Using Ampere's Law for these two different integration paths

[WannaLearnPhysics]

Homework Statement:: The magnetic field at every point on the path of integration
Relevant Equations:: The scenarios/situations are shown in the attached photo.

"Any conductors present that are not enclosed by a particular path may still contribute to the value of B field at every point, but the line integrals of their fields around the path are zero" This is the statement from Young and Freedman's University Physics with Modern Physics.

For situation A, I'm thinking that since no currents are enclosed, we can't really use Ampere's law to know the magnetic field at a certain point on the dotted circle. In this case (suppose that the conductors are very long, and straight), we should use B = μI/2πr and do vector addition. This means that the magnetic field at every point on the dotted circle is not the same. I'm sure about this, but I just want to verify if I'm correct.

My real problem is situation B. If I use ampere's law then that means that the magnetic field at every point on the path (dotted circle) is just B=μI(encl)/2πr. I don't think this is correct because just like the statement above, conductors not enclosed by the path may still contribute to the value of B. I'm thinking that this is similar to situation A wherein the magnetic field at every point on the dotted circle is not the same (considering the currents outside the circle and their distances to a certain point), which means I should follow the principle of superposition of magnetic fields.Additional question. Can I say "consider the conductors to be really heavy" to imply that they stay in place or that their distances to the points on the circle remain constant?

[Moderator's note: Moved to a technical forum since the questions are about general understanding..]

 

[phyzguy]

Ampere's law applies for any closed path, regardless of the shape of the path and whether or not it encloses conductors. However, it is only useful to calculate the B-field when you can use the symmetry of the problem to see (for example) that the B-field is constant along the path you have chosen. When there is no symmetry, like in the pictures you have uploaded, I don't see any way to use Ampere's law to calculate the B-field. In cases like these, you typically have to use numerical codes to calculate the B-field by numerically carrying out the Biot-Savart law integrals.