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Homework Help: Help! Easy Physics - Resistivity Ratios

  1. Oct 23, 2011 #1
    The resistivity of Aluminium is twice that of copper. However, the density of Aluminium is one-third that of Copper.

    a) For equal length and resistance, calculate the ratio:

    mass of aluminium/mass of copper​


    I'm thinking, the density ratio (Al:Cu) is 1:3 .....and the resistivity ratio is 2:1 ...so would the overall ratio be, 2:3 ? I have no idea...

    This is easy Physics compared to what else is on this site, but my physics isn't any good, so I joined this forum in hope that someone would be able to help me!

    Thank you :)
     
    Last edited: Oct 23, 2011
  2. jcsd
  3. Oct 23, 2011 #2

    PeterO

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    refer to the formula of resistance - often given as R = ρL/A - to see that one of the wires will have to be thicker - thus have a greater volume of metal, thus greater mass than if it was the same thickness.
     
  4. Oct 25, 2011 #3
    Okay, thank you. I understand that, i'm just not sure how to get a ratio from that, with no other information... I'm not even sure if I need to work out a ratio actually. Oh well, thanks anyway.
     
  5. Oct 25, 2011 #4

    PeterO

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    Homework Helper

    Firstly, your original answer 2:3 was correct - but your uncertainty indicated you were not sure why.

    When doing ratios, I just use the formulas and do a grand divide to produce the ratio.

    In this case we want the ratio of masses.

    Well I know density is mass/ volume [ σ = M/V] so

    M = Vσ

    Now the ratio: firstly put subscripts on the variable - I would use c for copper and a for aluminium

    Ma = Vaσa
    and
    Mc = Vcσc

    In ratio form:

    Ma/Mc = Va/Vc x σac

    We thus know

    Ma/Mc = Va/Vc x 1/3

    since we were given the ratio of the densities.

    So now we need the ratio of the Volumes to complete this.
    Each wire is effectively a cylinder

    V =πr2h

    For the wire, h = length of the wire - which is the same for both wires - so the ratio reduces to.

    Va/Vc = ra2/rc2

    So now we need the ratio of Radii [or diameters?]

    Resistance is given by"

    R = ρL/A

    Since Area here is that of the circular wire,

    R = ρL/∏r2

    transposing

    r2 = ρL/∏R

    This gives

    ra2 = ρaLa/∏aRa
    and
    rc2 = ρcLc/∏cRc


    Now for these wires, Length and resistance [and of course ∏] are the same, so the ratio simplifies to

    ra2/rc2 = ρac

    Substituting back into:

    Va/Vc = ra2/rc2

    gives

    Va/Vc = ρac

    Then back into:

    Ma/Mc = Va/Vc x 1/3

    gives

    Ma/Mc = ρac x 1/3

    which gives

    Ma/Mc = 2 x 1/3

    which is 2/3 or 2:3 if you like.

    While this has been lengthy to type out, when written it is much quicker.

    Note: Normally the line:

    ra2/rc2 = ρac

    would be expressed as

    ra/rc = √[ρac]

    but I knew my previous formula had ra2/rc2 in it so I left it as was.

    You can use this ratio technique to find the ratio of anything:

    eg Ratio of two accelerations

    F = ma → a = F/m

    so

    a1 = F1/m1
    and
    a2 = F2/m2

    a1/a2 = F1/F2 x m2/m1

    [Note that since m was in the denominator, is appears "upside down" as a ratio.]

    SO once we know the ratio of the forces, and the ratio of the masses we can work out the ratio of the accelerations, without calculating/knowing the actual value of each acceleration.
     
  6. Oct 26, 2011 #5
    Ahhhh okay, thanks so much for your help. Much appreciated. :)
     
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