electromagnetism

If I interpret your question correctly, it means that the rod is rotating with its transverse axis in parallel with the direction of magnetic field.

The basic principle for the production of induced emf is because of cutting of magnetic field lines by a moving conductor. Since the number of magnetic field lines cannot be easily quantitified, hence magnetic flux is used instead. Such representation is employed in Faraday's Law to evaluate the induced emf, but the basic principle should not be undermined.

When the rod is rotating, there is surely cutting of magentic field lines by the rod. The induced emf is thus proportional to the amount of field lines cut by the rod in unit time, which is proportional to the area swept by the rod in unit time.

Magnetic force along the rod
F = q (v x B)