# Physical meaning of magnetic vector potential [duplicate]

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Can anyone give me an intuition of why the magnetic vector potential $A$ is sometimes interpreted as the electromagnetic momentum ?

I don't know analytical mechanics, just classical electromagnetism

## marked as duplicate by Kyle Kanos, Brandon Enright, JamalS, John Rennie, NeuneckNov 5 '14 at 8:51

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• Who refers to the vector potential as momentum? That's nonsense, AFAIK. – ACuriousMind Nov 5 '14 at 3:07
• Yes I'm wrong. It's sometimes interpreted as electromagnetic momentum Edited – Keith Nov 5 '14 at 3:08
• – Cheeku Nov 5 '14 at 3:15
• I already check that one. The problem is that in the solution they show some things that I don't understand, like "canonical momentum ". I search for a more elementary answer – Keith Nov 5 '14 at 3:18
• You might find this helpful: "Potential Momentum, Gauge Theory, and Electromagnetism in Introductory Physics" arxiv.org/abs/physics/9803023 – Alfred Centauri Nov 5 '14 at 3:36

## 1 Answer

In Lagrangian mechanics, one usually introduces the "generalized momentum" as $$p_{\phi_i}:=\frac{\partial L}{\partial \phi_i}$$where $L$ is the Lagrangian depending on a bunch of generalized coordinates $\phi_1\dots\phi_n$. If you introduce electromagnetic forces, you can derive, that the Lagrangian in the presence of an electromagnetic field has a form like $$L=\frac{m}{2}\left(\vec{p}-\frac{q}{c}\vec{A}\right)^2+q\Phi$$ where $A$ is your vector potential and $\Phi$ your scalar potential, therefore your generalized momentum will be shifted by $A$, i.e. if you write down the respective Newton equation, instead of $\dot{p}$ you will have this shifted by $A$.

The physical meaning behind this formal stuff is, that the electromagnetic field actually carries a momentum, which can be translated into a mechanical momentum. For example, I remember an experiment, where you put two spherical shells into a magnetic field and then turn the field off, the shells will get an angular momentum (I searched for it, didnt find it yet, will update if found).

• Ok. Thanks for the answer. So that means that the total momentum of any particle interacting with a magnetic field is less by a factor of $A$. Why is this ? Any intuition , if the particle is on magnetostatic field is different from a changing current in time. Does this affect the energy of the particle By some other factor of $A$? – Keith Nov 5 '14 at 3:29