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Geometric perspective of the vector potential

  1. Jun 9, 2012 #1
    I'm struggling with trying to visualize the vector potential as in the identity:

    B = ∇⨯A

    For starters, how does A relate to, say, a uniform magnetic field, which is quite easy to visualize. Then, how about the magnetic field around a bar magnet -- where is A?
    Any help would be appreciated.
     
  2. jcsd
  3. Jun 9, 2012 #2
    The formula should look similiar to something you're already familiar with:

    [tex]\mu_0 j = \nabla \times B[/tex]

    If you can imagine the magnetic field that goes with a specific current density, then the same picture applies to the vector potential that goes with the magnetic field.
     
  4. Jun 10, 2012 #3

    vanhees71

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    It's not easy to visualize [itex]\vec{A}[/itex] or give it a physical meaning since it is a gauge-dependent quantity. What's physical is the magnetic field, [itex]\vec{B}[/itex] which is given by

    [tex]\vec{B}=\vec{\nabla} \times \vec{A}.[/tex]

    For a constant field, it's easy to get the vector potential. So let's consider

    [tex]\vec{B}=B_0 \vec{e}_z.[/tex]

    You have quite some freedom to choose the vector potential. You can take always one constraint since it is only defined from [itex]\vec{B}[/itex] up to the gradient of a scalar field. Here, I'd choose the spatial axial gauge

    [tex]A_z=0.[/tex]

    Then you have

    [tex]\vec{B}=\begin{pmatrix}
    0 \\ 0 \\ B_0
    \end{pmatrix} = \vec{\nabla} \times \vec{A}=\begin{pmatrix}
    -\partial_z A_y \\ \partial_z A_x \\ \partial_x A_y-\partial_y A_x
    \end{pmatrix}.[/tex]

    Obviously our constraint doesn't fix the solutions completely, and you have some more freedom. You can, e.g., set [itex]A_x=0[/itex] and [itex]A_y=B_0 x[/itex]. Then you have

    [tex]\vec{A} = B_0 x \vec{e}_y.[/tex]

    Around a bar magnet you have to solve the magnetostatic Maxwell equations for a given magnetization of your bar. You find some calculations about this in Sommerfeld's Lectures on Theoretical Physics, vol. III.
     
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