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Homework Help: Finding resistance (electric circuit)

  1. Sep 3, 2010 #1
    First of all, thank you taking a look at this. I hope to return to community in the future as I got help today.
    I'm assuming this belongs to introductory physics, so I'll post here, feel free to move if I posted in wrong section.

    1. The problem statement, all variables and given/known data
    A carbon wire and a Nichrome wire are connected in series, so that the same current exists in both wires. If the combination has a resistance of 10.0k[tex]\Omega[/tex] at 0celcius, what is the resistance of each wire at 0Celcius? so that the resistance of the combination does not change with temperature? The total or equivalent resistance of resistors in series is the sum of their individual resistances

    [tex]\rho[/tex] is rho which stands for resistivity
    [tex]\alpha[/tex] is alph which stands for Temperature coefficient

    [tex]\rho[/tex](carbon)=3.5x10^-5
    [tex]\alpha[/tex](carbon)=-0.5x10^-3
    [tex]\rho[/tex](nichrome)=1.5x10^-6
    [tex]\alpha[/tex](nichrome)=0.4x10^-3


    2. Relevant equations

    I honestly have no idea what equations to use here, but i'll put what can possibly be used

    Resistivity and Resistance changes by temperature
    [tex]\rho[/tex](0) stands for resistivity at 20celcius
    R(0) stands for Resistance at 20celcius
    [tex]\Delta[/tex]T is T(final)-T(0) and T(0) is 20celcius

    [tex]\rho[/tex]=[tex]\rho[/tex](0)[1+a*[tex]\Delta[/tex]T]
    R=R(0)[1+a*[tex]\Delta[/tex]T]

    I=electrical current in units Ampere
    V=Voltage in coulomb
    I=[tex]\Delta[/tex]V / R

    and general R=[tex]\rho[/tex] * l / A
    where...
    l= length
    A= area

    this equation may be manipulated like...
    R(carbon)=R(0carbon)[1+a(carbon)*[tex]\Delta[/tex]T]
    R(nichrome)=R(0nichrome)[1+a(nichrome)*[tex]\Delta[/tex]T]
    add these two equation and...
    R(total)=R(0carbon)[1+a(carbon)*[tex]\Delta[/tex]T] + R(0nichrome)[1+a(nichrome)*[tex]\Delta[/tex]T]
    and R(total)=10.0[tex]\Omega[/tex]

    3. The attempt at a solution

    I don't think none of the work I have so far is useful... if necessary, i'll post more info

    I'm saying I don't think they'll help because I don't think the work I have is toward the right direction xD

    but so far I added R(carbon)=R(0carbon)[1+a(carbon)*[tex]\Delta[/tex]T]
    R(nichrome)=R(0nichrome)[1+a(nichrome)*[tex]\Delta[/tex]T]
    add these two equation and...
    R(total)=R(0carbon)[1+a(carbon)*[tex]\Delta[/tex]T] + R(0nichrome)[1+a(nichrome)*[tex]\Delta[/tex]T]
    and R(total)=10.0[tex]\Omega[/tex]

    and since temperature does not change, R(nichrome)=R(0nichrome) and R(carbon)=R(0carbon)
    same goes for [tex]\rho[/tex]

    It's given that I (current) is the same and i'm guessing that voltage is the same as well opening room for...
    R(total) = V/I
    although this seems to be irrelevant...

    and to poke the problem from different angle...
    can I safely assume that [tex]\alpha[/tex](total)=C*[tex]\alpha[/tex](carbon) + N * [tex]\alpha[/tex](nichrome)
    C= percentage of carbon
    N= percentage of Nichrome
    assuming percentage is percentage of free electrons? (or should I try mass?)

    anyways, hardest part for this problem is setting up the problem. Even conceptual answer would be helpful. I need know what "directions" I should take =p right now I'm just poking the problem from every side aimlessly.
    thank you.

    edit: Answer is 5.56k[tex]\Omega[/tex], 4.44k[tex]\Omega[/tex] (i don't know which one's which, but I don think that'll be problem once I figure out how to solve this problem).

    edit2: I added
    R(carbon)=R(0carbon) * [1+[tex]\alpha[/tex](carbon)[tex]\Delta[/tex]T]
    R(nichrome)=R(0nichrome) * [1+[tex]\alpha[/tex](nichrome)[tex]\Delta[/tex]T]

    and canceling out lots of "R" (considering R(total)=R(0total)
    R(0total) is R, resistance, at 20celcius
    R(total) is resistance at 0 celcius

    I know that R(carbon)+R(nichrome)=R(total)=10.0 k Ohms
    but i forgot to use this as well since total resistance is always 10.0k Ohms regardless of temperature
    R(0carbon)+R(0nichrome)=R(0total)=10.0 k Ohms

    anyways after adding those two equations (right under edit:2) I got up to R(0carbon)=4.44k Ohms.
    Problem is, the problem asks for R(carbon) not R(0carbon)
    so I use R(carbon)=R(0carbon) * [1+[tex]\alpha[/tex](carbon)[tex]\Delta[/tex]T]
    and I get R(carbon)=4.40k Ohms... my R(0carbon) is rather closer to the answer ><
     
    Last edited: Sep 3, 2010
  2. jcsd
  3. Sep 4, 2010 #2

    ehild

    User Avatar
    Homework Helper

    You need this equations only. Collect the terms in R(total) which contain ΔT and make the coefficient cancel, so as the total resistance do not depend on T. Try.

    ehild
     
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