A sensitive magnetometer sits at the centre of a thin spherical mu-metal shield of…
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 sensitive magnetometer sits at the centre of a thin spherical mu-metal shield of radius a=7.0cm and wall thickness d=0.90mm (μr=5.0×104, conductivity σ=1.8×106Sm−1), standing near a mains transformer whose core does not confine all of its flux. At the shield's location the transformer superposes a 50Hz field of amplitude B0=8.0μT on the steady 50μT geomagnetic field. Treat the shield as thin (d≪a) and the mu-metal as linear (unsaturated) throughout.
((a)) Using the thin-wall static shielding factor Ssh≈1+2μrd/3a, find the part of the steady geomagnetic field that reaches the magnetometer.
((b)) Find the skin depth δ of the mu-metal at 50Hz (take μ=μ0μr in the skin-depth formula). Show that, for the wall treated as a slab of thickness d, ωτD=2(d/δ)2, and evaluate d/δ. State, with a reason, whether the static factor of part (a), applied to the 50Hz amplitude B0, over-estimates or under-estimates the field the magnetometer actually sees at 50Hz.
((c)) To see how quickly a conducting shield settles once the transformer is switched on, model a non-magnetic copper test shell of the same radius and wall thickness (σCu=5.96×107Sm−1), whose interior field obeys Bin=B0(t)−τsdBin/dt with τs=μ0σCuad/3. A steady field of amplitude B0 is switched on at t=0 with Bin(0)=0. Find Bin(t) and the time after which the transient part of the response has fallen below 1 per cent of B0.