Basic Permanent Magnetism:

From basic electromagnetics, we know the following fact: if you imagine a small rectangular loop of wire (of area A) carrying an electrical current i which is placed in a uniform magnetic field B, then a torque T will be developed which tends to rotate the plane of the current loop until it is perpendicular to the direction of B. This torque is calculated from:

The most fundamental microscopic property of any magnetic material is its magnetic dipole moment µ_m. Since this comprises electron spins, it may be likened to a microscopic current loop and defined via:

The torque that tends to rotate a magnetic dipole moment into alignment with an applied magnetic field is:

Selection of a permanent magnet material frequently involves consideration of the energy they store. On a microscopic level, the work done E in rotating a magnetic dipole moment into alignment with an applied magnetic field is:

Throughout a magnetic material, there is a large distribution of magnetic dipole moments. When these are all aligned with the applied field, the material is said to have reached saturation. In a small enough "elemental" volume, adjacent moments will be identical, and it is possible to define a macroscopic property of the magnetic material, its magnetization M, as the dipole moment per unit (elemental) volume:

In an elemental volume in which there is only a z-directed magnetization M_z, the sum of the moments is:

Also, from our definition of magnetic dipole moment,

which relates the magnetization M_z in the element to an equivalent current i flowing around its perimeter:

In the adjacent elemental volume in which there is also only a z-directed magnetization M'_z with its equivalent current i', the change in current is found to be:

It is as if there is a "wall" from M_z to M'_z which carries an equivalent current i - i', in other words a current density J_y:

So far, we have shown that a material's magnetization is equivalent to an electrical current density.

It is also apparent from this derivation that the equivalent current only exists across a boundary where there is a changing tangential component of magnetization. If a permanent magnet has uniform uni-axial magnetization throughout its volume as shown in (a), then the equivalent current density only exists at the side boundaries as shown in (b).

In general, of course, the magnetization in an elemental volume may have a general direction M (rather than just M_z as assumed above), in which case there will be a contribution to J_y from a change in M_x as well as M_z. Furthermore, the current density will also have a general direction J_m, with contributions from J_x and J_z in addition to J_y. We use the suffix m here to indicate that this current density is equivalent to magnetization, as opposed to the density of a real electrical current. As was suggested in the last diagram, it is the change, or rotation, of the material's magnetization that gives rise to an equivalent current density, which is expressed in vector shorthand as:

This really isn't a surprising finding, because it is just like the familiar magnetic occurrence of a magnetic field B rotating around a conductor carrying a real electrical current density J. Known as Ampère's Law, this rotation is expressed in vector shorthand as

While this version of Ampère's Law refers to real electrical current density J, it may equally well refer to a magnet's equivalent current density J_m, or to a system in which there exists both electrical currents and magnetic materials, in which case:

Substituting for J_m and rearranging to separate the "electrical" the "magnetic" terms:

Finally in this Section, we need to complete our definitions of the fundamental magnetic parameters we shall need for permanent magnet design. We have derived magnetization M, and we understand the concept of electrical current density J, but as yet we have been a little vague about the magnetic "field" B. As illustrated in the last diagram, for "field" circulating around a current-carrying conductor, B is just a convenient concept that was devised to quantify the observed electromagnetic effects that occur around electrical current flow. B is therefore defined as flux density, the density of magnetic flux which flows around a magnetic circuit, and is governed by the foregoing equations.

The last equation quantifies the flux density that is caused by real electrical currents and material magnetization. In a permanent magnet, we are interested in the flux density B that can be produced by the magnet's magnetization M, that is, we are interested in the term (B-µ_oM). You might say, the more B a magnet's M can produce the better. This notion leads to the definition of a magnetizing force H, which is proportional to (B-µ_oM):

Also, the more general version of Ampère's Law may now be written to include material magnetization M with real current density J:

This still says that magnetization does not have to exist (M=0) for there to be magnetic field; electrical current alone can obviously provide a magnetizing force H, which then causes a magnetic flux to flow, of flux density B=µ_oH. However, neither does current have to exist (J=0) for there to be magnetic field; a material's permanent magnetization M may also provide H and hence a flux density B=µ_o(H+M).

In the next section, we shall explain more about the physical relationship between these three basic parameters, M, H and B, of permanent magnetism. And, we shall introduce the energy of a permanent magnet material.

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