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Amperes law: magnetic field inside an infinite slab containing a volume current.

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data
    an infinite slab is centered on the xy plane. the top of the slap is at z=a/2, the bottom of the slab is at z= -a/2. A volume current (J) is set up within the slab. find B everywhere

    J = k (1 - z^2/a^2) [itex]\hat{y}[/itex]
    2. Relevant equations

    amperes law

    3. The attempt at a solution

    the only thing I am confused about is finding B inside the slab.

    I was going to set up my amperian loop as a rectangle, which is oriented perpendicular to the volume current.

    The rectangle would extend a distance z above and below z=0 (z still inside the slab)

    however my text book mentions B is discontinious at a surface current, so i am uncertain if my amperian loop takes that into account.

    But given the stuff we have learned so far, I am very confident that we are supposed to apply amperes law to this situation.

    Can anyone explain if you can still apply amperes law for points inside a volume current?
  2. jcsd
  3. Jan 31, 2012 #2

    rude man

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    The only way i see is to
    (1) detrmine the vector potential A = (1/4π)∫idv/r
    where i = current density across a surface in elemental volume dv anywhere within the slab,
    and r = distance from dv to the point of observation. Since i has a y component only, so will A. So A = Ay y.

    dv = dx*dy*dz and extends from -∞ < x < ∞, -∞ < y < ∞, and -a/2 < z < a/2
    for the part inside the slab. For the outside, integration is limited to |z| > a/2.
    r is the distance from each dv to (x,y,z).

    This looks like a horrible pair of volume integrals to me!

    (2) If you did solve for Ay you could then find B from
    B = μ*curl (Ay y).

    Sorry this is the best I could do.
  4. Jan 31, 2012 #3
    so I think I may have left out the most important piece of information to this problem:

    the slab is made of a diamagnetic material (linear media, magnetization antiparallel to B)

    then inside the slab, you apply amperes law for the auxilary field H


    where the magnetization M =χ[itex]_{m}[/itex]H

    So this discontinuity I was so worried about is due to the magnetization?
    (M should produce a surface current, which is exactly the source of B's discontinuity)

    I am pretty comfortable doing these types of problems mathematically, but is my reasoning sound?
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