We provide an analytical model to show that disassembling an assembly pair using a large external magnet is very unlikely. We first consider a simple case, and then we generalize the case for more complex configurations.
In the simple case, we fix one NdFeB assembly magnet (1.2 × 1 × 0.5 mm) where its center position is at the origin of the Cartesian coordinate system (x = y = z = 0). Its dipole position points toward the positive x direction ( with magnetic dipole moment ). Now, we consider placing a dipole magnet on the left in the x coordinate (position: , with magnetic dipole moment ), as shown in Supplementary Fig. 7a.
The magnetic force between the two dipole magnets is: where is the magnetic permeability in a vacuum and is the relative position vector between the centers of the two dipole moments and .
After plugging in all the relevant values, the force between two magnets can be expressed as
Now let us consider that there are two magnets with opposite dipole directions, as shown in Supplementary Fig. 7b. In this case, two magnets ( and ) generate opposite force directions on magnet . We would like to use this case to find the scaling effect of magnet that can destabilize the assembly pair and .
If we consider both and to be NdFeB assembly magnets (1.2 × 1 × 0.5 mm), then we consider the third magnet with an opposite dipole direction with magnetic moment , which provides a repulsion force. If we consider that is sufficiently large that the force can balance the attraction force for , it needs to be balanced; as a result:
The required needs to be large as at position to balance the magnetic force generated by . Now, if we assume that is a cube magnet with an edge size of a, we can rewrite the equation as where is the magnetization of the NdFeB magnet, which is equal to A m^(-1). If we now consider D as a variable and consider how a needs to scale with D, we will find
It shows that with an increasing distance D, the third magnet needs to increase rapidly in size ( ) to match the force. This means that the magnet needs to be larger than a to destabilize the assembly pair between and . If the magnet is touching , as in the assembly pair, the required size of can be so large that it is physically impossible to fit on the left side. If one changes the direction, it will only decrease the magnetic repulsion force. The above scaling law provides an important insight that the magnetic gradient generated by a nearby permanent magnet is very unlikely to destabilize the magnetic assembly pair. The analysis results also resonate with our experimental observations. Therefore, we can conclude that the assembly will be stable in our envisioned applications regardless of the neighboring NdFeB magnet configurations.
Gu H., Möckli M., Ehmke C., Kim M., Wieland M., Moser S., Bechinger C., Boehler Q, & Nelson B.J. (2023). Self-folding soft-robotic chains with reconfigurable shapes and functionalities. Nature Communications, 14, 1263.