Comparison of closed-form solutions to experimental magnetic force between two cylindrical magnets

Abstract

The force between permanent magnets implemented in many engineering devices remains an intriguing problem in basic physics. The variation of magnetic force with the distance x between a pair of magnets cannot usually be approximated as x-4 because of the dipole nature and geometry of magnets. In this work, the force between two identical cylindrical magnets is accurately described by a closed-form solution. The analytical model assumes that the magnets are uniformly magnetized along their length. The calculation, based on the magnetic field exerted by one magnet on the other along the direction of their orientation, shows a reduction in the magnetic force with the distance x and a dependence on the size parameters of magnets. To verify the equation, the experiment was set up by placing two cylindrical neodymium iron boron type magnets in a vertical tube. The repulsive force between the identical upper and lower magnets of 2.5 cm in diameter and 7.5 cm in length was measured from the weight on the top of the upper magnet. The resulting separation between the magnets was recorded as x. The forces measured at x=0.004-0.037 m differ from the values calculated using the analytic solution by -0.55 % to -13.60 %. The calculation also gives rise to a practical remnant magnetic field of 1.206 T. When x is much large than the equation of force is approximated as a simple form proportional to 1/x-4. The finding can be directly used in magnetic levitation as well as applied in calculating magnetic fields and forces in other systems incorporating permanent magnets.

Design and analysis of a d33 mode piezoelectric energy generator for vehicle braking system

“…Following the method in [29], the magnetic repulsion F rpl between two cylindrical magnets as a function of their separation gap can be written as: where x is the separation gap between the two magnets during the braking process, we assume x to be 1 mm; r and t are the radius and thickness of cylindrical magnet, respectively; B 0 is the flux density in tesla (T), B 0 is between 1 to 1.5 for Neodymium and B 0 is 1.45 for N52 grade neodymium magnet which will be used for our study [30]; µ 0 is the permeability of space, which equals 4π × 10 −7 Tm A −1 . However, for small values of x, the result from equation ( 2) becomes inaccurate and large [31].…”

Performance of closed-form equations for force between cylindrical magnets over wide range of volume, aspect ratio, and force

“…where, M -magnetization, V = πR 2 L -volume of the cylindrical magnet, R -cylinder radius, L -cylinder length. If the separation x between magnets is much greater than their size ( x >> L, but also x >> R) then the size of dipoles can be ignored ( ie treated as "points") and the force can be approximated [9][10][11][12][13],…”

“…The following equation, from [13] (Cheedket) behaves more correctly, providing a value for x = 0 ( ie contact force F 0 ). However in [11], the value of B r was fitted to the experimental data, so the predictive capability of this equation was not proved, despite seemingly good agreement of the curve shape with experiment over the investigated range.…”