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How to interpret the magnetic vector potential?

In electromagnetism, we can re-write the electric field in terms of the electric potential, and the magnetic vector potential. That is:

$E = -\nabla\phi - \frac{\partial A}{\partial t}$, where A is such that $B = \nabla \times A$.

I have an intuitive understanding of $\phi$ as the electric potential, as I am familiar with the formula $F = -\nabla V$, where $V$ is the potential energy. Therefore since $E = F/q$, it is easy to see how $\phi$ can be interpreted as the electric potential, in the electrostatic case.

I also know that $F = \frac{dp}{dt}$, where $p$ is momentum, and thus this leads me to believe that $A$ should be somehow connected to momentum, maybe like a "potential momentum". Is there such an intuitve way to understand what $A$ is physically?