1. Limited time only! Sign up for a free 30min personal 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!

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


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    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


    Then you have

    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

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook