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Current density of a wire.

  1. Oct 17, 2013 #1
    1. Problem statement
    The current density from the axis in the wire of radius R is given by J = cr3/2

    A) draw the current density as a function of r graph
    B) determine the constant c in terms of Itotal

    C) determine the current as a function of r.

    D) draw the graph of the current as a function of r.

    2. Known equations

    3. Attempt

    For part a) I think the graph would just look like a gradually increasing graph, since r grows exponentially.

    For part b) I am just confused on this one, I really don't know just how to start this, I don't know which equation to include to give me c in terms of I

    Any input is greatly appreciated, thanks!
     
  2. jcsd
  3. Oct 17, 2013 #2

    UltrafastPED

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    You need to calculate I_total; this is the integral of J over the cross sectional area - this will relate c and I_total.
     
  4. Oct 17, 2013 #3
    Ok so I am using the equation.. J = I / A,

    The area, I think would be r * d(theta) since we are looking at the area of a wire, and using the J as given above, I will get

    Cr3/2 = I / r * d(theta)

    I move r * d(theta) to other side, then integrate from 0 to 2pi, then I solve for c leaving me

    C = I / 2pi * r5/2
     
  5. Oct 17, 2013 #4

    UltrafastPED

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    But r is the radial distance from the wire axis, and R is the wire radius; to get the total current:
    I_total = ∫J r dr dθ

    The angular factor is, as you found, 2 pi.
    So I_total = 2 pi ∫(cr^3/2 x r) dr = 2 pi c∫r^5/2 dr.

    After you integrate this you can solve for c in terms of I_total.
     
  6. Oct 17, 2013 #5
    Oo I see, the bounds for the integral will then be from 0 to R correct?

    Now since I know the equation of I_total isn't that the answer for part c aswell?

    I(R) = 4pi * c * R^(7/2) / 7
     
  7. Oct 17, 2013 #6

    UltrafastPED

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    You should be okay with the rest.
     
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