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Homework Help: Two closely related questions about current density

  1. Jan 7, 2012 #1
    *The problem statement

    The second question kinda builds on the first and I am not too confident my solution to the first is correct.

    1: A disk has a uniform surface charge density and is rotated at a rate 'w'. Find 'K'.

    2: A uniformly charged sphere ('ρ') is spun. Find the current density.

    *Relevant equations
    K=surface current density = σ*v
    J = volume current density = ρ*v


    *The attempt at a solution

    1: ω= the linear (tangental?) velocity of a point on this disk / radius, so v=wr

    then K = ωr*σ , and is oriented in the ∅ direction

    2: for a spinning sphere, the velocity of the charge is the same as above except v=0 on the axis of rotation for any r.

    so if i let the z axis be the axis of rotation, and let r = r(cos∅+sin∅), and v=wr(cos∅+sin∅)

    then J = ρwr(cos∅+sin∅), oriented in ∅ direction


    yea thats my attempt at the solution. I think my answers make sense, but most homework problems I am assigned require more than 2 lines of work so it is not unlikley that i am missing something.

    thanks in advance for any help
     
  2. jcsd
  3. Jan 8, 2012 #2

    vela

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    That's correct.

    The speed of a point in the sphere depends on its distance from the axis of rotation. If the axis of rotation is the z-axis, what's the distance of a point (r, θ, φ) from the z-axis?
     
  4. Jan 8, 2012 #3

    rude man

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    Kind of a strange question.
    The current di passing by an arbitrary elemental radial length dr at r, 0 < r < R is
    di = σrω dr
    so the corresponding elemental current crossing dr is a function of r
    and the only meaning for current density must be the current per unit radial length, or
    di/dr = σrω which has units of current per unit length.

    The total current is i = ∫di = σω∫r dr from 0 to R = σω(R^2)/2 so you could say an average current density is σω(R^2)/2R = σωR/2. The last also has units of current per unit length, the same units as the OP's "K".

    But really current density varies from 0 to σωR as we go from the center (r=0) to the edge (r=R) of the disk.

    I won't even try for the 2nd part.
     
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