# 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 (x) 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

-rcosθ++(rcosθ)

H

0

∑

n

A

n

n

r

B

n

-n-1

r

P

n

P

n

H=-gradψ

A

n

B

n

r>b

a<r<b

r<a

r=a

r=b

μ

0

μ

r

μ

0

H

0

H

0

B=μH

μ=

μ

0

μ=

μ

r

μ

0