Is the Differentiation of Power in a Circuit Correct?

[Joza]

Power of an electrical circuit is equal to current squared times resistance.

My expression for current is: E/(R + r), where E is emf of battery.

So, my expression for Power, P, is:

P=((E^2)R)/(R + r)^2

I would like to differentiate P with respect to R, to find where the slope is zero, ie., the maximum power.
I am having some trouble though. I used the quotient rule and got:

((R + r)E^2 - 2RE^2)/(R + r)^3

Is this correct? I need to know where this is zero, but I am a bit unsure about where to go from here.

 

[HallsofIvy]

Yes, that is correct. But I think you will make the problem much easier by combining those "R²"s!
[tex]\frac{(R+r)E^2- 2RE^2}{(R+r)^3}= E^2\frac{r- R}{(R+r)^3}[/itex]
and, of course, a fraction is only 0 where the numerator is 0.

 

[Joza]

Ah, excellent!

I guess I was a bit too tired to realize I should expand it.

And of course it's zero then... why didn't I see that!

That agrees perfectly, because it's 0 when r=R. And that is what I was looking for!

Brilliant mate, cheers!