I’m trying to explain to a friend of mine the concept of electric displacement vector fields (D-fields) and magnetic intensity fields (H-fields). I’m so used to using them that I haven’t really thought of the conceptual definition for so long, but I guess it’s quite tricky to grasp because unlike electric, magnetic, polarization and magnetization fields, D and H do not have immediately clear physical meanings. We only introduce them to take bound charge and current respectively into account.

That’s why I’ve been finding it difficult to articulate in words precisely what D and H are. See what you think:

D, or the electric displacement field takes into account the fact that within a dielectric media, an electric field will induce a polarization density (P). The induced electric dipoles will tend to create a field which opposes the field that created them, thus reducing the density of electrostatic lines of flux that permeate the region of space in question. The source of these opposing fields are referred to as bound charges. The degree to which a dielectric media will reduce electrostatic flux lines is measured by its permittivity. Usually, however, instead of referring to absolute permittivity of a dielectric media, we refer to the relative permittivity. That is, the permittivity of the media as a ratio to that of a vacuum. Now consider an isotropic dielectric media as the simplest case. In such a case, there will be uniform density, hence the divergence of polarization will be zero. In this case, there is no net bound charge density and so the electric field is uniformly reduced by a factor of epsilon throughout the field due to the polarization. In this case, it is helpful to think of the electric displacement field as the “electric intensity”. In other words, it is directly related to the density of electrostatic flux lines that would be produced if the dielectric media was gone. The direct relation is a universal constant, the vacuum permittivity. Because of this, D and E do not have the same units.

You can use a variation of this line of thought in anisotropic media as well. If we have a field of non-uniform density, there will be a net divergence and hence a net bound charge. Because of this, the electric field is reduced by different factors in different regions. Since the divergence of P is a function of which is independent of the electric field, the polarization field is not a scalar multiple of E (in an isotropic media, it is, and the multiple is called electric susceptibility). Because of this, the only thing that makes physical sense is for epsilon to be a second rank tensor, which it is. However, in these cases, even though epsilon is a function of position, the same ratio still holds. Examining an infinitesimal volume with dielectric media (which, in the limit of volume, we assume is isotropic over that volume) we find that the field is still reduced by the same factor (epsilon). Everywhere in the field, D characterizes the intensity of the field by effectively removing the dissipation of flux lines by the media as a factor. This is useful because it means we can separate the effects of bound and free charges in producing electric fields in dielectric media. This only holds up to a point because as the electric field through a dielectric approaches the break-down point, the polarization no longer varies linearly with the electric field, and once it reaches break-down, the whole concept of a dielectric becomes meaningless anyway, but supposing that E is much less than the breakdown strength, you should be fine.

In magnetic fields, the same principle holds for the H field, but unfortunately, it can get much more complicated. In a manner analogous to bound and free charges producing fields, in magnetic materials, we can consider the loops of current due to electrons orbiting the nucleus as creating small permanent dipoles. It is helpful for you to think of the magnetization density field as a vector field whose vectors point in the direction of the generated south pole at that point from an orthogonal loop of current. It is a relatively simple exercise to demonstrate that the magnetization current (the bound current resulting from orbital motion) is the curl of the magnetization field. In other words, there will be no bound current in an isotropic magnetic media. This is exactly analogous to electric polarization density, which only generates bound charge in anisotropic media.

Please note that in most materials, there is no magnetism associated with the electron spin because of Pauli exclusion. With filled subshells, such materials have exactly zero net magnetic moment. We will consider paramagnetism and diamagnetism first, before ferromagnetism (which is more complicated). In general, the existence of magnetic domains in magnetizable materials will have the same effect as dielectric media responding to electric fields. Depending on electron arrangement, the media can either reinforce (paramagnetize) or reduce (diamagnetize) the applied field. Please note that because of Lenz’s Law, all materials will diamagnetize to some extent, including those that non-physicists think of as non-magnetic. Materials with unfilled shells will paramagnetize due to net moment. They will also diamagnetize, but paramagnetization is nearly always much stronger than diamagnetization, and overpowers it. In these materials, there is a relatively simple relationship between the magnetic field strength we permeate a material with and the degree to which they respond. A relatively simple way to understand this is to consider the magnetization density field (M) as denoting the relative dipole strength in a region. The net dipoles will align with the magnetic field and strengthen it by increasing the total flux density through the material.

