Amperian Loop and Biot-Savart Law: Can It Detect a Non-Enclosed Perpendicular Current?

Consider an amperian loop that looks like the outline of a sector of a washer and a current element directed perpendicular to the plane of the amperian loop. This current is NOT enclosed by the amperian loop and does NOT extend to infinity. In addition, the current intersects the plane of the loop at the center shared by the two arcs of the amperian loop.

It appears to me that when evaluating the integral form of Amperes law the inner arc of the amperian loop contributes more (in an absolute value sense) than the outer arc to the greater amperes law expression being evaluated.

My reasons for this observation is that:

1.according to the biot-savart law, the B-field produced by the mentioned current element is proportional to (radius)^(-2), so, due to the position and direction of the current element, the inner arc ( with radius = r1) sees a B-field proportional to (r1)^(-2) over a length of 2*r1*sector_angle and the outer arc (with radius = r2) sees a B-field proportional to (r2)^(-2) over a length of 2*r2*sector_angle.

2. The remaining two "sides" of the amperian loop are directed perpendicularly to the B-field produced by the current element

3. After adding together the contributions from each segment of the amperian loop, the answer is non-zero.



Could someone please prove me wrong?

Thanks in advance:)