Finding vector potential

Hi PF members,

I have a question about how to find the vector potential from a given electric field. For example,

[itex]\textbf{E}=-\nabla\phi-\partial\textbf{A}/\partial t[/itex] and [itex]\textbf{B}=∇\times\textbf{A}[/itex]

Given [itex]\textbf{E}=E_{0}\hat{x}[/itex], electrostatic potential may be 0 and [itex]\textbf{A}=-E_{0}t\hat{x}[/itex]

or A may be 0 and [itex]\phi=-E_{0}x[/itex] or may be it can be [itex]\textbf{A}=-\frac{E_{0}}{2}t\hat{x}[/itex] and [itex]\phi=-\frac{E_{0}}{2}x[/itex] or any other combinations of these two.

So how can I know which is the correct one?

Thanks in advance..