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A Electric field in an arbitrary number of dimensions

  1. Sep 17, 2016 #1
    I am looking to use Gauss's law to find the electric field in ##1+1## dimensional spacetime:

    ##\int \vec{E}\cdot d\vec{A}=\frac{Q}{\epsilon_{0}}##

    Now, for a point charge in ##1+1## dimensional spacetime, the Gaussian surface is the two endpoints (a distance ##r## away from the point charge) along which the electric field points outwards. How do I account for ##d\vec{A}## of the two endpoints?
     
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  3. Sep 17, 2016 #2

    mfb

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    The integral gets a sum over the two endpoints.
     
  4. Sep 17, 2016 #3
    I know that, but I am finding it difficult to form a quantitative value for the infinitesimal area of the two points.
     
  5. Sep 17, 2016 #4

    mfb

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    Just add the two electric field strengths? Your field will have different dimensions anyway, so it fits.
     
  6. Sep 17, 2016 #5
    I don't exactly understand what you mean when you say that the field will have different dimensions.
     
  7. Sep 17, 2016 #6

    mfb

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    Sorry, I didn't mean the field itself, I meant the equation. To work, ##\epsilon_0## needs different units, which means the whole equation has different units.
     
  8. Sep 17, 2016 #7
  9. Sep 17, 2016 #8

    robphy

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    The more interesting question is the magnetic field in an arbitrary number of dimensions....
     
  10. Sep 18, 2016 #9
    Well, ##\nabla\cdot{\vec{B}}=0## generalises to ##\partial_{\mu}B^{\mu}=0##, I suppose, which means that ##B## has inverse dimensions of the length in any spacetime.

    Am I right?
     
  11. Sep 18, 2016 #10

    mfb

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    But how is it generated? ;)
     
  12. Sep 18, 2016 #11
    It's generated by a changing electric field, which comes from Faraday's Law.

    ##\epsilon_{ijk}\partial_{i}E_{i}=-\partial_{t}B_{i}## is Faraday's law in 3 dimensions.

    I guess the epsilon symbol ought to have more (or less) spatial indices as the number of spatial dimensions increases (or decreases)?
     
  13. Sep 18, 2016 #12

    mfb

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    There is no obvious generalization of this law to N dimensions.
     
  14. Sep 18, 2016 #13
    Is this why the magnetic field cannot be generalised to more than 3 dimensions?
     
  15. Sep 18, 2016 #14

    robphy

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    Crudely...

    The electric field is "time-space part" of the [antisymmetric] field tensor, orthogonal to the observer's 4-velocity. In (n+1)-spacetime, it has n independent components. In 4+1, Ex, Ey, Ez, Ew.

    The magnetic field is the remaining part... also orthogonal to the observer's 4-velocity. In (n+1)-spacetime, it has n(n-1)/2 independent components... Counting the number of strictly lower triangular space-space components. In 4+1, Bxy, Byz, Bzx, Bxw, Byw, Bzw (up to signs). The epsilon symbol or its equivalent will appear.

    For n=3, the electric and magnetic field each have 3 components.
     
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