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Ampere's Law: Determining magnetic fields of a shell conductor

  1. Mar 25, 2013 #1
    1. The problem statement, all variables and given/known data

    A current of constant density, J0, flows through a very long cylindrical conducting shell with inner radius a and outer radius b. What is the magnetic field in the regions r < a, a < r < b, and r > b? (Use any variable or symbol stated above along with the following as necessary: μ0.)

    2. Relevant equations

    Ampere's Law
    [itex]\oint B \bullet ds[/itex] = μ0ienc


    3. The attempt at a solution

    [itex]\oint B \bullet ds[/itex] = μ0ienc

    Solving for B:

    B[itex]\oint ds[/itex] = μ0ienc
    B2∏r = μ0ienc

    B = μ0ienc/2∏r

    At r < a:
    B = 0 because ienc at this point = 0

    At a < r < b:

    B = μ0ienc/2∏r

    I don't know how to get ienc. I know it has something to do with the current density.

    At r > b:

    ienc = itotal, but I would need an expression for the volume of the cylinder.

    V = ∏b2h - ∏a2h
    V = ∏h(b2-a2)
    Ienc = J0V [itex]\Rightarrow[/itex] Ienc = J0∏h(b2-a2)

    B = μ0ienc/2∏r

    B = μ0J0h(b2-a2)/2r

    However, by including h, I am introducing variables that the problem hasn't allowed me to use.
     
    Last edited: Mar 25, 2013
  2. jcsd
  3. Mar 25, 2013 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    Current = current density times cross-sectional area. What is the cross-sectional area of your cylinder at a < r < b?
    No. Volume does not enter the picture. Cross-section does.
     
  4. Mar 25, 2013 #3
    That makes sense.

    So at a < r < b:

    B = μ0J0(r2-a2)/2r

    At r > b:

    B = μ0J0(b2-a2)/2r

    That worked. Thanks for your help.
     
    Last edited: Mar 25, 2013
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