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

[ugenetic]

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?

 

[maajdl]

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.

 

[ugenetic]

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)

 

[sandy.bridge]

Can you provide more information regarding the question?

 

[maajdl]

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)

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) e_Θ .
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.

 

[ugenetic]

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 scenario 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)