Finding scalar potential from E-field and vector potential

[Denver Dang]

Homework Statement

Hi.

This one I really am lost on :/
In my mind it seems rather easy, but I still can't figure it out.
I have been given the E-field:
$$\mathbf{E}\left( t,\,\,\vec{r} \right)=\frac{\kappa }{{{\varepsilon }_{0}}}\left[ \begin{matrix} ctx+{{x}^{2}}-{{y}^{2}} \\ cty+{{y}^{2}} \\ ctz+{{z}^{2}}-{{y}^{2}} \\ \end{matrix} \right]$$
And then I have calculated the vector potential, which gives:
$$\mathbf{A}=\frac{\kappa t}{{{\varepsilon }_{0}}}\left[ \begin{matrix} {{y}^{2}} \\ 0 \\ {{y}^{2}} \\ \end{matrix} \right]$$
And now I need to calculate the scalar potential

Homework Equations

I'm thinking this one:
$$\mathbf{E}=-\nabla V-\frac{\partial \mathbf{A}}{\partial t}$$

The Attempt at a Solution

My problem is, that I'm not entirely sure about what to do with the scalar potential $V$.
Taking the derivative of $A$ is no problem, and adding $E$ and $A$ together is easy as well. But how is it get the scalar potential to stand alone ? Differentiate with $\nabla$ on both sides doesn't make sense to me.

I'm guessing it's pretty simple, but again, at the moment, I'm kinda lost :/

So any help would be appreciated.Thanks in advance.

 

[haruspex]

You should be able to write out three partial differential equations (expressing ∇V = some vector). If one of those equations looks like, say, ∂V/∂x = f(x), what could you write down as the general form of V?

 

[pdxautodidact]

i would write out i=x, y, z and consider them all at once.

$$E_i = -\frac{\partial}{\partial x^i}\phi -\frac{\partial A_i}{\partial t} $$