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Vector potential confusion

  1. Sep 12, 2011 #1
    1. The problem statement, all variables and given/known data
    The concept of a scalar potential is reasonably straight forward. It is the energy needed to move to a point from some arbitrary reference point, the reference point being the origin for most mechanical problems and infinity for most electromagnetic problems.And of course this will produce a scalar field.

    The physical meaning of a vector potential, on the other hand, is alluding me. All of my texts seem to be very vague at this point. Mathematically I can say that B = del cross A where A is a vector potential but what does that mean physically.

    Any clarification of this point will be sincerely appreciated.

    Gary R



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 12, 2011 #2

    diazona

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    It's true that the physical significance of a vector potential is not at all as clear as it is for a scalar potential. Just as the scalar potential describes a difference between points, the vector potential describes a difference between paths. For example, when you move a charge from one point to another, it gains a certain amount of energy, and the scalar potential lets you figure out how much. Similarly, when you move a current from one path to another (imagine you have a wire carrying current from point A to point B, and you bend it into a different shape), the current gains a certain amount of momentum, and the vector potential lets you figure out how much.

    In practice, it's more useful to talk about the vector potential around a loop, rather than along an arbitrary open path. In the example above, if you take the original path and the new path (in reverse), you form a loop. In more advanced physics, this ties in to the interpretation of the vector potential (and the scalar potential) as the connection of a gauge covariant derivative: essentially it describes the transformations you have to make on a quantum field as you go from one point in space to another.
     
  4. Sep 12, 2011 #3
    Thanks diazona

    Just what I needed. It's nice to know that I'm not just dense.

    Gary R
     
  5. Sep 12, 2011 #4

    vela

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    In volume two of the Feynman Lectures, Feynman notes that in magnetostatics the energy of currents in a magnetic field is given by[tex]U= \frac{1}{2}\int \vec{j}\cdot\vec{A}\,dV[/tex]In comparison, for electrostatics, you have[tex]U = \frac{1}{2}\int \rho\phi\,dV[/tex]But then he points out the idea of the vector potential as potential energy for currents doesn't turn out to be very useful.

    He also discusses how A fits into quantum mechanics. It's worth a read if you get a chance.
     
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