Final answer:
Magnetic flux can be zero when the magnetic field is not zero if the magnetic field lines and the surface's area vector are oriented such that they are either parallel, antiparallel, or perpendicular to one another, resulting in zero flux due to the angle's sine or cosine being zero.
Step-by-step explanation:
The magnetic flux through a surface can be zero even if the magnetic field is not zero. This situation arises when the angle between the magnetic field lines and the surface's area vector is either 0° or 180°, making them parallel or antiparallel, or when the angle is 90°, perpendicular to each other.
Case (a): If the angle is 0°, the magnetic field lines are parallel to the surface's area vector, and the sine of the angle is zero, resulting in zero magnetic flux.
Case (d): Conversely, if the angle between the magnetic field and the area vector is 90°, then the cosine of the angle is zero, which also results in zero flux, because the magnetic field is effectively 'glancing' off the surface without penetrating it.
Therefore, the key to having zero magnetic flux despite a non-zero magnetic field comes down to the orientation of the magnetic field relative to the surface area through which the flux is being calculated.