Final answer:
To achieve the greatest mutual inductance between two identical flat coils, align their axes parallel to each other. For the least mutual inductance, align their axes perpendicular to each other.
Step-by-step explanation:
To achieve the greatest mutual inductance between two identical flat coils, you would align their axes parallel to each other. This means that their planes would be facing the same direction. This orientation allows for the maximum amount of magnetic field lines produced by one coil to pass through the other coil, resulting in a higher mutual inductance.
On the other hand, to achieve the least mutual inductance, you would align the axes of the coils perpendicular to each other. In this configuration, the magnetic field lines produced by one coil would be mostly orthogonal to the other coil, resulting in minimal interaction between the coils and a lower mutual inductance. Contrarily, to achieve the least mutual inductance, these coils should be aligned with their axes perpendicular to each other. When positioned thus, the magnetic field lines produced by one coil cut the edge rather than the area of the second coil, leading to a minimized inductive coupling between them.