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Electric resistance in the truncated rotating cone

  1. May 19, 2016 #1
    1. The problem statement, all variables and given/known data
    Homogeneous body with the shape of a truncated rotating cone has a base shaped like a

    circle. The radius of the lower base is R2 = 8 cm and radius of the upper base is R1 =

    4 cm. The height h = 8 cm (see figure). Calculate the total electric

    resistance between the base surfaces of the body, when made of a material of

    resistivity ρ = 4*10 ^(-2) Ohm*m. Assume that during the transport of electric charge

    through the body volume, the current density changes.

    2. Relevant equations
    see the picture

    3. The attempt at a solution

    idea:
    triple integration..one through the height...and the remaining ones through the radius and the perimeter....but does this take into account the changing of the radius at different heights?
     

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    Last edited: May 19, 2016
  2. jcsd
  3. May 19, 2016 #2

    TSny

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    Only a single integration is needed. Hint: thin circular disks.
     
  4. May 19, 2016 #3

    cnh1995

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    Write a general expression for any intermediate radius in terms of R1,R2 and h. Use it for calculation of resistance of thin circular discs and integrate along proper path.
     
  5. May 19, 2016 #4
    hm, this gives me the integral of PI*r^2 with boundries of R2 and R1...?
     
  6. May 19, 2016 #5

    TSny

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    This is not what you need to integrate. The entire truncated cone can be thought of as made of many thin slices (circular disks). Each disk will have a small resistance dR that will depend on the radius and thickness of the disk. These resistances are in "series", so the total resistance is the sum (or integral) of the individual resistances.
     
  7. May 19, 2016 #6

    cnh1995

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    As you can see the resistances are in series, your limits of integration will be from 0 to h. Try writing a general expression for intermediate radius in terms of R1, R2 and h.
     
  8. May 19, 2016 #7
    could you clarify this term pls, not even google helped me
     
  9. May 19, 2016 #8
    am I getting closer? dR = (PI*r^2*dh)/dI where I is electric current
     
  10. May 19, 2016 #9

    cnh1995

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    The radius varies from x=0 to x=h. You need to find a general expression for radius at a distance x in terms of R1, R2 and h. That's what I meant by intermediate radius. Limits of x will be from 0 to h when you'll integrate.
     
  11. May 19, 2016 #10

    cnh1995

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    There's no need to involve current.
     
  12. May 19, 2016 #11

    cnh1995

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    R=ρ*l/A.. For a small disc at a distance x from the leftmost end, what is the length and area of cross section in terms of x?
     
  13. May 19, 2016 #12

    TSny

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    In the figure below, you see a disk located at position x along the axis of the cone. The disk has a thickness dx and an area A(x) which depends on x. ( You will need to express A(x) explicitly in terms of the position x.) How would you express the resistance, dR, of the disk in terms of A(x) and dx?
     

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  14. May 19, 2016 #13

    cnh1995

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    Radius varies linearly with x. At x=0, r=R1 and at x=h, r=R2. Using this data, can you write a general expression for r in terms of x?
     
  15. May 19, 2016 #14
    The concept behind seems to be clear, the second thing is to write it down...what about: ....l (radius of a given element) = r1 + ((r2-r1)/h)*x
     
  16. May 19, 2016 #15

    cnh1995

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    It's like a straight line passing through (0, R1) and (h, R2). How will you write r as a function of x from this data? It is in the form r=ax+b.. Now a and b can be found using known quantities R1, R2 and h.
     
  17. May 19, 2016 #16
    r1 + ((r2-r1)/h)*x could it be?
     
  18. May 19, 2016 #17
    dR = ρ * (r1 + (((r2-r1)/h)*x)/dA) ?
     
  19. May 19, 2016 #18

    cnh1995

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    Right!
     
  20. May 19, 2016 #19
    great, I thank you so much!
    now all I need is to express dA and then I can turn to mathematics, yes?

    what about, dA = PI*r^2*dx?
     
  21. May 19, 2016 #20

    cnh1995

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    The expression you got is for radius r.
    R=ρ*l/A..
     
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