Imagine I have two current-carrying wires with the current going in the same direction. The magnetic field looks like the sketch below on the left: current carrying wire magnetic fields

Now imagine I squish the wires very close together but not touching. Then, I can draw an amperian loop between both wires, which does not enclose either wire, where the integral $\oint \vec{B}\cdot\vec{dL} \neq 0, \rightarrow I_{encl}\neq 0$ , because the 'vertical' bit is elongated while the 'horizantal' bit is foreshortened, so my amperian loop looks like an oval, with lots of overlap with B fields circulating with the loop and less overlap with B fields anti-circulating with the loop.

That means there is an extra current which appears between the two wires!

What gives?

  • 3
    $\begingroup$ I think you are making too big of an assumption based on what you think the magnetic field lines would be doing by imagining them in your head. Remember that the magnetic field lines, if drawn correctly, are drawn from the same equation that ensures amp's law is followed, so it's logically impossible for correctly drawn lines to violate amp's law. $\endgroup$
    – Señor O
    Mar 25 '19 at 3:48
  • $\begingroup$ Your assessment is incorrect. The integral you wrote is zero. It is easier to see using the differential rather than the integral form $\endgroup$
    – Dale
    Mar 25 '19 at 3:53
  • $\begingroup$ The more you squeeze the wires close to each other, indeed the "opposing" horizontal segments become shorter and shorter. But at the time, if you are right in the middle, the magnetic field along any of the verticals, becomes smaller and smaller. Notice that this vertical field, is a superposition of two opposing fields. At the limit, vertical field is zero and horizontal length is also zero. $\endgroup$
    – npojo
    Mar 25 '19 at 16:27

It has to be hand waving but hopefully you will see that your concentration on just the top and bottom of your oval Amperian loop has missed out the contributions due to the "middle" part of the loop.

I have reproduced your left-hand diagram, included two extra magnetic field lines and added an Amperian loop $ABCDA$ which is shown in green.

enter image description here

I have added labels in red to show the local direction of the magnetic field $B$ where there are positive $(+)$ and negative $(-)$ contributions to the line integral $\int \vec B \cdot d\vec L$ when moving anticlockwise around the loop.
I have confidence that if I could do such an integration over the whole loop the result would be zero and that Ampere's law is still valid in such a situation!


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