For some materials, the response will be to reduce the field, and for others it will be to strengthen it, but in either case there is a constant linking the degree to which a field will be changed, and that is the relative permeability. For isotropic paramagnetic media, the number of dipoles aligned with the material is directly proportional to the flux density of the field that goes through it. Obviously, for a diamagnetizing response, this will be <1 (the material will retard the permeation of B), and for a paramagnetizing response, it will be >1 (the material will increase the permeation of B). In any case, just as with the D-field, we can in these cases view H as a “magnetic intensity” field. It is directly related to the magnetic field strength that would exist if the media was not there (the direct relation is the vacuum permeability, so H does not have teh same units as D). In these cases, it is helpful to think of B as the flux density through a region and therefore as a function of H. This is useful for engineering problems but is not advisable when studying electromagnetism in physics because in actuality, B is a more fundamental quantity than H. Of course, if the media is anistropic, than the permeability constant will be replaced by a rank 2 tensor, but the same relationship still holds. Remember that because of Stokes’ Theorem, the total bound current due to a uniform magnetization field will be equal to the line integral around the surface of the material.

 It might be visually helpful to consider the anisotropic media is being divided into infinitesimal blocks of isotropic media to represent a variable density field hence a variable magnetization field. In these cases, we can consider the magnetic intensity (H) as an independent quantity and the flux density (B) as a function of the intensity and depending on the media it permeates. But if we examine each individual little isotropic volume, we still find that B is related to H by this tensor which directly measures the degree to which the region will strengthen or weaken the field. Please note that permeability does not have to be either a constant or a tensor. It could be a non-constant scalar function of position. This will occur if a variable within a media does not alter the direction of the field lines through it but does alter it with varying intensity.

What makes magnetism more complicated than general electric materials is the property of ferromagnetism. Ferromagnetism is helpful in terms of understanding the H-field because ferromagnetic materials exhibit hysteresis. That is, the domains do not necessarily release once the magnetic field through them is released. So, we can increase the magnetic intensity through a ferromagnetic  material (H) and observe the resulting flux density through the material (B). This is where the quantity H becomes very useful because the magnetic field within a ferromagnetic material has no simple relationship with the magnetizing force. This is because the domains can align with each other and not just external fields and have a tendency to become stuck together, a tendency which varies with density, temperature, humidity, etc. That is why the magnetization of ferromagnets is in general, quite complicated. The hysteresis loop is useful here because it enables us to quickly grasp properties of the material we are magnetizing. If we apply a certain magnetizing force to a thoroughly demagnetized ferromagnet, then we will get a curve where the flux density increases with H. This is not linear because the increase in dipole alignment starts to slow down with increase in magnetizing force until it reaches the magnetic saturation. At this point there is no way to channel more magnetic flux through a material because all the dipoles are aligned with the field. If we start to ramp down the magnetic intensity to zero, the material will still have a flux through it because the domains have become permanently aligned. The amount of flux left in the material at the point where the intensity has been ramped down to zero is called the magnetic retentance and is a useful measure of permanent magnetization. In order to force the aligned dipoles to demagnetize, we must switch the direction of magnetic intensity until we reach a flux of zero. The magnetic intensity at this point is called the coercivity and is a useful measure of how strong a field we have to put through a magnet to demagnetize it.

If we continue to intensify H in the other direction past the coercivity point then we will reach a point of magnetic saturation again. This is equal and opposite to that in the other direction, and as we take the magnetic intensity to zero, we should find the flux to be equal and opposite to the coercivity.

I hope this helps.