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Kirchoff's Law and Critical Points

  1. Mar 11, 2009 #1

    JJBladester

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    Gold Member

    1. The problem statement, all variables and given/known data[/b]

    What is/are the critical points of Kirchoff's Law:

    L[tex]\left(\frac{di}{dt}\right)[/tex] + Ri = E

    3. The attempt at a solution

    I solved the differential equation above and got the following solution (which I verified to be correct):

    i = [tex]\left(\frac{E}{R}\right)[/tex] + C[tex]e^{-\left(\frac{R}{L}\right)t}[/tex]

    If I remember correctly, the critical points would be when [tex]\left(\frac{di}{dt}\right)[/tex] = 0.

    [tex]\left(\frac{di}{dt}\right)[/tex] = [tex]\left(\frac{E}{L}\right)[/tex] - [tex]\left(\frac{R}{L}\right)[/tex]i so you have a critical point when

    [tex]\left(\frac{E}{L}\right)[/tex] = [tex]\left(\frac{R}{L}\right)[/tex]i

    Is this correct or am I on the wrong path?
     
  2. jcsd
  3. Mar 11, 2009 #2

    berkeman

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    Sorry, what is the definition of a critical point in this context? Do they mean critical damping of this series RL circuit, or something else?
     
  4. Mar 12, 2009 #3

    JJBladester

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    I asked my professor about the definition of a critical point in this context. She wrote back:

    The critical point of a first order DEQ is the value of the dependent variable found by setting its derivative to zero.

    So, my crack at an answer is:

    L(di/dt) + Ri = E

    di/dt + (R/L)i = E/L

    0 + (R/L)i = E/L

    i = E/R <----- since "i" is the dependent variable, by setting di/dt = 0, we have a critical point at E/R, or voltage/resistance.

    How does this sound?
     
  5. Mar 12, 2009 #4

    berkeman

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    Staff: Mentor

    Seems like you did the math right, I'm just not able to intuit what it means physically.
     
  6. Mar 12, 2009 #5

    JJBladester

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    Current equals voltage over resistance. This is Ohm's law, if I'm not mistaken. If di/dt = 0 then the change in current with respect to time is zero, which means if you have a circuit running at constant current, that current can be measured as voltage/resistance.
     
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