Basic Permanent Magnetism:

self-demagnetizing field.

by Dr. Peter Campbell

In Section 1C, we illustrated the relationship between B, H and M for a particular condition across a magnet/air boundary, for which all of these vectors are assumed to be normal to the boundary. If the principal of flux conservation discussed in Section 4A is applied to a narrow region enclosing the boundary, this dictates that B must be continuous across this boundary, which in turn requires that H be discontinuous.

We can now use this to demonstrate that the shape of a permanent magnet is critical to its ability to provide stable magnetic field.

Using the subscripts m and o to denote parameters which exist within and outside the magnet, this boundary condition can be expressed as:

An ellipsoidal-shaped magnetic material has the unique ability to sustain uniform field along either its major or minor axis, and of course a sphere is a particular case of ellipsoid. Now, the spherical magnet shown in this diagram is placed in a uniform applied field B_o, H_o, while the field within the magnet B_m, H_m is also uniform. Intuitively one knows that field is drawn into the magnetic material from the surrounding air, so that:

eliminating the inequality by introducing 0< N <1. Notice that N=1 is the particular condition which describes the planar surface of the first diagram in which B_m = B_o, while smaller values of N allow for more surface curvature. N is simply a number, which may be determined theoretically for any shape of magnetic material, although ellipsoids present a simpler case. For the purposes of this demonstration, suffice it to say that for extreme ellipsoids, N tends to unity value when viewed across the thickness (minor axis) of a very wide, thin plate magnet, while N tends to zero along the (major) axis of a very long needle-shaped magnet. N is therefore known as the demagnetizing factor, describing the shape of a magnet, while NM is its self-demagnetizing field.

The equation B_m = µ_o(H_m+M) represents the demagnetization curve of a permanent magnet, but another characteristic is required to determine at what point on the demagnetization curve the magnet is actually operating. This "load line" (which is properly defined in Section 4A) will introduce the physical dimensions of the magnetic circuit, which in this derivation are incorporated within the calculation of N. We already expressed H_m in terms of N in the previous equation, and as H_m was related to H_o, so too can B_m be:

Combining the last two expressions into a "load line", and setting H_o=0 to represent the magnet alone providing field:

Now let's consider two field orientations through an "extreme" ellipsoid:

1: N = 1

{short description of image} We stated above that N could be calculated for an ellipsoid, and it tends to unity value when viewed across the thickness (minor axis) of a very wide, thin plate magnet (as approximated by the elongated ellipsoid shown in this diagram).

The self-demagnetizing field NM is very strong in this case, as indicated by the close proximity of the free North and South poles on the opposite surfaces of the magnet. As shown by the last equation, the "load line" tends to a slope of - zero for this field orientation.

2: N = 0

{short description of image} A calculation of N for an ellipsoid shows that its value tends to zero along the length (major axis) of a very long needle-shaped magnet (as approximated by the elongated ellipsoid shown in this diagram).

The self-demagnetizing field NM is very weak in this case, as indicated by the remote proximity of the free North and South poles on the opposite surfaces of the magnet. As shown by the last equation, the "load line" tends to a slope of - infinity for this field orientation.

This field orientation through this shape of magnet clearly provides a more stable magnetic field.

The intercept between the demagnetization curve and the load line gives a unique operating point with a specific B_m and H_m for a magnet of a particular material and of a particular shape. The change in operating point due to field orientation relative to the magnet's shape is illustrated as the load line moves around the B-H diagram below.

To increase the field of a permanent magnet (and improve the magnetic stability), magnet length should be increased relative to its cross-sectional area.