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Power in a simple electric circuit

  1. Feb 13, 2015 #1
    1. The problem statement, all variables and given/known data
    A circuit consists of a battery with internal resistance r and a resistor with resistance R.
    It has been found that the maximum power generated is when R=r.
    Prove that in those conditions ##P_{max}=\frac{ε^2}{4r}##

    2. Relevant equations
    Power invested in a resistor: P=i2R
    Power invested between 2 points in a circuit: P=Vi

    3. The attempt at a solution
    The current: ##i=\frac{V}{2r}##
    Voltage drop on the internal resistance: ##V=ir=\frac{V}{2}##
    Power between the terminals of the battery: ##P=iV=\frac{V}{2r} \frac{V}{2}=\frac{V^2}{4r}##
    But the power loss on one resistor (the internal or the external):
    $$P=i^2r=\frac{V^2}{4r^2}\cdot r=\frac{V^2}{4r}$$
    It's identical to the total power, and there are 2 resistors so they consume double the power, how come?
     
  2. jcsd
  3. Feb 13, 2015 #2

    Doc Al

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    Staff: Mentor

    The total power delivered by the battery is the current times the battery EMF (which is V), not the terminal voltage (V/2).
     
  4. Feb 13, 2015 #3
    So:
    $$P_{max}=\varepsilon\cdot i=\varepsilon\cdot \frac{\varepsilon}{2r}=\frac{\varepsilon^2}{2r}$$
    The book asked about the maximum power of the battery and i guess this is the answer. the answer of the book is:
    $$P_{max}=\frac{\varepsilon^2}{4r}$$
     
  5. Feb 13, 2015 #4

    Dick

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    Science Advisor
    Homework Helper

    The maximum power the question is asking about is the maximum power delivered to the load, that's the external resistor only.
     
  6. Feb 13, 2015 #5
    Thanks, i guess you are right. but is there, in general, a meaning to the multiplication of the terminals of the battery voltage and the current P=V⋅i? not in this specific problem but in general. i guess it's the power delivered to the rest of the circuit, right?
     
    Last edited: Feb 13, 2015
  7. Feb 13, 2015 #6

    NascentOxygen

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    Staff: Mentor

    Yes. That's the power that the battery is delivering [to the external circuit]. In this problem, and in general.
     
  8. Feb 14, 2015 #7
    Je Suis Charlie also, thanks
     
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