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Homework Help: Using stokes theorem to find magnetic field

  1. May 10, 2010 #1
    1. The problem statement, all variables and given/known data

    magnetic field is azimuthal B(r) = B(p,z) [tex]\phi[/tex]
    current density J(r) = Jp(p,z) p + Jz(p,z) z
    = p*exp[-p] p + (p-2)*z*exp[-p] z

    use stokes theorem to find B-filed induced by current everywhere in space

    2. Relevant equations

    stokes - {integral}dS.[curl A] = {closed integral}dl.A
    curl B(r) = J(r)

    3. The attempt at a solution

    ={integral}dS.[curl B(r)] = {closed integral}dl.B(r)
    ={integral}dS.J(r) = {closed integral}dl.B(p,z) [tex]\phi[/tex]

    {integral}dS.J(r) = {integral}dS.p*exp[-p] p + (p-2)*z*exp[-p] z = {closed integral}dl.B(p,z) [tex]\phi[/tex]

    dl = pd[tex]\phi[/tex] [tex]\phi[/tex]
    dS = pd[tex]\phi[/tex]dz p + pd[tex]\phi[/tex]dp z

    So:
    {closed integral}pd[tex]\phi[/tex].B(p,z) [tex]\phi[/tex] - with limits 0-2pi
    = B(p,z)*2pi*p

    {integral}dS.p*exp[-p] p + (p-2)*z*exp[-p] z
    do in 2 parts:
    {integral}pd[tex]\phi[/tex]dz.p*exp[-p] p - with limits 0-2pi and 0-R
    = 2pi*p2*R*exp[-p]

    {integral}pd[tex]\phi[/tex]dp.(p-2)*z*exp[-p] - with limits 0-2pi and 0-r
    = -2pi*r2*exp[-r]

    so B(p,z)*2pi*p = 2pi*p*R*exp[-p] - 2pi*r2*exp[-r]
    = B(p,z) = p*R*exp[-p] - r2/p*exp[-r]

    is this answer right?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 10, 2010 #2

    vela

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    Staff Emeritus
    Science Advisor
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    What's the difference between R, r, and p (by which I think you mean ρ)? That's the first thing you need to clear up.

    You didn't say what surface S you're integrating over. In any case, I think your evaluation of

    [tex]\oint_S (\nabla\times\textbf{B})\cdot d\textbf{S}[/tex]

    is incorrect.
     
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