1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
     
  2. jcsd
  3. Sep 17, 2016 #2

    mfb

    User Avatar
    2017 Award

    Staff: Mentor

    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

    User Avatar
    2017 Award

    Staff: Mentor

    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

    User Avatar
    2017 Award

    Staff: Mentor

    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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    User Avatar
    2017 Award

    Staff: Mentor

    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

    User Avatar
    2017 Award

    Staff: Mentor

    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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted