Working with Electric Field E, not Vector Potential A

[WeiJie]

We commonly have E and B defined as:

But how can I work in electric field E, instead of vector potential A?

 

[Orodruin]

I am sorry, but it is unclear what you are asking. Can you be more specific?

 

[RPinPA]

We commonly have E and B defined as:

But how can I work in electric field E, instead of vector potential A?

That's not Maxwell's Equations. That is the relationship between the potentials and $\mathbf E$ and $\mathbf B$.

Maxwell's Equations are commonly written in terms of $\mathbf E$ and $\mathbf B$:
$$\nabla \cdot \mathbf E = \frac \rho {\varepsilon_0} \\
\nabla \cdot \mathbf B = 0 \\
\nabla \times \mathbf E = - \frac {\partial \mathbf B} {\partial t} \\
\nabla \times \mathbf B = \mu_0 \mathbf J + \mu_0 \varepsilon_0 \frac {\partial \mathbf E} {\partial t}$$

Those can of course be rewritten in terms of the potentials by substitution. But the usual thing you find when you search for "Maxwell's Equations" is in terms of the fields.