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Homework Help: Vectorpotential as a function of (t- x/c)

  1. Sep 11, 2010 #1
    1. The problem statement, all variables and given/known data

    I am working through chapter 47 of the Landau Lifschitz. And there is the following argument:

    The vector potential is a function of [tex] t - \frac{x}{c} [/tex]
    From the defining equations for the electric and magnetic fields:
    [tex] \vec{E} = - \frac{1}{c} \frac{\partial \vec{A}}{\partial t}, \vec{B} = \nabla \times \vec{A} [/tex]
    follows
    [tex] \vec{E} = - \frac{1}{c} \vec{A'} [/tex]
    [tex] \vec{B} = \nabla \times \vec{A} = \nabla (t- \frac{x}{c}) \times \vec{A'} = - \frac{1}{c} \vec{n} \times \vec{A'} [/tex]
    [tex] \vec{B} = \vec{n} \times \vec{E} [/tex]

    I can't follow his argument.
    Why did the equation for the electric field change from a time derivative of A to a derivative over (t- x/c).
    And where does that [tex] \nabla (t - x/c) [/tex] come from?
    Finally where does that vector n come from?

    Any help would be greatly appreciated!
     
  2. jcsd
  3. Sep 13, 2010 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    Let's define [itex]t_r\equiv t-\frac{x}{c}[/itex]. Then apply the chain rule:

    [tex]\begin{aligned}\frac{\partial}{\partial t}\textbf{A}(\textbf{x},t_r) &= \frac{\partial t_r}{\partial t}\frac{\partial}{\partial t_r}\textbf{A}(\textbf{x},t_r)+ \frac{\partial \textbf{x}}{\partial t}\cdot\mathbf{\nabla}\left(\textbf{A}(\textbf{x},t_r)\right) \\ &= (1)\textbf{A}'(\textbf{x},t_r)+(\mathbf{0})\cdot\mathbf{\nabla}\left(\textbf{A}(\textbf{x},t_r)\right) \\ &= \textbf{A}'(\textbf{x},t_r)\end{aligned}[/tex]

    Again, use the chain rule: In index notation w/ Einstein summation convention,

    [tex]\begin{aligned}\mathbf{\nabla}\times\textbf{A}(\textbf{x},t_r) &= \textbf{e}_i\epsilon_{ijk}\partial_j A_k(\textbf{x},t_r) \\ &= \textbf{e}_i\epsilon_{ijk}\left[(\partial_j\textbf{x})\cdot\mathbf{\nabla}A_k(\textbf{x},t)\right]_{t=t_r} + \textbf{e}_i\epsilon_{ijk}\frac{\partial}{\partial t_r} A_k(x_m\textbf{e}_m,t_r) \partial_j t_r \\ &= \textbf{e}_i\epsilon_{ijk}\left[\textbf{e}_j\cdot\mathbf{\nabla}A_k(\textbf{x},t)\right]_{t=t_r}+ (\mathbf{\nabla}t_r)\times\textbf{A}(x_m\textbf{e}_m,t_r) \\ & = \left[\mathbf{\nabla}\times\textbf{A}(x_m\textbf{e}_m,t)\right]_{t=t_r} -\frac{1}{c} (\mathbf{\nabla}x)\times\textbf{A}(x_m\textbf{e}_m,t_r) \\ & =\end{aligned}[/tex]

    I don't have the text with me, but it looks like Landau is describing a case where radiation is emitted in the [itex]\textbf{n}\equiv \mathbf{\nabla}x[/itex] direction, and you are only interested in a region where the non-retarded magnetic field is zero.
     
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