Articles about
Electromagnetism
Electricity Magnetism Optics History Computational Textbooks Phenomena
Electrostatics Charge density Conductor Coulomb law Electret Electric charge Electric dipole Electric field Electric flux Electric potential Electrostatic discharge Electrostatic induction Gauss's law Insulator Permittivity Polarization Potential energy Static electricity Triboelectricity
Magnetostatics Ampère's law Biot–Savart law Gauss's law for magnetism Magnetic dipole Magnetic field Magnetic flux Magnetic scalar potential Magnetic vector potential Magnetization Permeability Right-hand rule
Electrodynamics Bremsstrahlung Cyclotron radiation Displacement current Eddy current Electromagnetic field Electromagnetic induction Electromagnetic pulse Electromagnetic radiation Faraday's law Jefimenko equations Larmor formula Lenz's law Liénard–Wiechert potential London equations Lorentz force Maxwell's equations Maxwell tensor Poynting vector Synchrotron radiation
Electrical network Alternating current Capacitance Current density Direct current Electric current Electric power Electrolysis Electromotive force Impedance Inductance Joule heating Kirchhoff's laws Network analysis Ohm's law Parallel circuit Resistance Resonant cavities Series circuit Voltage Watt Waveguides
Magnetic circuit AC motor DC motor Electric machine Electric motor Gyrator–capacitor Induction motor Linear motor Magnetomotive force Permeance Reluctance (complex) Reluctance (real) Rotor Stator Transformer
Covariant formulation Electromagnetic tensor Electromagnetism and special relativity Four-current Four-potential Mathematical descriptions Maxwell equations in curved spacetime Relativistic electromagnetism Stress–energy tensor
Scientists Ampère Biot Coulomb Davy Einstein Faraday Fizeau Gauss Heaviside Helmholtz Henry Hertz Hopkinson Jefimenko Joule Kelvin Kirchhoff Larmor Lenz Liénard Lorentz Maxwell Neumann Ohm Ørsted Poisson Poynting Ritchie Savart Singer Steinmetz Tesla Thomson Volta Weber Wiechert
v t e

Thedemagnetizing field, also called thestray field (outside the magnet), is themagnetic field (H-field)^[1] generated by themagnetization in amagnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to anyfree currents ordisplacement currents. The termdemagnetizing field reflects its tendency to act on the magnetization so as to reduce the totalmagnetic moment. It gives rise toshape anisotropy inferromagnets with asingle magnetic domain and tomagnetic domains in larger ferromagnets.

The demagnetizing field of an arbitrarily shaped object requires a numerical solution ofPoisson's equation even for the simple case of uniform magnetization. For the special case ofellipsoids (including infinite cylinders) the demagnetization field is linearly related to the magnetization by a geometry dependent constant called thedemagnetizing factor. Since the magnetization of a sample at a given location depends on thetotal 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. (Seemagnetic hysteresis.)

In general the demagnetizing field is a function of positionH(r). It is derived from themagnetostatic equations for a body with noelectric currents.^[2] These areAmpère's law

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

1

andGauss's law

${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0.}$[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 thepermeability of vacuum andM is themagnetisation.

The general solution of the first equation can be expressed as thegradient of ascalarpotentialU(r):

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

4

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

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

5

Outside the body, where the magnetization is zero,

${\displaystyle {\mathrm{\nabla }}^2{U}_{\text{out}}=0.}$

6

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

• The component ofHparallel to the surface must becontinuous (no jump in value at the surface).
• The component ofBperpendicular to the surface must be continuous.

This leads to the followingboundary conditions at the surface of the magnet:

7

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

The outer potentialU_out must also beregular at infinity: both |r U| and |r²U| must be bounded asr goes to infinity. This ensures that the magnetic energy is finite.^[10] Sufficiently far away, the magnetic field looks like the field of amagnetic dipole with the samemoment as the finite body.

Any two potentials that satisfy equations (5), (6) and (7), along with regularity at infinity, have identical gradients. The demagnetizing fieldH_d is the gradient of this potential (equation4).

The energy of the demagnetizing field is completely determined by an integral over the volumeV 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 magnetizationsM₁ andM₂. The energy of the first magnet in the demagnetizing fieldH_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

Thereciprocity 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

${\displaystyle U(\mathbf{r})=-\frac{1}{4\pi }{\int }_{\text{volume}}\frac{{\mathrm{\nabla }}^{\prime }\cdot \mathbf{M}\left({\mathbf{r}}^{\prime }\right)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{V}^{\prime }+\frac{1}{4\pi }{\int }_{\text{surface}}\frac{\mathbf{n}\cdot \mathbf{M}\left({\mathbf{r}}^{\prime }\right)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{S}^{\prime },}$

10

wherer′ 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 avolume pole) is analogous to a bulkbound electric charge in the body whilen ·M (called asurface 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 thepole-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 insingle-domain ferromagnets. This still leaves the surface poles, so division intodomains reduces the poles further^{[clarification needed]}. However, very small ferromagnets are kept uniformly magnetized by theexchange 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 ofmagnetic anisotropy calledshape anisotropy.

If the ferromagnet is large enough, its magnetization can divide intodomains. 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 componentsn ·M are the same but opposite in sign). Domains configured this way are calledclosure 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 fieldH₀ the internal magnetic fieldH is uniform:

${\displaystyle \mathbf{H}={\mathbf{H}}_0-\gamma {\mathbf{M}}_0,}$

11

whereM₀ is the magnetization of the sphere andγ is called the demagnetizing factor, which assumes values between 0 and 1, and equals1/3 for a sphere in SI units.^[5][6][11] Note that in cgs unitsγ assumes values between 0 and4π.

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

${\displaystyle H_k=(H_0{)}_k-{\gamma }_k(M_0{)}_k,k=x,y,z.}$

12

Other important examples are an infinite plate (an ellipsoid with two of its axes going to infinity) which hasγ = 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 hasγ = 0 along its axis and1/2 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]

• Electric and magnetic fields in matter
• Magnetostatics
• Potentials

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles needing clarification from April 2023