Finding scalar potential from E-field and vector potential

  • #1
148
1

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:
[tex]
\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]
[/tex]
And then I have calculated the vector potential, which gives:
[tex]\mathbf{A}=\frac{\kappa t}{{{\varepsilon }_{0}}}\left[ \begin{matrix}
{{y}^{2}} \\
0 \\
{{y}^{2}} \\
\end{matrix} \right]
[/tex]
And now I need to calculate the scalar potential


Homework Equations


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


The Attempt at a Solution


My problem is, that I'm not entirely sure about what to do with the scalar potential [itex]V[/itex].
Taking the derivative of [itex]A[/itex] is no problem, and adding [itex]E[/itex] and [itex]A[/itex] together is easy as well. But how is it get the scalar potential to stand alone ? Differentiate with [itex]\nabla[/itex] 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.
 

Answers and Replies

  • #2
haruspex
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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?
 
  • #3
i would write out i=x, y, z and consider them all at once.

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

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