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Relationship between changing a wire's diameter and change in temp

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Hello! I have a problem where I need to develop and justify a hypothesis surrounding the relationship between the thickness (or diameter) of a nichrome wire and the temperature of water that it's submersed in over a time of 5 minutes. (proportionality statement and equations)


    2. Relevant equations

    So far, I know that resistance is inversely proportional to cross-sectional area, and therefore:
    R ∝ 1/A
    R ∝ 1/∏r^2
    R ∝ 1/∏(d/2)^2
    and because 2 and ∏ are constant
    R ∝ 1/d^2 (resistance is inversely proportional to diameter squared) (I think!)

    Now, I've also researched Joule's law, which eventuates into:
    Heat produced = Pt = VIt = V^2/Rt = I^2Rt

    3. The attempt at a solution

    Now, I'm trying to get a relationship between diameter and ΔT, and I thought, so I have R ∝ 1/d^2, and P = V^2/R where V is a constant so P ∝ 1/R
    and since P is the change in heat:
    ΔT ∝ 1/R


    OR


    since thickness makes resistance go down
    and less resistance makes ΔT (somehow)
    then is R ∝ ΔT
    and substituting so therefore:
    ΔT ∝ 1/d

    and I don't have much idea how to reason out what the graph will look like/why...

    I think I've tied myself in knots... any help would be much appreciated! Thank you!:)
     
    Last edited: Sep 20, 2012
  2. jcsd
  3. Sep 20, 2012 #2

    CWatters

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    Science Advisor
    Homework Helper

    I think you have to assume the wire is connected to a constant voltage source and I'd also ignore any change in resistance of the wire with temperature.

    Therefore your approach is correct..

    R ∝ 1/∏(d/2)2
    R ∝ 1/d2

    Then use..

    Power = V2/R

    Power = V2 * d2.........................................(1)

    If the water is in an insulated container then the energy it contains (E) is given by

    E = Specific heat capacity * mass * Δ Temperature

    Rearrange to give..

    Δ Temperature = E / Specific heat capacity * mass

    Then..

    E = Power * Time

    so

    Δ Temperature = Power * Time / Specific heat capacity * mass ...............(2)

    Put (1) into (2)...

    Δ Temperature = V2 * d2 * Time / Specific heat capacity * mass

    Voltage, Time, Specific heat capacity and mass are all constant so

    Δ Temperature ∝ d2

    eg The change in temperature over 5min (not the actual temperature) is proportional to d2

    Temperature = Initial Temperature + Δ Temperature

    So actual temperature is proportional (but not directly proportional) to d2.

    http://en.wikipedia.org/wiki/Proportionality_(mathematics [Broken])
     
    Last edited by a moderator: May 6, 2017
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