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.^[2] These are Ampère's law

${\displaystyle \nabla \times \mathbf {H} =0,}$[3]

(1)

and Gauss's law

${\displaystyle \nabla \cdot \mathbf {B} =0.}$[4]

(2)

The magnetic field and flux density are related by^[5][6]

${\displaystyle \mathbf {B} =\mu _{0}\left(\mathbf {M} +\mathbf {H} \right),}$[7]

(3)

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):

${\displaystyle \mathbf {H} =-\nabla U.}$[5][6]

(4)

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

${\displaystyle \nabla ^{2}U_{\text{in}}=\nabla \cdot \mathbf {M} .}$[8]

(5)

Outside the body, where the magnetization is zero,

${\displaystyle \nabla ^{2}U_{\text{out}}=0.}$

(6)

At the surface of the magnet, there are two continuity requirements:^[5]

• 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:

${\displaystyle {\begin{aligned}U_{\text{in}}&=U_{\text{out}}\\{\frac {\partial U_{\text{in}}}{\partial n}}&={\frac {\partial U_{\text{out}}}{\partial n}}+\mathbf {M} \cdot \mathbf {n} .\end{aligned}}}$

(7)

Here n is the surface normal and ${\textstyle \partial /\partial n}$ is the derivative with respect to distance from the surface.^[9]

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.^[10] 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:

${\displaystyle E=-{\frac {\mu _{0}}{2}}\int _{\text{magnet}}\mathbf {M} \cdot \mathbf {H} _{\text{d}}dV}$

(7)

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

${\displaystyle E=\mu _{0}\int _{\text{magnet 1}}\mathbf {M} _{1}\cdot \mathbf {H} _{\text{d}}^{(2)}dV.}$

(8)

The reciprocity theorem states that^[9]

${\displaystyle \int _{\text{magnet 1}}\mathbf {M} _{1}\cdot \mathbf {H} _{\text{d}}^{(2)}dV=\int _{\text{magnet 2}}\mathbf {M} _{2}\cdot \mathbf {H} _{\text{d}}^{(1)}dV.}$

(9)

Magnetic charge and the pole-avoidance principle

Formally, the solution of the equations for the potential is

${\displaystyle U(\mathbf {r} )=-{\frac {1}{4\pi }}\int _{\text{volume}}{\frac {\nabla '\cdot \mathbf {M\left(r'\right)} }{|\mathbf {r} -\mathbf {r} '|}}dV'+{\frac {1}{4\pi }}\int _{\text{surface}}{\frac {\mathbf {n} \cdot \mathbf {M\left(r'\right)} }{|\mathbf {r} -\mathbf {r} '|}}dS',}$

(10)

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.^[9]

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 tries to reduce the poles as much as possible.^[9]

Single domain

Illustration of the magnetic charges at the surface of a single-domain ferromagnet. The arrows indicate the direction of magnetization. The thickness of the colored region indicates the surface charge density.

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

Illustration of a magnet with four magnetic closure domains. The magnetic charges contributed by each domain are pictured at one domain wall. The charges balance, so the total charge is zero.

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 n · M are the same but opposite in sign). Domains configured this way are called closure domains.