Linear stability of magnetic rings consisting of permanently magnetized particles p of the same magnetization is studied. Employing Lagrange’s approach and multipole moments, we determine the stability criterion in closed form. We then apply this criterion to characterize the stability for two scenarios in which the ring is i) compressed by dipolar loading due to a central point dipole and ii) compressed by mechanical loading. In contrast to previous works which show break up of magnetic rings under homogenous external magnetic field when instability occurs, we find that the ring does not break up when instability arises in both situations considered. However, the loading scenario decides instability modes via which the magnetic ring first loses its stability. For the first scenario, the ring deforms into planar but noncircular shapes via in-plane instability modes, regardless of ring sizes. For the other scenario, the ring deforms into planar but noncircular or nonplanar shapes via in-plane or out-of-plane instability modes for, respectively, small ring sizes or sufficiently large ring sizes. Finally, a simple experiment for the case of dipolar loading is proceeded and experimental results show good agreement with theoretical predictions.

研究了由具有相同磁化强度的永久磁化粒子p组成的磁环的线性稳定性。利用拉格朗日的方法和多极矩，我们确定了封闭形式的稳定性判据。然后，我们应用此标准来表征两种情况的稳定性，在这种情况下，环i）由于中心点偶极子而受到偶极载荷压缩，并且ii）受机械载荷压缩。与先前的研究表明在发生不稳定性时在均匀的外部磁场下磁环破裂时，我们发现在考虑的两种情况下，当发生不稳定时磁环都不会破裂。但是，加载方案决定了磁环首先失去稳定性的不稳定性模式。对于第一种情况，无论环的大小如何，环都会通过面内不稳定性模式变形为平面但非圆形的形状。对于另一种情况，对于较小的环尺寸或足够大的环尺寸，环分别通过平面内或平面外不稳定性模式变形为平面但非圆形或非平面形状。最后，对偶极加载情况进行了简单的实验，实验结果与理论预测吻合良好。