Ampere Law in a hole Should give zero?

[Neverthelessif]

Hi all,

I just did an exercice and there is one point that I don't understand (about Ampere's law).

Homework Statement

The exercice was about a cable with given diameter a, in which a small hole with diameter b is drilled in. The axis of the cable and the hole are parallel and the distance between the axis is given as r. There is a uniform steady current density flowing through it.
The question is to calculate the magnetic field at the center of cable and center of the hole.

2. Useful equations

Ampere's Law


Don't worry, I won't ask you to answer the exercice for me since I more or less finished it, I was just wondering: if I choose as Ampere's loop to be a circle centered in the hole, and with a radius smaller than b, why can't we say that the magnetic field equals ZERO because there is no current flowing through that circle anyhow?

What is to be done is to add up both magnetic fields by superposition and therefore we indeed have the sum of the magnetic filed of the hole being ZERO, plus the magnetic field of the cable which is NOT zero at that point.


So here is the question, black on white: How come we can have a non-zero magnetic field at a point in space around which we can draw an Ampere loop/surface through which there is zero current going through?


Thank you very much for your time :)

 

[Bhumble]

It seems to me like you already answered your own question.

"What is to be done is to add up both magnetic fields by superposition and therefore we indeed have the sum of the magnetic filed of the hole being ZERO, plus the magnetic field of the cable which is NOT zero at that point."

You are interested in the sum of individual contributions. Think about if you had a wire with current flow and were trying to determine the resultant magnetic field. You could have any number of ampere loops in which the wire is not enclosed and their resulting contribution is still zilch.
In your problem with the ampere loop from r = 0 to r = b is telling you that there is no contribution to the magnetic field from this portion of space since there is no current flow there.