Magnetic Assembly Stability Analysis

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 ( ${\mathbf{x}}_1={\left[\mathrm{0,\; 0,\; 0}\right]}^T$ with magnetic dipole moment ${\mathbf{m}}_1={\left[m_1,\mathrm{0,\; 0}\right]}^T$ ). Now, we consider placing a dipole magnet on the left in the x coordinate (position: ${\mathbf{x}}_2={\left[-L,\mathrm{0,\; 0}\right]}^T$ , with magnetic dipole moment ${\mathbf{m}}_2={\left[m_2,\mathrm{0,\; 0}\right]}^T$ ), as shown in Supplementary Fig. 7a.
The magnetic force between the two dipole magnets is: ${\mathbf{F}}_{m_1-m_2}=\frac{3{\mu }_0}{4\pi {∣\mathbf{r}∣}^4}\left\{\left(\widehat{\mathbf{r}}\times {\mathbf{m}}_1\right)\times {\mathbf{m}}_2+\left(\widehat{\mathbf{r}}\times {\mathbf{m}}_2\right)\times {\mathbf{m}}_1-2\widehat{\mathbf{r}}\left({\mathbf{m}}_1\cdot {\mathbf{m}}_2\right)\right)\left(+5\widehat{\mathbf{r}}\left[\left(\widehat{\mathbf{r}}\times {\mathbf{m}}_1\right)\cdot \left(\widehat{\mathbf{r}}\times {\mathbf{m}}_2\right)\right]\right\}$ where ${\mu }_0$ is the magnetic permeability in a vacuum and $\mathbf{r}$ is the relative position vector between the centers of the two dipole moments ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ . $\mathbf{r}={\mathbf{x}}_1-{\mathbf{x}}_2={\left[L,0,0\right]}^{\mathrm{T}}$
After plugging in all the relevant values, the force between two magnets can be expressed as ${\mathbf{F}}_{m_1-m_2}=\frac{3{\mu }_0}{4\pi {∣\mathbf{r}∣}^4}{m}_1{m}_2\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]$
Now let us consider that there are two magnets with opposite dipole directions, as shown in Supplementary Fig. 7b. In this case, two magnets ( ${\mathbf{m}}_2$ and ${\mathbf{m}}_3$ ) generate opposite force directions on magnet ${\mathbf{m}}_1$ . We would like to use this case to find the scaling effect of magnet ${\mathbf{m}}_3$ that can destabilize the assembly pair ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ .
If we consider both $m_1$ and $m_2$ 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 ${\mathbf{m}}_3={\left[{-m}_3,\mathrm{0,\; 0}\right]}^T$ , which provides a repulsion force. If we consider that ${\mathrm{m}}_3$ is sufficiently large that the force can balance the attraction force for ${\mathrm{m}}_2$ , it needs to be balanced; as a result: ${\mathbf{F}}_{m_1-m_2}+{\mathbf{F}}_{m_1-m_3}=0$ $\frac{3{\mu }_0}{4\pi L^4}{m}_1{m}_2-\frac{3{\mu }_0}{4\pi D^4}{m}_1{m}_3=0$
The required ${\mathrm{m}}_3$ needs to be large as $m_3=\frac{D^4}{L^4}{m}_2$ at position ${\mathbf{x}}_3={\left[-D,\mathrm{0,\; 0}\right]}^T$ to balance the magnetic force generated by ${\mathrm{m}}_2$ . Now, if we assume that ${\mathrm{m}}_3$ is a cube magnet with an edge size of a, we can rewrite the equation as $M_{NdFeB}{a}^3=\frac{D^4}{L^4}{m}_2$ where $M_{NdFeB}$ is the magnetization of the NdFeB magnet, which is equal to $1.08\times {10}^6$  A m^(-1). If we now consider D as a variable and consider how a needs to scale with D, we will find $a=\sqrt[3]{\frac{m_2}{M_{NdFeB}{L}^4}}{D}^{\frac{4}{3}}$
It shows that with an increasing distance D, the third magnet needs to increase rapidly in size ( ${~D}^{\frac{4}{3}}$ ) to match the force. This means that the magnet needs to be larger than a to destabilize the assembly pair between ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ . If the ${\mathrm{m}}_2$ magnet is touching ${\mathrm{m}}_1$ , as in the assembly pair, the required size of ${\mathrm{m}}_3$ can be so large that it is physically impossible to fit on the left side. If one changes the ${\mathrm{m}}_3$ 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.