3. Magnetic screening: (total 30 points) Conductors can shield electric field. Can magnetic field be shielded? We study

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3 Magnetic Screening Total 30 Points Conductors Can Shield Electric Field Can Magnetic Field Be Shielded We Study 1 (118.12 KiB) Viewed 96 times

3. Magnetic screening: (total 30 points) Conductors can shield electric field. Can magnetic field be shielded? We study this by analyzing the following magnetostaties problem. As shown in Figure 3, a spherical shell has two concentric surfaces, an inner surface at radius r = a, and an outer surface at radius r = b. The shell is made of magnetically susceptible material, and its interior can be characterized by a linear constitutive equation B = u H, where u > O is the permeability constant. The shell is placed in a uniform magnetic field, Bea = -Bo ż, which is sourced by a far away device. The regions atr <a and r > b are free space, and there are no free currents anywhere. Since the whole setup is axi-symmetric about the z axis, it is convenient to adopt a spherical coordinate system that centers at the center of the spherical shell with the polar axis aligned with the axis. You may find the following mathematical results useful: the general axi-symmetric solution to the Laplace's equation in terms of the spherical coordinates in the separation-of-variable form is: В, = Pe(cose), (2) V (5,0) = (der+ + . -0 Here P(x) is the Legendre polynomial of order l; the lowest order polynomials are given by P.(x) = 1, P.(x) = x, and P2(x) = (3x2 - 1)/2. The orthonormal relations of the Legendre polynomials are La dx P(x) P(x) = 24 +7 der 1 2 (3) 3 uniform external magnetic field B--B, Figure 3: A spherical shell made of magnetically susceptible material, placed in a uniform external magnetic field. (a) (5 points) Consider the auxiliary field H(r). Explain why in this problem it is possible to introduce a single scalar potential function W (r) which satisfies H=-TW. (b) (5 points) Write down the equation W satisfies in each of the three regions: r <a,a<r<b, and r>b. (c) (10 points) At each of the two surfaces, r = a and r = b, write down a set of two appropriate boundary conditions, one for the value of Witself, and another for its normal derivative. (d) (5 points) Apply separation of variables in the spherical coordinate system to solving W. Since W is determined up to an additive constant, you are free to and should set W = 0 at the center of the spherical shell (r = 0) to simplify this calculation. For this part, you just need to derive a complete set of linear equations that are adequate for determining all the separation-of-variables coefficients - do NOT attempt to explicitly solve the linear equations. (Hint: Consider how the solution matches the uniform external field at large distances. You shall find that only the dipole terms (t = 1) are needed in the full expansion in terms of the Legendre polynomial. This will significantly simplify the form of the solution] (e) (5 points) Imagine that the shell material is strongly paramagnetic, i.e. Mr = k/Ho >> 1. Find explicit solutions to the separation-of-variables coefficients in Part(d) in this limit. Then calculate B in the shielded regionr <a. Show that in that region B is uniform, but with a strength that is suppressed by a factor 1/a, relative to that of the external field. [This shows that strongly paramagnetic material can be used to screen external magnetic field]