Sphere in a Uniform Magnetic Field
A sphere of linear, uniform, homogeneous magnetic material is placed in an otherwise uniform magnetic field B0. What is the resulting magnetic field inside the sphere?
QUESTIONS
(Score is number right minus number wrong.)
When the sphere is placed in the magnetic field and comes to rest, a free current that continues to flow will have been induced on the surface of the sphere.
True
False
If there is no free current anywhere, Maxwell says that the curl of H must vanish everywhere.
True
False
Inside linear materials, the divergence of H vanishes as a consequence of Equation (3) and Equation (6).
True
False
If the curl and divergence of H vanish everywhere near the sphere, the perturbation to the otherwise uniform auxiliary field H caused by the sphere probably vanishes everywhere.
True
False
In this problem, the divergence of H cannot vanish identically
at large z (infinity).
everywhere at the surface of the sphere.
everywhere in the space surrounding the sphere.
inside the sphere.
The term in the divergence of M that cannot vanish at least somewhere in this problem is
Equation (8a).
Equation (8b).
Equation (8c).
If the curl of H vanishes everywhere near the sphere, but the divergence not, the Helmholtz Theorem suggests that lines of the H field will have sources and sinks that will perturb the H field near the sphere so that we expect,
Equation (9a).
Equation (9b).
Equation (9c).
Equation (9d).
Where the divergence of the magnetization vanishes, the divergence of H vanishes. That means that the scalar potential from which H is derived will satisfy Laplace's equation there.
To solve Laplace's equation (Equation (7)), which of the following can be used to generate a boundary condition at the surface of the sphere that can be applied given what is known about the problem?
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
In addition to the continuity of the scalar potential, an applicable boundary condition at the surface of the sphere is
Equation (11a).
Equation (11b).
Equation (11c).
Equation (11d).
At large distance (z), the correct asymptotic boundary condition is
Equation (12a).
Equation (12b).
Equation (12c).
Equation (12d).
Because Legendre polynomials form a 'complete set', the only term in the sums of Equation (7) for the INSIDE and OUTSIDE solutions that can fit the asymptotic boundary condition (large z) is
Equation (13a).
Equation (13b).
Equation (13c).
Equation (13d).
The magnetic field B inside the sphere is given by
Equation (14a).
Equation (14b).
Equation (14c).
Equation (14d).
EQUATIONS
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