Uniqueness of the demagnetizing field Any two potentials that satisfy equations (), () and (), along with regularity at infinity, have identical gradients. The demagnetizing field is the gradient of this potential (equation ). Energy The energy of the demagnetizing field is completely determined by an integral over the volume of the magnet: Suppose there are two magnets with magnetizations and . The energy of the first magnet in the demagnetizing field of the second is The reciprocity theorem states that Magnetic charge and the pole-avoidance principle Formally, the solution of the equations for the potential is where is the variable to be integrated over the volume of the body in the first integral and the surface in the second, and is the gradient with respect to this variable. Qualitatively, the negative of the divergence of the magnetization (called a volume pole) is analogous to a bulk bound electric charge in the body while (called a surface pole) is analogous to a bound surface electric charge. Although the magnetic charges do not exist, it can be useful to think of them in this way. In particular, the arrangement of magnetization that reduces the magnetic energy can often be understood in terms of the pole-avoidance principle, which states that the magnetization tries to reduce the poles as much as possible. Effect on magnetization Single domain One way to remove the magnetic poles inside a ferromagnet is to make the magnetization uniform. This occurs in single-domain ferromagnets. This still leaves the surface poles, so division into domains reduces the poles further. However, very small ferromagnets are kept uniformly magnetized by the exchange interaction. The concentration of poles depends on the direction of magnetization (see the figure). If the magnetization is along the longest axis, the poles are spread across a smaller surface, so the energy is lower. This is a form of magnetic anisotropy called shape anisotropy. Multiple domains If the ferromagnet is large enough, its magnetization can divide into domains. It is then possible to have the magnetization parallel to the surface. Within each domain the magnetization is uniform, so there are no volume poles, but there are surface poles at the interfaces (domain walls) between domains. However, these poles vanish if the magnetic moments on each side of the domain wall meet the wall at the same angle (so that the components are the same but opposite in sign). Domains