Magnetic Shielding

A spherical shell of high permeability μ is placed in a uniform magnetic field.

a) Compute the attenuation (shielding) produced by the sphere in terms of μ and the inner and outer radii a and b, respectively, of the shell.

b) Take the limit at μ >> 1 and estimate the shielding for μ = 10⁵, a = 0.5 m, b = 0.55 m.

SOLUTION

a) By analogy with electrostatics, we assume that the shell can be described by a magnetic dipole placed in the center of the shell for r > a, and try to satisfy boundary conditions for H and B. We can write

(1)

(2)

(3)

where (1), (2), and (3) are written for areas 1, 2, and 3 outside the shell, at b > r > a, and inside the shell, respectively (see Figure 1.1); α, β, γ, and δ are numerical factors that we shall find from

Figure 1.1

the constitutive relation B = μH; and n is a unit vector parallel to r. From (1)–(3), we can impose conditions for the normal components of B and the tangential components of H, taken at the same angle θ

(4)

(5)

(6)

(7)

where (4) and (5) apply to interface 1–2, while (6) and (7) apply to interface 2–3. Dividing out the H₀ cos θ and H₀ sin θ, appropriately, we obtain

(8)

(9)

(10)

(11)

This system of four equations for the four numerical coefficients may be easily solved. Using (10) and (11), we find

and from (8) and (9), we obtain

Now we can calculate δ, which is the attenuation factor we seek. Isolating δ from (10) and (11) and substituting β, we have

b) In the limit of high permeability μ >> 1, we arrive at

For a = 0.5 m, b = 0.55 m, and μ = 10⁵