Magnetic Shielding Effect of a Spherical Shell
Magnetic Shielding Effect of a Spherical Shell
Consider a spherical shell of linear magnetic material with relative permeability placed in a uniform magnetic field . The magnetic fields in this region can be described by a magnetic potential . Selecting the direction of as the axis of spherical coordinates , is given by , where is a Legendre function. The magnetic field at any point is . The coefficients , in the regions with (1) , (2) , and (3) are determined by considering the boundary conditions at and , taking into account the permeability in each region: (1) and (3) and (2) . As the result, the field in (1) is that of superimposed with the contribution of a magnetic dipole. The field in (3) turns out to be uniform, with magnitude considerably lower than . The magnetic induction or B field is obtained by , where or depending on the region.
μ
r
H
0
ψ
H
0
(r,θ,ϕ)
ψ
-Hrcosθ+Ar+BrP(rcosθ)
0
∑
n
n
n
n
-n-1
n
P(x)
n
H=-gradψ
A
n
B
n
r>b
a<r<b
r<a
r=a
r=b
μ
0
μμ
r
0
H
0
H
0
B=μH
μ=μ
0
μ=μμ
r
0