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Resistance of a cone section

  1. Oct 8, 2013 #1
    1. The problem statement, all variables and given/known data
    A piece of conically-shaped material is placed in a circuit along the x-axis. The resistivity of this material varies as rho=(6*10^6)*x^4 (where x is measured in meters and rho is measured in ohm*meters), and its radius varies linearly as a function of x, ranging from r1=xinitial=3 cm to r2=xfinal=8.4 cm.


    2. Relevant equations
    R=rho*L/A


    3. The attempt at a solution

    Well current is flowing through the cone section so the cross sectional area is a circle, with the radius increasing proportional to x, in fact the are equal.

    So dR=rho*dx/(2*pi*r) and dx=dr and x=r so plugging in for r

    dR=(6*10^6)*r^4/(2*pi*r) dr = (3/pi * 10^6)*r^3 dr

    which integrating from 0.03 to 0.084 m i got 11.69 Ohm, doesn't seem quite right did I mess up somewhere?
     
  2. jcsd
  3. Oct 8, 2013 #2

    gneill

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    Staff: Mentor

    You sure that ##A = 2 \pi r## ?
     
  4. Oct 8, 2013 #3
    Would it be something else? A bunch of little rings expanding out would make a circle yes?
     
  5. Oct 8, 2013 #4

    gneill

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    Staff: Mentor

    Shouldn't A be the cross-sectional area of the cone? It didn't appear that you were integrating in the radial direction...
     
  6. Oct 8, 2013 #5
    Okay so the cross sectional area would be a circle, A=pi*r^2 , and since r varies linearly with the length of the cone A=pi*x^2

    And integrating with respect to x..

    (6/pi*10^6)*x^2 dx from xi to xf?
     
  7. Oct 8, 2013 #6

    gneill

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    Staff: Mentor

    Yup, that looks better.
     
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