What is the impact of superposition on Amper's Law?

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Superposition impacts Amper's Law by determining the net current enclosed by Amperian loops. When analyzing a system with two opposing currents, the loop outside the cylinder encloses zero current, resulting in a zero magnetic field. Conversely, the loop inside the cylinder encloses the current I, leading to a magnetic field equivalent to that of the wire. The distinction arises because the conductors are considered infinite, allowing for the application of this method. Understanding these principles is crucial for correctly applying Amper's Law in complex scenarios.
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what is your question?
 
this is the wrong answer... the question is where is my mistake...?
 
It's been awhile since I've done a problem like this, so please forgive me if I'm wrong.

The first thing you should do is draw two amperian loops. One inside the cylinder curling in the azimuthal direction with it's center being right where the wire is. The second one should be outside the cylinder with the same direction and same center as the first one.

Now the current in ampere's law is the current ENCLOSED by the amperian loops. So, for the one outside of everything the total current enclosed is zero (since the currents are in opposite directions). For the amperian loop inside the cylinder the current enclosed is I.

Therefore, the answer is, the field is zero outside the cylinder and inside the cylinder it is just the first field you gave (you called it the Wire's magnetic field). Is that the answer you were looking for?

Now, you might wonder why your answer is wrong. And I think they would be right if the conductors were not infinite, but they are and since they are we can use this amperian trick.
 
Cheers... very helpful...
 
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