Final answer:
To determine the magnetic field in all space regions around a wire with current and a hole, we apply Ampère's Law and superposition, accounting for the displacement of the hole and the symmetrical current distribution.
Step-by-step explanation:
To find the magnetic field in all regions of space around a long straight wire with a current and a circular hole, we can apply Ampère's Law and the superposition principle. For a point outside the wire, we would consider the symmetrical distribution of current and use Ampère's Law to find the magnetic field.
Inside the wire, we'd subtract the field due to the missing section (hole with radius b) from the field it would have if the hole weren't there. If the hole's center is displaced by distance d from the center of the wire, the situation is mathematically equivalent to a wire with no hole plus a wire of radius b carrying an opposite current and whose center is at the position of the hole. By calculating and combining these contributions, it is possible to obtain an expression for the magnetic field B at any point in space.