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Homework Help: Magnetic Field inside plate

  1. Mar 12, 2015 #1
    1. The problem statement, all variables and given/known data
    What is the magnetic field outside and inside of a large, flat conducting plate if the current density decreases linearly with depth z inside the plate. J = J0(1-az). The plate thickness is 1/a.

    2. Relevant equations
    *see below

    3. The attempt at a solution
    [itex] \oint \vec B \cdot d \vec l = \mu_0 I[/itex]

    So outside the plate,
    [itex] 2Bl = \mu_0 \int J_0(1-az)ldz[/itex] *The integral is taken from from z=0 to z=1/a (the whole plate)
    [itex]B=\frac{\mu_0 J_0}{4a} [/itex]

    Which agrees with the back of the book solution.

    Now inside is where I am having trouble...
    I tried applying Ampere's Law again but I just got
    [itex] \oint \vec B \cdot d \vec l = \mu_0 I[/itex]
    [itex] 2Bl = \mu_0 \int J_0(1-az)ldz[/itex] *taking the integral from 0 to z this time
    [itex] B=\frac{\mu_o J_0}{2} (z-\frac{az^2}{2})[/itex]

    but the back of the book says the answer is [itex] B = \mu_0 J_0 (\frac{az^2}{2} - z +\frac{1}{4a}) [/itex]

    So I'm not sure where the [itex] \frac{1}{4a} [/itex] came from... Does anyone know how i should go about finding the magnetic field inside the plate?
    Last edited: Mar 12, 2015
  2. jcsd
  3. Mar 12, 2015 #2


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    Homework Helper

    At some point z, you have a field above and a field below. You should account for both of them.
  4. Mar 13, 2015 #3
    Got it - thanks for the help
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