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Differentiation maximum problem

  1. Dec 5, 2007 #1
    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.
  2. jcsd
  3. Dec 5, 2007 #2


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    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.
  4. Dec 5, 2007 #3
    Ah, excellent!!!

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

    And of course it's zero then....:tongue: 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!!!:biggrin:
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