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Magnetic field in asymmetric hollow cylinder

  1. Mar 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Cylinder of radius [itex]a[/itex] and a cylindrical hole [itex]b < a[/itex] is displaced a distance [itex]d[/itex] in x-direction. Current density [itex]\textbf{J}=J_z\textbf{e}_z[/itex]. Show that a uniform magnetic field inside the hole is

    [itex]\textbf{B}=\frac{\mu_0}{2}J_zd\textbf{e}_y[/itex]


    2. Relevant equations
    Using previous result of whole cylinder

    [itex]\textbf{B}=\frac{\mu_0}{2}J_z(-y\textbf{e}_x + x\textbf{e}_y)[/itex]

    and that the cylindrical hole can be modelled as a charge density of [itex]-J_z\textbf{e}_z[/itex].


    3. The attempt at a solution

    I tried superposition of fields so

    [itex]\textbf{B}_1=\frac{\mu_0}{2}J_z(-y\textbf{e}_x + x\textbf{e}_y)[/itex]

    [itex]\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x + d)\textbf{e}_y)[/itex]

    [itex]\textbf{B}= \textbf{B}_1 + \textbf{B}_2[/itex] to which I get

    [itex]\textbf{B}=-\frac{\mu_0}{2}J_zd\textbf{e}_y[/itex]

    Any ideas, is this the right approach?
     
  2. jcsd
  3. Mar 8, 2012 #2
    Sorted, silly mistake

    [itex]\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x + d)\textbf{e}_y)[/itex]

    should be

    [itex]\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x - d)\textbf{e}_y)[/itex]
     
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