Two wires carrying equal but opposite currents are twisted together in the construction of a circuit. why does this technique reduce stray magnetic fields?

The key is to think about it using Integral form of Ampere's Law:


We use this equation like so, create an circular surface in which the wire will go through as shown in the illustration. The current in the wire will also go through the surface. Ampere's law states then that if the current traverses the surface the the current enclosed will be equal to the line integral of the magnetic field H ( in this case) along the closed path that encloses such surface. Due to symmetry the magnetic field will no vary so we end up having a constant magnetic field which we can take out of the integral. We end up having the following:

The field will point in the direction given by the right hand rule. We we assumed in the second step using the right hand rule that the field and the vector tangent to the path are parallel. The path is a cirlce so the integral over the path element is the circumference of that cirlcle.
For a parallel just next to the first wire but with current -I will produce a magnetic field Equal to our first in magnitude but with opposite direction, effectively nullifying the first one. Now it way sound bold but twisting the wires together locally means that both wires are always parallel to each other all along them. This implies both magnetic fields will negate as long as the wires are twisted.

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