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Lorentz and Coulomb gauges

  1. Dec 6, 2007 #1
    1. The problem statement, all variables and given/known data
    For the potentials:

    [tex] V(\vec{r}, t) = ct [/tex]
    [tex] \vec{A}(\vec{r}, t) = -\frac{K}{c} x \^x[/tex]

    c being velocity of light in a vacuum, determine the constant K assuming the potentials satisfy the Lorentz gauge.
    b) Do these potentials satisfy the Coulomb gauge as well?
    c) Show that for a set of potentials the Coulomb and Lorentz gauges can be simultaneously satisfied if V does not vary with time.
    d) Is this condition sufficient for the two gauges not to be mutually exclusive?
    2. Relevant equations
    For the Coulomb gauge:
    [tex]\bigtriangledown \cdot \vec{A} = 0[/tex]

    For the Lorentz gauge:

    [tex] \bigtriangledown \cdot \vec{A} = -\mu_0 \epsilon_0 \frac{\partial V}{\partial t}[/tex]

    Also:

    [tex] \bigtriangledown^2 + \frac{\partial}{\partial t} (\bigtriangledown \cdot \vec{A}) = -\frac{1}{\epsilon_0} \rho[/tex]

    [tex] (\bigtriangledown^2 \vec{A} - \mu_0 \epsilon_0 \frac{\partial^2 \vec{A}}{\partial^2 t}) - \bigtriangledown(\bigtriangledown \cdot \vec{A} + \mu_0 \epsilon_0\frac{\partial V}{\partial t}) = -\mu_0 \vec{J}[/tex]

    Which contain all the information in Maxwell's equations.

    3. The attempt at a solution

    I solved the first part, found that the constant K is

    [tex] K = c^2 \mu_0 \epsilon_0[/tex]

    My question is, how do I show that these potentials "satisfy" a gauge? Do I just plug the potentials into the condition for the divergence of A or is it something else?
     
  2. jcsd
  3. Dec 6, 2007 #2

    Avodyne

    User Avatar
    Science Advisor

    You plug them into the appropriate "For the ____ gauge" equation, and see if the equation is true or false for those particular potentials.
     
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