A demonstration is built to show eddy-current braking with the falling magnet…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetic materials · Electromagnetic induction and alternating-current circuits

The Boundary-Value Problem, Ch Magnetic diffusion, shielding and flux exclusion

Problem

A demonstration is built to show eddy-current braking with the falling magnet described in the worked example (a neodymium sphere of magnetic moment ∣m∣=0.50 A m2|\mathbf{m}| = 0.50\,\mathrm{A\,m^2} and mass 3.9 g3.9\,\mathrm{g}). Two candidate tubes are on hand: a narrow aluminium pipe, mean radius a1=7.15 mma_1 = 7.15\,\mathrm{mm}, wall d1=0.70 mmd_1 = 0.70\,\mathrm{mm}, conductivity σ1=3.5×107 S m−1\sigma_1 = 3.5\times10^{7}\,\mathrm{S\,m^{-1}}; a wider copper pipe, mean radius a2=14.3 mma_2 = 14.3\,\mathrm{mm}, wall d2=0.70 mmd_2 = 0.70\,\mathrm{mm}, conductivity σ2=5.96×107 S m−1\sigma_2 = 5.96\times10^{7}\,\mathrm{S\,m^{-1}}. ((a)) Using the drag law F=βvF = \beta v with β=45μ02∣m∣2σd/1024a4\beta = 45\mu_0^2|\mathbf{m}|^2\sigma d/1024a^4, find the terminal speed of the sphere in each pipe. ((b)) State which pipe brakes the sphere more strongly and explain the result in terms of how β\beta depends on σ\sigma and aa. ((c)) The builder also wants to know whether mounting a sensitive magnetic sensor inside either pipe would protect it from the laboratory's static stray field. Using the diffusion-time argument for a conducting shell, decide whether either pipe shields a steady field, and state what kind of shield would be needed instead.

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