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Magnetic Shielding Effect of a Spherical Shell

​
a / b
0.5
μ
r
10
display field
E
B
number of field lines
N
f
10
20
40
a / b = 0.5
μ
r
= 10
Consider a spherical shell of linear magnetic material with relative permeability
μ
r
placed in a uniform magnetic field
H
0
. The magnetic fields in this region can be described by a magnetic potential
ψ
. Selecting the direction of
H
0
as the axis of spherical coordinates
(r,θ,ϕ)
,
ψ
is given by
-
H
0
rcosθ+
∑
n
A
n
n
r
+
B
n
-n-1
r
P
n
(rcosθ)
, where
P
n
(x)
is a Legendre function. The magnetic field at any point is
H=-gradψ
. The coefficients
A
n
,
B
n
in the regions with (1)
r>b
, (2)
a<r<b
, and (3)
r<a
are determined by considering the boundary conditions at
r=a
and
r=b
, taking into account the permeability in each region: (1) and (3)
μ
0
and (2)
μ
r
μ
0
. As the result, the field in (1) is that of
H
0
superimposed with the contribution of a magnetic dipole. The field in (3) turns out to be uniform, with magnitude considerably lower than
H
0
. The magnetic induction or B field is obtained by
B=μH
, where
μ=
μ
0
or
μ=
μ
r
μ
0
depending on the region.
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