Methods for generating directional magnetic fields and magnetic apparatuses thereof

A magnetic apparatus for generating a directional magnetic field towards a target direction on a first side thereof, the magnetic apparatus comprising:

• a first layer along a first surface perpendicular to the target direction, the first layer comprising one or more first-layer magnets interleaved with one or more first-layer ferromagnetic blocks;
• and
• a second layer on a second side opposite to the first side, the second layer comprising one or more second-layer magnets interleaved with one or more spacers;
• wherein a total number of the one or more first-layer magnets and the one or more second-layer magnets is greater than or equal to three, and a total number of the one or more first-layer magnets and the one or more first-layer ferromagnetic blocks is greater than or equal to three;
• wherein the one or more first-layer magnets are in an end-to-end arrangement on the first plane with alternating polarities along the first plane such that for each adjacent pair of the first-layer magnets, a pair of the ends thereof have a same pole and are at a distance smaller than that of the other pair of the ends thereof;
• wherein the one or more second-layer magnets are in a side-by-side arrangement with the polarities thereof parallel to the target direction, each of the one or more second-layer magnets overlaps one of a set of first ferromagnetic blocks of the one or more first-layer ferromagnetic blocks along the target direction, and each of the one or more spacers overlaps one of the one or more first-layer magnets;
• and
• wherein, in a first state, for each of the set of first ferromagnetic blocks, a pole of the adjacent second-layer magnet adjacent thereto is same as pole or poles of the adjacent first-layer magnet(s) adjacent thereto.