Electrodynamics of a Magnet Moving through a Conducting Pipe

Abstract

The popular demonstration involving a permanent magnet falling through a conducting pipe is treated as an axially symmetric boundary value problem. Specifically, Maxwell's equations are solved for an axially symmetric magnet moving coaxially inside an infinitely long, conducting cylindrical shell of arbitrary thickness at nonrelativistic speeds. Analytic solutions for the fields are developed and used to derive the resulting drag force acting on the magnet in integral form. This treatment represents a significant improvement over existing models which idealize the problem as a point dipole moving slowly inside a pipe of negligible thickness. It also provides a rigorous study of eddy currents under a broad range of conditions, and can be used for precision magnetic braking applications. The case of a uniformly magnetized cylindrical magnet is considered in detail, and a comprehensive analytical and numerical study of the properties of the drag force is presented for this geometry. Various limiting cases of interest involving the shape and speed of the magnet and the full range of conductivity and magnetic behavior of the pipe material are investigated and corresponding asymptotic formulas are developed.