The demagnetizing field, also called the stray field (outside the magnet), is the magnetic field (H-field) generated by the magnetization in a

magnet A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nicke ...

. 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 any free currents or

displacement current In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...

s. The term ''demagnetizing field'' reflects its tendency to act on the magnetization so as to reduce the total

magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...

. It gives rise to ''shape anisotropy'' in

ferromagnets Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...

with a single magnetic domain and to

magnetic domains A magnetic domain is a region within a magnetic material in which the magnetization is in a uniform direction. This means that the individual magnetic moments of the atoms are aligned with one another and they point in the same direction. When ...

in larger ferromagnets. The demagnetizing field of an arbitrarily shaped object requires a numerical solution of

Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...

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 Magnetic hysteresis occurs when an external magnetic field is applied to a ferromagnet such as iron and the atomic dipoles align themselves with it. Even when the field is removed, part of the alignment will be retained: the material has become '' ...

Magnetostatic principles

Maxwell's equations

In general the demagnetizing field is a function of position . It is derived from the magnetostatic equations for a body with no

electric currents An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...

. These are Ampère's law
and

Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...

The magnetic field and flux density are related by
where

\mu_0

is the

permeability of vacuum The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...

and is the magnetisation.

The magnetic potential

The general solution of the first equation can be expressed as the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

of a

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...

potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...

: Inside the magnetic body, the potential is determined by substituting () and () in (): Outside the body, where the magnetization is zero, At the surface of the magnet, there are two continuity requirements: *The component of

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

to the surface must be

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

(no jump in value at the surface). *The component of

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...

to the surface must be continuous. This leads to the following

boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...

s at the surface of the magnet: Here is the surface normal and

\partial/ \partial n

is the derivative with respect to distance from the surface. The outer potential must also be ''regular at infinity'': both and must be bounded as 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 In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...

with the same moment as the finite body.

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

File:SingleDomainMagneticCharges.svg, 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. defaul
direct SVG link
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 In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classic ...

. 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 In condensed matter physics, magnetic anisotropy describes how an object's magnetic properties can be different depending on direction. In the simplest case, there is no preferential direction for an object's magnetic moment. It will respond ...

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 A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls are also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model or models with pol ...

) 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 configured this way are called ''closure domains''.

Demagnetizing factor

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 the internal magnetic field is uniform: where is the magnetization of the sphere and is called the demagnetizing factor and equals for a sphere. 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: Other important examples are an infinite plate (an ellipsoid with two of its axes going to infinity) which has 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 along its axis and perpendicular to its axis. 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.