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Homework Help: Magnetic scalar potential and function expansion

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider two long, straight wires, parallel to the z-axis, spaced a distance [itex]d[/itex] apart and carrying currents [itex]I[/itex] in opposite directions. Describe the magnetic field [itex]\mathbf{H}[/itex] in terms of the magnetic scalar potential [itex]\Phi[/itex], with [itex]\mathbf{H}=-\nabla \Phi[/itex]. If the wires are parallel to the z-axis with positions [itex]x=\pm d/2,\; y=0[/itex] show that in the limit of small spacing, the potential is approximately that of a two dimensional dipole
    [tex]
    \Phi\approx -\frac{Id\sin\phi}{2\pi \rho}+\mathcal{O}(d^2/\rho^2)
    [/tex]


    2. Relevant equations
    For 2D, the general solution for a polar coordinates problem is
    [tex]
    \Phi(\rho,\phi)=a_0 + b_0\ln\rho + \sum_{n=1}^{\infty}a_n \rho^n \sin(n\phi+\alpha_n)+\sum_{n=1}^{\infty}b_n \rho^{-n} \sin(n\phi +\beta_n)
    [/tex]



    3. The attempt at a solution
    Well, I already have the solution for this problem doing it a different way... but I was thinking about it some more and I was wondering if it's possible to solve it using the equation above as a solution to Laplace's equation
    [tex]
    \nabla^2 \Phi=0
    [/tex]
    and writing down a solution as a series solution. But I don't know how to properly implement the BCs (if any...) to begin solving it this way. Is it possible to get a solution to Laplace's equation this way, and if so, is it close to (or faster) that simply taking the scalar potential of a wire and using superposition? Also, in the event that its practical, can someone give me a hand in setting it up?

    Thanks,
     
  2. jcsd
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