Magnetic drag in the quasi-static limit: A computational method

Abstract

The method of successive approximations is applied to Maxwell’s equations to calculate the magnetic drag on a conducting disk rotating under the influence of a localized nonuniform magnetic field. An expression for the damping torque produced by the magnetic field is obtained in the low-velocity (quasi-static) limit of the disk’s motion: The damping force, in the case of rectilinear motion, is also calculated. When the theoretical expression for the damping torque is specialized to the case of a uniform magnetic field, the result is found to be identical with that of an existing textbook treatment. In the Appendix, a simplified treatment of the magnetic drag problem suitable for an introductory-level laboratory class is given. This treatment yields a final expression for the damping torque which is identical in form to the rigorous result except for a scaling factor.

Magnetic braking revisited

“…In the experimental setup of Marcuso et al, 5,6 a/R d ϭ0.3 or larger, which prevents us from using our calculations to explain quantitatively the decreasing trend of the torque near the border, because then the footprint will encompass the border and the cylindrical symmetry inside the disk will be broken. However, the tail of the displayed curve may explain qualitatively the decreasing of the torque near the border that they observed experimentally but could not explain, even qualitatively, with the analytical expression they obtained.…”

“…Marcuso et al 5,6 made use of a method of successive approximations to solve Maxwell's equations in order to compute the braking force on a rotating disk under the action of a static external nonuniform magnetic field. The braking torque was compared with the experimental results for disks of aluminum and copper, obtaining very good agreement except near the disk border.…”

Design of a magnetic braking system

“…Heald [12] replace this simplification by a more realistic theory that allows for the fact that the eddy currents tend to concentrate toward the ends of the footprint and are not orthogonal to the velocity of the conducting sheet. Marcuso et al [13] computed the braking force on a rotating disk under non-uniform magnetic field. Experimental results are close to the theoretical predictions except near the disk edge [14].…”

Analytical results for rotating and linear magnetic brakes

“…Depending on the geometries of the conductor and the magnetic field, these space charges may be accompanied by an electric current density J, as in the case of the Faraday's first dynamo [9,10] and the induction motor [11]. References to these space charges on a magnetic brake made up of a disk rotating under the magnetic field of a single pole can be found in literature, [12,13,14,15,16,17] in the context of analytical calculations of forces and torques.…”

“…Previous analytical works on this subject either completely ignore the space charges induced on the disk, due to the technique used in the solution [14,18,19,20] or due to the simplicity of the model [5], or calculate these space charges only partially. According to the convenience, only ρ F is calculated, ignoring the surface charge density σ F [17] or, conversely, only σ F is calculated ignoring the charge density ρ F [12,13,15,16].…”