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Conceptual origin of the magnetic vector potential...?

  1. Sep 22, 2015 #1
    In Griffiths, it seems that the conceptual introduction of the magnetic vector potential to electrodynamics was justified based on the fact that the divergence of a curl is zero; so we can define a magnetic field as the curl of another vector A and still maintain consistency with Maxwell's equations.

    Further, curl-less components could be added to A (introducing the concept of different gauges) and still obtain the same results as well.

    My question is, basically: was it a purely mathematical justification for introducing the physical concept of the magnetic vector potential? I.e., was it just a question of noticing "Hey, I can make B the curl of another vector!"?

    So essentially I guess I'm asking: did the physics drive the mathematics or vice versa?

    My assumption is that the mathematical relationship was noticed first, followed by the introduction of physical concepts, gauges, etc.; but wondering if others may know differently?

  2. jcsd
  3. Sep 23, 2015 #2


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    Staff: Mentor

    I was curious about your question myself, because the textbooks I've used don't go into much detail on the history of classical electrodynamics. So I did a Google search on "magnetic vector potential history" and this turned up on the first page:

    Evolution of the Concept of the Vector Potential in the Description of Fundamental Interactions (A. C. T. Wu, U of Michigan; C. N. Yang, Chinese U of Hong Kong and Tsinghua U of Beijing)

    This struck my eye because I remember Dr. Wu from when I was a grad student at U of M, and Dr. Yang is a Nobel Prize winner. So it might be worth your reading...
  4. Sep 23, 2015 #3

    Thanks, pretty much exactly what I was looking for...
  5. Sep 24, 2015 #4


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    2017 Award

    Wu and Yang have marvelous papers. One of my favorites is

    T. T. Wu and C. N. Yang. Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D, 12:3845, 1975.

    For classical electrodynamics the potentials (or relativistically spoken the four-vector potential) are auxilliary quantities to simplify the solution of the Maxwell equations. For given charge-current distributions they reduce a first-order set of differential equations for the 6 components of the electromagnetic field to a second-order set plus a gauge-fixing constraint. They are not physical, because they are only defined up to a gauge transformation, i.e., a physical situation is represented by an entire class of four-vector potentials, all connected by an appropriate gauge transformation. The choice of the appropriate gauge constraint for a given problem can be the key idea of its solution. The physical meaning of the solution is, however, given by the electromagnetic field, not immediately by the potentials.