# magnetic vector potential question

The divergence of a magnetic field is zero.

$$\vec{\nabla}.\vec{B}=0$$

i.e. $$\vec{B}=\vec{\nabla} \times \vec{A}=\vec{\nabla} \times (\vec{A'}+\vec{\nabla}f)$$

Here:

$\vec{B}$ is magnetic field.

$\vec{A}$ is ambiguous magnetic vector potential.

$f$ is arbitrary function of $(x,y,z)$. It indeed is also ambiguous.

What is $\vec{A'}$ called? Is it also ambiguous?

• Do you hear the rusty gears creak as I answer? Anyway, if you have an $A'$ that has zero curl, then you can add an arbitrary value to $A$ and get the same $B$. Did your calculus tell you something with zero curl?
– user93146
Jun 29 '18 at 16:49
• The magnetic potential itself is a meaningless just as the electrostatic potential. They both are ambiguous, if you like. Jun 29 '18 at 17:12
• $\vec{A}'$ is the vector potential in a different gauge. Jun 29 '18 at 19:02

In the literature it is common to call both $\vec A$ and $\vec A'$ magnetic vector potential (or Gauge potentials). Either one is as valid as the other and one as such does not have a different name from any other.
That said, it is common to apply a "Gauge condition" to $\vec A$ such as: $$\vec \nabla \cdot \vec A=0$$ in which case we would call $\vec A$ for example the "magnetic vector potential in the Coulomb gauge".
The commonly accepted point of view is that any theory involving electromagnetism should be gauge invariant, that is, invariant under the substitution $\phi, \vec A \rightarrow \phi - \partial \chi / \partial t, \vec A + \vec \nabla \chi$ where $\chi$ is arbitrary. However there are arguments for a critical position. The Aharonov-Bohm effect implies that $\vec A$ is a physical quantity. Classical mechanics, quantum mechanics and quantum electrodynamics cannot be formulated in terms of $\vec E$ and $\vec B$, but require $V$ and $\vec A$. Only in the Lorenz gauge the wave equations for $\phi, \vec A$ are compliant with causality. There are numerous paradoxes involving the electromagnetic conservation laws. In my paper I argue that an alternative formulation is possible that resolves the paradoxes and respects causality. It states the relation known as the Lorenz gauge as the consequence of charge-current conservation.