I have a short question about the Coulomb potential.
Let $\vec{E}$ and $\vec{B}$ be the electric field and magnetic field respectively. The electric field and magnetic fields are described by the scalar potential $V$ and vector potential $\vec{A}$ respectively. As we know, there are a lot of scalar potentials and vector potentials describing the same electric field and magnetic field. My question is, can we $\textbf{always}$ find, in this case, the vector potential $\vec{A}$ that describes the given field $\vec{B}$ and also satisfy $$\nabla \cdot \vec{A} = 0~?$$
And is this also the case for Lorenz gauge, $$\nabla \cdot \vec{A} = -\mu_{0}\varepsilon_{0}\frac{\partial V}{\partial t}~?$$