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Finding scalar potential from E-field and vector potential

  1. Jun 14, 2013 #1
    1. The problem statement, all variables and given/known data
    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


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


    3. 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.
     
  2. jcsd
  3. Jun 14, 2013 #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?
     
  4. Jun 14, 2013 #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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