A magnetometer is to sit at the centre of a thin closed spherical enclosure

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 magnetometer is to sit at the centre of a thin closed spherical enclosure, mean radius

a = 4.0\,\mathrm{cm}

and wall thickness

d = 1.2\,\mathrm{mm}

. Two wall materials are available: copper (

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

) and mu-metal (

\mu_r = 5.0\times10^{4}

\sigma = 1.8\times10^{6}\,\mathrm{S\,m^{-1}}

). The sensor must be protected from the Earth's steady field,

50\,\mu\mathrm{T}

, and from a

50\,\mathrm{Hz}

interference field of amplitude

3.0\,\mu\mathrm{T}

from a nearby transformer. ((a)) Using the thin conducting shell result

\tau_s = \mu_0\sigma ad/3

and

|B_{\rm in}/B_0| = 1/\sqrt{1+\omega^2\tau_s^2}

, find what fraction of the

50\,\mathrm{Hz}

field reaches the sensor through a copper enclosure, and state what happens to the steady Earth's field once enough time has passed. Decide whether copper alone is adequate. ((b)) Using the thin-wall static shielding factor

S_{\rm sh} \approx 1 + 2\mu_rd/3a

, find the static shielding factor of a mu-metal enclosure of the same dimensions and the residual amplitude of both the Earth's field and the

50\,\mathrm{Hz}

field inside it (treat the static value as a floor for the

50\,\mathrm{Hz}

shielding, because eddy currents in the mu-metal wall add their own attenuation, provided its permeability at

50\,\mathrm{Hz}

stays near the static value). ((c)) Before the sensor is fitted, a small magnetised calibration bead (magnetic moment

0.020\,\mathrm{A\,m^2}

, mass

0.50\,\mathrm{g}

) is accidentally dropped down a long vertical copper tube of the same mean radius and wall thickness, from a height of

30\,\mathrm{cm}

. Using the eddy-current drag law

F = \beta v

with

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

, estimate whether the bead's fall is significantly slowed, and hence estimate the speed with which it strikes the bottom.

Related theory

Magnetic materials

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