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Find Magnetic Field of Infinite Slab with constant current density

  1. Jan 26, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the magnetic field from an infinite slab with constant current density, Jo, in the x direction.

    ρ(z) = ρ1 x_hat for -b<z<b
    ρ(z) = 0 for |z| >= b


    2. Relevant equations
    Ampere's Law.

    3. The attempt at a solution

    I draw a rectangular prism inside the slab with -b<z<b with length 2z.

    I use Ampere's Law for the 4 sides and two sides are 0 due to being perpendicular to the current density.

    My remaining line integrals give me: 2zB + 2zB = μInet

    Inet = Itotal (A of amperian loop / A total slab)
    Inet = Itotal (2zxy / 2bxy)
    Inet = Jo dot dS (z/b)

    I do not know what to do with that integral of Jo dot dS.

    Am I on the right track?

    Thank you!
     
  2. jcsd
  3. Jan 26, 2013 #2

    mfb

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    The magnetic field is constant and [edit:] parallel to your sides, if you chose them properly. This allows to evaluate the integral.
     
    Last edited: Jan 26, 2013
  4. Jan 26, 2013 #3
    I don't understand. I've already cancelled two due to being perpendicular. Are you saying all 4 sides will be perpendicular?

    I've already solved the left side of the Ampere's law. Are you saying the line integrals are wrong?
     
  5. Jan 26, 2013 #4

    mfb

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    Sorry, that should be parallel.
    I don't understand your notation.
     
  6. Jan 26, 2013 #5
    I am given Jo, the current density.
    Inet = Current enclosed by amperian loop
    Itotal = Total current in slab.
    the slab is infinite in x and y direction but stops at -b< z <b in z direction.
    A is area of slab on XY plane
    Itotal = integral of J DOT dS

    I use ampere's law and create a rectangular prism inside the slab. Two of the lines are 0 due to being perpendicular. The other two with length 2z add up and become

    4zB =μ (Inet)

    Inet = Itotal (Volume of amperian Loop / Volume of Slab)
    Inet = Itotal (2zA / 2bA)
    Inet = integral J DOT dS (z/b)
    Inet = integral Jo DOT dS (z/b)

    I do not know if I am doing the right side correctly.

    Thanks for the replies!
     
  7. Jan 26, 2013 #6

    mfb

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    Can you add a sketch of your rectangle?

    Those are not well-defined.
     
  8. Jan 26, 2013 #7
    z_hat points up. x_hat right. y_hat into the board. slab goes from -infinity< x and y < infinity

    -b< z < b. I draw a ampere rectangular prism starting at the origin going to infinity in x and y direction. I go z1 up and -z1 down where -b < z1 < b.

    I do the loop in one face since current is pointing in the +x_hat direction. Using Right hand rule, B is coming out in -y direction at z >= b and beyond. B is coming into the board at y_hat direction at z <= -b
     
  9. Jan 26, 2013 #8

    mfb

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    You have a magnetic field in the slab as well, but that is not relevant here.
    I understand the geometry of the problem statement, I just don't understand what you are trying to calculate.
     
  10. Jan 26, 2013 #9
    I am trying to find the magnetic field in and outside of the slab.
     
  11. Jan 26, 2013 #10

    mfb

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    That is not what I meant. I did understand the problem statement based on the first post, there is no need to repeat.

    Apparently you have made some attempt to solve it - but I have no idea what you did there, and I don't think this will change without a sketch or a very clear explanation how you came to those calculations.
    If it involves any infinite length, current or whatever: This is wrong.
     
  12. Jan 26, 2013 #11
    I don't understand what you cannot follow. If I'm doing something wrong, I would love to be pointed in the right direction.

    I calculated the line integral on the left side of Ampere's Law by calculating the rectangle where two sides cancel and the remaining sides are on the z axis. Since these sides are length 2z, they add up to be 4zB. I do not know how to calculate the right side.

    You could throw a dog a bone and show me a little of what to do.
     
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