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Directions of currents in Kirchhoff's 2nd law problems

  1. Feb 20, 2014 #1
    1. The problem statement, all variables and given/known data

    Apply Kirchhoff's laws to find the current at point X in the circuit shown. What is the direction of the current?

    ElectricityProblem.jpg

    2. Relevant equations

    V = IR
    Kirchhoff's 1st law: ∑Currents entering junction = ∑Currents leaving junctions
    Kirchhoff's 2nd law: ∑EMFs in a loop = ∑PDs in a loop

    3. The attempt at a solution

    This problem is simple enough but my issue is how to tell initially which directions the currents are going in. My initial approach was as follows:

    ElectricityProblemIncorrectCurrents.jpg

    However, this leads to incorrect answers:

    By Kirchoff's 1st law, I2 = I0 + I1
    By Kirchoff's 2nd law, the sum of the emfs in the top loop must equal the sum of the pds:
    EMF = 4V
    PDS:
    V = IR, V = 20 x I0
    V = IR, V = 20 x I1
    So 20 x I0 + 20 x I1 = 4
    Using I2 = I0 + I1 from earlier,
    20 x I2 = 20 x I0 + 20 x I1
    substituting this into 20 x I0 + 20 x I1 = 4
    gives 20 x I2 = 4
    I2 = 4 / 20 = 0.2A

    Applying Kirchhoff's 2nd law to the big loop (ie ignoring middle resistor):
    EMF = 10 + 4 = 14V
    ∑PDs = 80 x I2 + 20 x I0
    80 x I2 + 20 x I0 = 14
    80 x 0.2 + 20 x I0 = 14
    I0 = -0.1A

    I2 = I0 + I1
    I1 = 0.2 -- 0.1 = 0.3 A which is not the correct answer.

    Why should labelling the currents in this way not give the correct answer?
     
  2. jcsd
  3. Feb 20, 2014 #2

    SteamKing

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    Staff Emeritus
    Science Advisor
    Homework Helper

    In these problems, if the assumed direction of the current is incorrect, the value obtained should just be the negative of the actual current magnitude. I haven't checked the set up of your equations, but there may be other problems there.
     
  4. Feb 21, 2014 #3

    NascentOxygen

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    Staff: Mentor

    This equation is not correct. You have a sign wrong.

    Once you have assigned directions to currents, you must pay close attention to polarities of associated potentials.
     
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