Stability Analysis of an Array of Magnets: When Will It Jump?

A bidimensional array of magnets whose magnetic moments share the same vertical orientation, and lying on a planar surface, can be gradually compacted. As the density reaches a threshold, the assembly becomes unstable, and the magnets violently pop out of plane. In this Letter, we investigate experimentally and theoretically the maximum packing fraction (or density) of a bidimensional planar assembly of identical cylindrical magnets. We show that the instability can be attributed to local fluctuations of the altitude of the magnets on the planar surface. The maximum density is theoretically predicted assuming dipolar interactions between the magnets and is in excellent agreement with experimental results using a variety of cylindrical magnets.

Figure 1

Time sequence showing the dramatic out-of-plane burst of magnets assembly as the critical density is reached. The first magnet bursting at $t=0$ is highlighted in red. Magnetic dipoles $\mu$ appear as blue arrows, while the forces from the first neighbors of the hexagonal pattern are shown as orange arrows on the left.

Figure 2

Schematics of an ideally planar system (a) and realistic situation with slighted misaligned magnets (b). Model cases in which (c) only the altitude $z$ of the central magnet varies (creating a net vertical force $F_z$) and (d) in which the central magnet is inclined by $\theta$ (creating a torque $\tau$).

Figure 3

Energy landscapes. (a) Corresponds to situations with $\theta =0$ and only the potential of vertical magnetic force and (b) to situations with $z=0$ with only the potential of magnetic torque. The red lines are the sum of gravitational and magnetic potential energies.

Figure 4

Popup distance measured experimentally and predicted range{ see (3) with $z_c=[0.1-1]\mathrm{mm}$}. (a) Minimum distance $D$ for a variety of magnets. Lines: for vertical force (left, blue) and torque (right, yellow) Black points: experimental values. (b) Same experimental points plotted as function of distance $L_0$. Limits are only for vertical force.