Kelvin's formula is used to calculate forces acting on a permanent magnet in the presence of an external magnetic field from a second permanent magnet. This approach is used to derive explicit analytical solutions for the axial and lateral forces between cuboidal and cylindrical permanent magnets as functions of magnet dimensions and separation. While exact solutions can be found for cuboidal magnets, a hypergeometric expansion is used to approximate the elliptic integrals in solving for the fields and forces for the cylindrical magnets. The resulting equations are applied over a range of magnet sizes and geometries to explore scaling laws and other geometrical effects. It is shown that cuboidal magnets provide larger forces than equivalently sized cylindrical magnets. Also, the aspect ratio of the magnets significantly affects the forces. These results are intended to benefit the design and optimization of sensors, actuators and systems that rely on magnetic forces, particularly at the microscale.
In this work, samarium cobalt (SmCo) powder is utilized for the fabrication of micromagnetic structures embedded in silicon wafers. The fabrication process involves the dry packing of raw, magnetic powder (mean particle diameter of 5-10 m) into etched cavities within the wafer. After the wafer is loaded with the powder, a 6 m layer of polyimide is spun over the wafer's surface to seal the magnetic powder in place and permit the development of additional structures around the embedded magnets. The feature sizes achieved during the investigation range in thickness from 15 m to the entire depth of the wafer (~ 500 m), and in lateral dimensions from 150 m to 600 m. One set of processed micromagnets demonstrated a coercivity, H c = 141 kA/m (1.8 kOe), remanence, B r = 0.52 T (5.2 kG), and maximum energy product, (BH) max = 23 kJ/m 3 (2.9 MGOe).
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