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Maxwell's 4th equation. BUT, curl of H is zero everywhere

  1. Jun 5, 2014 #1
    please go to "Wolfram Alpha" website, (google the term) and copy paste the following formula:

    curl ( (-y/(x^2+y^2), x/(x^2+y^2), 0)

    you can see the result is zero

    I think that's the expression of H surrounding a current. what am I missing?
     
  2. jcsd
  3. Jun 6, 2014 #2

    maajdl

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    Gold Member

    Try this:

    curl ( I(Sqrt(x^2+y^2)) (-y/(x^2+y^2), x/(x^2+y^2), 0) / (2 Pi) )

    Where I(r) = I(Sqrt(x^2+y^2)) represents the current flowing within radius r.
     
  4. Jun 6, 2014 #3
    I don't think your equation was correct. and I think curl of H outside of the wire SHOULD be zero. it should be I / Sqrt (x2 + y2) and x direction's expression should be -y/ sqrt(x2 + y2)
     
  5. Jun 6, 2014 #4
    Can you provide more information regarding the question?
     
  6. Jun 7, 2014 #5

    maajdl

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    My suggestion was to assume a current distribution I(r) that is not concentrated on x=y=0, but spread around x=y=0.
    Moreover, I suggest to assume any distribution of current I(r) (total current within radius r).
    That leads to a magnetic field H = I(r)/(2 ∏ r) eΘ .
    Replacing r as a function of (x,y,z) and taking the curl on Wolfram Alpha will help you understand what happens.

    The clue is that with a continuous function I(r), the curl will give you the current density j = I'/(2πr) .

    The reason why you got zero, is obviously because the derivation on Wolfram Alpha applies only on the domain of continuous functions. But you magnetic field is singular in x=y=0.

    Have a check!
    Simply paste this on Wolfram Alpha:

    curl ( I(Sqrt(x^2+y^2)) (-y/(x^2+y^2), x/(x^2+y^2), 0) / (2 Pi) )

    And analyze the result:

    I'(sqrt(x^2+y^2)) / (2 pi sqrt(x^2+y^2)) e_z

    =

    I'(r) / (2 pi r)

    You will recognize there the current density j.
    Note also that, in the problem you wanted to solve, I'(r) is zero everywhere except in x=y=0.

    If you already followed a course on the theory of distribution you could take for I a Gaussian distribution with a variable width w.
    In the limit of w->0, you will verify that j=curl(H) will tend to the Dirac delta distribution.
     
    Last edited: Jun 7, 2014
  7. Jun 7, 2014 #6
    Excellent answer, thank you

    I did check, but the notation of i (r) threw me off a bit, I just assumed that we are talking about the air portion of the H.

    so, maxwell's 4th equation in point form means: the rotating tendency of an Magnetic field (H) at a POINT = current density at THAT point plus if there are any changing E field.
    that's why in my infinitesimally small current will produce curl(H) = 0 every where, a senario not possible in the real world. But, it is good to know that H generated by current will have curl of zero in air. (not counting the displacement current business)
     
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