Demagnetizing field

The demagnetizing field of an arbitrarily shaped object requires a numerical solution of Poisson's equation even for the simple case of uniform magnetization. For the special case of ellipsoids (including infinite cylinders) the demagnetization field is linearly related to the magnetization by a geometry dependent constant called the demagnetizing factor. Since the magnetization of a sample at a given location depends on the total magnetic field at that point, the demagnetization factor must be used in order to accurately determine how a magnetic material responds to a magnetic field. (See magnetic hysteresis.)

Magnetostatic principles

Maxwell's equations

In general the demagnetizing field is a function of position H(r). It is derived from the magnetostatic equations for a body with no electric currents. These are Ampère's law

and Gauss's law

The magnetic field and flux density are related by

where ${\displaystyle {\mu }_0}$ is the permeability of vacuum and M is the magnetisation.

The magnetic potential

The general solution of the first equation can be expressed as the gradient of a scalar potential U(r):

Inside the magnetic body, the potential U_in is determined by substituting (3) and (4) in (2):

Outside the body, where the magnetization is zero,

At the surface of the magnet, there are two continuity requirements:

• The component of H parallel to the surface must be continuous (no jump in value at the surface).
• The component of B perpendicular to the surface must be continuous.

This leads to the following boundary conditions at the surface of the magnet:

Here n is the surface normal and $\mathrm{\partial }/\mathrm{\partial }n$ is the derivative with respect to distance from the surface.

The outer potential U_out must also be regular at infinity: both |r U| and |r² U| must be bounded as r goes to infinity. This ensures that the magnetic energy is finite. Sufficiently far away, the magnetic field looks like the field of a magnetic dipole with the same moment as the finite body.

Uniqueness of the demagnetizing field

Any two potentials that satisfy equations (5), (6) and (7), along with regularity at infinity, have identical gradients. The demagnetizing field H_d is the gradient of this potential (equation 4).

Energy

The energy of the demagnetizing field is completely determined by an integral over the volume V of the magnet:

Suppose there are two magnets with magnetizations M₁ and M₂. The energy of the first magnet in the demagnetizing field H_d⁽²⁾ 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 r′ 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 − ∇ · M (called a volume pole) is analogous to a bulk bound electric charge in the body while n · M (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 affects poles by limiting the poles (tries to reduce them as much as possible).