Vol 6 - Landau+Lifschitz - Fluid Mechanics · Problem 52.4

The key is recognizing that a uniformly magnetized sphere creates a demagnetizing field that opposes the magnetization everywhere inside. Unlike an infinite medium, the sphere's finite boundary means field lines escape, producing a uniform internal field proportional to $-\mathbf{M}$. Use the demagnetizing factor (which equals $1/3$ for a sphere) to find how the external field is modified, then apply the relation between $\mathbf{B}$, $\mathbf{H}$, and $\mathbf{M}$ to determine the actual magnetization.

The key is recognizing that a uniformly magnetized sphere creates a demagnetizing field that opposes the magnetization everywhere inside. Unlike an infinite medium, the sphere's finite boundary means field lines escape, producing a uniform internal field proportional to $-\mathbf{M}$. Use the demagnetizing factor (which equals $1/3$ for a sphere) to find how the external field is modified, then apply the relation between $\mathbf{B}$, $\mathbf{H}$, and $\mathbf{M}$ to determine the actual magnetization.