Elongation of a ferrofluid droplet near a permanent magnet: A tidal or magnetic energy effect?

Ferrofluid droplets falling toward a permanent magnet elongate as they approach the magnet. This phenomenon has been proposed as a way to visualize how tidal (nonuniform) forces can cause stretching/elongation of objects, such as the bulging of Earth's oceans under the nonuniform force of the Moon's gravity. In this manuscript, we analyze a ferrofluid in the nonuniform magnetic field of a permanent magnet and compare a simple model for the tidal stretching mechanism with the magnetic energy mechanism, which is known to stretch a ferrofluid droplet even in a uniform magnetic field. Far from the magnet, both mechanisms display power-law behavior as a function of the distance to the magnet z; for the tidal mechanism, the droplet's elongation is proportional to 1/z8, while that due to the magnetic energy mechanism is proportional to 1/z6. Thus, the elongation of a droplet falling toward a permanent magnet is initially dominated by the magnetic energy effect. The tidal effect overtakes the magnetic energy effect as the droplet approaches the magnet, and at a critical distance, a tidal disruption event occurs. This system can be studied in the laboratory and could be used as an exploratory laboratory for student-designed experimentation.