Details

Two models are used for the calculations: the point-dipole model of a magnet [1–3] (model 1) and the exact cylindrical magnet model [4, 5] (model 2).

The cylindrical magnet model contains a double integral; therefore, calculation is very slow.

The results of both models coincide when and both equal 1.6.

System parameters:

= pipe length,

= outer diameter of the pipe,

= thickness of the pipe wall,

= radius of the magnet,

= height of the magnet,

= volume of the magnet,

= remnant field.

Also:

,

,

, terminal velocity,

, drag coefficient for infinite pipe (point-dipole models),

, , inner and outer radius of pipe,

volume of magnet,

= conductivity of metallic pipe,

, drag force.

, drag force for finite pipe (point-dipole model).

, drag coefficient for infinite pipe (point-dipole model).

The velocity was found as the solution of the equation of motion by an iterative method:

.

References

[1] B. A. Knyazev, I. A. Kotelnikov, A. A. Tyutin and V. S. Cherkassky, "Braking of a Magnetic Dipole Moving with an Arbitrary Velocity through a Conducting Pipe," Physics-Uspekhi, 49(9), 2006 pp. 937–946. doi:10.1070/PU2006v049n09ABEH005881.

[2] A. K. Thottoli, M. Fayis, T. C. Mohamed, T. Amjad, P. T. Shameem and M. Mishab, "Study of Magnet Fall through Conducting Pipes Using a Data Logger," SN Applied Sciences, 1(9), 2019 1050. doi:10.1007/s42452-019-1086-z.

[3] C. S. MacLatchy, P. Backman and L. Bogan, "A Quantitative Magnetic Braking Experiment," American Journal of Physics, 61(12), 1993 pp. 1096–1101. doi:10.1119/1.17356.

[4] N. Derby and S. Olbert, "Cylindrical Magnets and Ideal Solenoids," American Journal of Physics, 78(3), 2010 pp. 229–235. doi:10.1119/1.3256157.

[5] Q. L. Peng, S. M. McMurry and J. M. D. Coey, "Axial Magnetic Field Produced by Axially and Radially Magnetized Permanent Rings," Journal of Magnetism and Magnetic Materials, 268(1–2), 2004 pp. 165–169. doi:10.1016/S0304-8853(03)00494-3.