats where my i am my maximum power okay so to take this derivative. Im going to use the quotient rule and ill just assume that you know the quotient rule. So im going to take the derivative of the numerator.

So thats a it with respect to our so. Thats going to be e squared times. The then you times.

It by the denominator so r squared plus 2 r capital r. Plus r capital r. Squared minus.

I now i take the derivative of the denominator lets get it with respect to r. So this is this is zero. Because thats just a constant this is going to go to two two lowercase r.

And then this will go to plus 2 r. Squared or 2 r. Rather.

I just took the derivative with respect to capital r of this bottom. Part. And then im going to multiply.

That by e. Squared. Times.

R. Ya. E.

Squared. Times r. The the numerator.

Alright. And then im going to divide by this is just the quotient rule. The denominator squared.

So i got to square that okay so im asking myself when is the numerator equal to zero. Because when the numerator is equal to zero. Then the whole thing is equal to zero like i dont have to worry about this bottom part in other words.

I could multiply both sides by this bottom part. And that would make that disappear so when is this numerator equal to zero. Thats when the whole thing is equal to zero.

And so maybe ill bring this term on the other side. So i got a squared times r. Squared.

Plus. Two i wont i wont say it out. I think thats helping you anyways.

So its when that equals. I brought this to the other side. So.

E. Squared. R.

To r. Plus. Two capital r.

Okay. Let. Me get rid of the e squared.

And then let me throw. This are in there. So i got ill just throw that are in there so r squared plus 2 plus r squared equals.

2r r. Plus 2r squared. I feel pretty good about this so far because all my units are the same.

I cant add meters to meters squared. So each term as is meter squared or excuse. Me own square.

Okay. So look. We have one of these on both sides.

So lets get rid of it and when i bring this r on the other side. I bring that are on the other side. I do believe that that cancels out one of those rs yeah.

So we just get r squared is equal to r squared so its when r equals capital r yeah so when r equals capital r. Thats when we have maximum power and what will the maximum power be then well ill just throw this back into the equation. Now so ill just go ahead and throw this back in the equation.

So it will be when this was our original equation. Power. Is equal to a east.

E. Squared r. All over r.

Plus r. Squared. Okay.

But if r. When big r equals little r. Then that turns into this so that turns into this this is squared.

So so what we have is the maximum power is going to be e squared over 4 r okay so when when will you get the maximum power youll get the maximum power. When this is equal to that all right thanks right .

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