Demagnetizing field

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 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.

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).

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

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 affects poles by limiting the poles (tries to reduce them as much as possible).^[9]

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^{[clarification needed]}. 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.

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.

An arbitrarily shaped magnetic object has a total magnetic field that varies with location inside the object and can be quite difficult to calculate. This makes it very difficult to determine the magnetic properties of a material such as, for instance, how the magnetization of a material varies with the magnetic field. For a uniformly magnetized sphere in a uniform magnetic field H₀ the internal magnetic field H is uniform:

${\displaystyle \mathbf {H} =\mathbf {H} _{0}-\gamma \mathbf {M} _{0},}$

11

where M₀ is the magnetization of the sphere and γ is called the demagnetizing factor, which assumes values between 0 and 1, and equals 1/3 for a sphere in SI units.^[5][6][11] Note that in cgs units γ assumes values between 0 and 4π.

This equation can be generalized to include ellipsoids having principal axes in x, y, and z directions such that each component has a relationship of the form:^[6]

${\displaystyle H_{k}=(H_{0})_{k}-\gamma _{k}(M_{0})_{k},\qquad k=x,y,z.}$

12

Other important examples are an infinite plate (an ellipsoid with two of its axes going to infinity) which has γ = 1 (SI units) in a direction normal to the plate and zero otherwise and an infinite cylinder (an ellipsoid with one of its axes tending toward infinity with the other two being the same) which has γ = 0 along its axis and 1/2 perpendicular to its axis.^[12] The demagnetizing factors are the principal values of the depolarization tensor, which gives both the internal and external values of the fields induced in ellipsoidal bodies by applied electric or magnetic fields.^{[13] [14] [15]}