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.

Any two potentials that satisfy equations (5), (6) and (7), along with regularity at infinity, have identical gradients.

The demagnetizing field Hd is the gradient of this potential (equation 4).

The energy of the first magnet in the demagnetizing field Hd(2) of the second is The reciprocity theorem states that[9] 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.

[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.

However, very small ferromagnets are kept uniformly magnetized by the exchange interaction.

If the magnetization is along the longest axis, the poles are spread across a smaller surface, so the energy is lower.

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] 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.

Comparison of magnetic field (flux density) B , demagnetizing field H and magnetization M inside and outside a cylindrical bar magnet . The red (right) side is the North pole, the green (left) side is the South pole.
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.
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.
Plot of B field, i.e., μ 0 ( H + M ) , for a uniformly magnetized sphere in an externally applied zero magnetic field H 0 = 0 . For such a case, the internal B and H are uniform with values B = +2 μ 0 M /3 and H = − M /3 .