Curl H outside a long, thin wire of constant current

[rude man]

Homework Statement

Per Maxwell's equation and my H. H. Skilling EM textbook, curl H = 0 in the absence of current density.

But consider a long, thin wire along the z axis carrying time-invariant current I. By Ampere's law, at a point (x,0) outside the wire, H = I/2πx j . But curl H computed by Cramer's rule from the curl determinant is not zero.

2. Relevant equations

curl H = J + ∂D /∂t
2πxH = I

The Attempt at a Solution

Since I is time-invariant there is no time-varying electric field so ∂D/∂t = 0.

Since H = H j = I/2πx j at (x,0),
curl H = ∂H/∂x k - ∂H/∂z i
but for a long wire, ∂H/∂z = 0 so we wind up with
curl H = dH/dx k
= -I/2π x² k ≠ 0.

So how is the paradox resolved, please? If you're familiar with the concept of little "paddle wheels" (Skilling uses) indicating curl, even there it seems curl H ≠ 0 outside the wire.

All glory and thanks to the resolver!

 

[TSny]

If you want to work in Cartesian coordinates, then first work out the general expression for the components of H in terms of x and y. Since you are going to take derivatives, you don't want to let y = 0 until after taking the derivatives.

 

[rude man]

Thanks, T. Will explore! I had a faint idea I shouldn't ignore the x component of H but never paid it enough attention.

 

[rude man]

I switched to cylindrical coordinates, and lo & behold ...

I won't bother with the cartesian but obviously am now a Believer! Thanks again, will get to sleep a bit faster now ...

 

[paranormal]

Thank you for your inquiry. The apparent paradox can be resolved by considering the assumptions and limitations of Ampere's law and Maxwell's equations.

First, it is important to note that Ampere's law is a simplified version of Maxwell's equations that is valid only in certain situations. It assumes that the current is steady and that there are no time-varying electric fields. In the case of a long, thin wire carrying a constant current, this assumption holds true.

However, as you have correctly pointed out, there is a non-zero curl of H outside the wire. This is because Ampere's law does not take into account the displacement current, which is the time-varying electric field produced by a changing magnetic field. In the case of a long, thin wire, the displacement current is negligible compared to the conduction current, but it is still present and contributes to the curl of H.

In general, Maxwell's equations provide a more complete and accurate description of electromagnetic phenomena, but they can be more complex to solve. In certain situations, such as the one described in your question, Ampere's law can be a useful approximation, but it is important to remember its limitations.

I hope this explanation helps to resolve the paradox. If you have any further questions, please let me know. Thank you for your interest in electromagnetic theory.