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Deriving a Magnetic Field in a Sphere Using Laplace's Equation

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This solution derives the expression for the magnetic field inside the spherical shell with magnetic permeability mu and with an inner and outer radius of a and b, respectively. This object is placed in a uniform and constant magnetic field Ha. As this is a steady-state problem, no currents exist and the magnetic field H can be derived using the Laplace's equation as a starting point. For this types of problems, the solution can generally be expressed in terms of Legendre polynomials.

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Solution Summary

The magnetic field inside a spherical shell with magnetic permeability mu and with an inner and outer radius of a and b, respectively, is derived. This field outside the spherical shell is uniform and has a constant magnetic field of Ha. As no currents exist, the magnetic field H can be derived using the Laplace's equation and the solution is expressed in terms of Legendre polynomials.

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