1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?