A spherical neodymium magnet of diameter

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Electricity, Magnetism & Electromagnetism · Kinematics and Newtonian dynamics of particles

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

Problem

20\,\mathrm{mm}

, remanence

B_r = 1.2\,\mathrm{T}

and density

7500\,\mathrm{kg\,m^{-3}}

falls on the axis of a long vertical copper pipe of outer diameter

28\,\mathrm{mm}

and wall thickness

1.0\,\mathrm{mm}

(bore

26\,\mathrm{mm}

, mean radius

a = 13.5\,\mathrm{mm}

), conductivity

\sigma = 5.96\times10^{7}\,\mathrm{S\,m^{-1}}

. Treating the magnet as a point dipole of moment

|\mathbf{m}| = (B_r/\mu_0)\times\tfrac{4}{3}\pi r^3

(with

r

the magnet's radius), the drag on the magnet is

F = \beta v

with

\beta = \frac{45\mu_0^2|\mathbf{m}|^2\sigma d}{1024\,a^4}

for a thin-walled pipe of wall thickness

d

. ((a)) Find

|\mathbf{m}|

and the magnet's mass. ((b)) Find the terminal speed

v_t

of the fall.

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