Final answer:
The magnetic field in a coaxial cable varies depending on the region: it is zero inside the solid inner conductor, determined by Ampère's law in the region between the conductors, and zero in the region outside the outer conductor due to equal and opposite currents cancelling out.
Step-by-step explanation:
To determine the magnetic field at a distance r from the axis in a coaxial cable for various regions, we apply Ampère's law. We consider a coaxial cable with a solid inner conductor of radius R1, and an outer hollow cylindrical conductor with an inner radius R2 and outer radius R3. Both conductors carry equal but opposite currents I0 distributed uniformly over their cross-sections.
- r < R1: The magnetic field is zero inside the solid inner conductor because the symmetry of the current distribution implies Ampère's loop will not enclose any current.
- R1 < r < R2: The magnetic field in this region can be found by integrating the current enclosed by Ampère's loop, applying the formula B = (μ0*Ienc)/(2π*r), where Ienc is the current enclosed by the loop.
- R2 < r < R3: For this region, we apply Ampère's law again, but now the loop encloses the current in both conductors. As they carry equal and opposite currents, their magnetic fields cancel each other out in the region outside R3.
- r > R3: In the region outside the outer conductor, the net current enclosed is zero, so the magnetic field is also zero.