# 4. Magnetic Materials

## 4.1 Definitions and General Relations

### 4.1.1 Fields, Fluxes and Permeability

| There are many analogies between dielectric and magnetic phenomena; the big difference being that (so far) there are no magnetic "point charges", so-called magnetic monopoles, but only magnetic dipoles.
|  | The first basic relation that we need is the relation between the magnetic flux density B and the magnetic field strength H in vacuum. It comes straight from the Maxwell equations:
B | = | µo · H
--- | --- | ---
|  | The symbols are: B = magnetic flux density or magnetic induction, µo = magnetic permeability of the vacuum = 4p · 10–7 Vs/Am = 1,26 · 10–6Vs/Am H = magnetic field strength .
|  | The units of the magnetic field H and so on are [H] = A/m [B] = Vs/m2, with 1Vs/m2 = 1 Tesla. B and H are vectors, of course.
|  | 103/4p A/m used to be called 1 Oersted, and 1 Tesla equales 104 Gauss in the old system.
|  | Why the eminent mathematician and scientist Gauss was dropped in favor of the somewhat shady figure Tesla remains a mystery.
| If a material is present, the relation between magnetic field strength and magnetic flux density becomes
B = µo · µr · H
|  | with µr = relative permeability of the material in complete analogy to the electrical flux density and the dielectric constant.
|  | The relative permeability of the material µr is a material parameter without a dimension and thus a pure number (or several pure numbers if we consider it to be a tensor as before). It is the material property we are after.
| Again, it is useful and conventional to split B into the flux density in the vacuum plus the part of the material according to
B = µo · H + J
|  | With J = magnetic polarization in analogy to the dielectric case.
| As a new thing, we now we define the magnetization M of the material as
M | = | J µo
--- | --- | ---
|  | That is only to avoid some labor with writing. This gives us
B | = | µo · (H + M)
--- | --- | ---
| Using the independent definition of B finally yields
M | = | (µr - 1) · H
--- | --- | ---
M | := | cmag · H
|  | With cmag = (µr – 1) = magnetic susceptibility.
|  | It is really straight along the way we looked at dielectric behavior; for a direct comparison use the link
| The magnetic susceptibility cmag is the prime material parameter we are after; it describes the response of a material to a magnetic field in exactly the same way as the dielectric susceptibility cdielectr. We even chose the same abbreviation and will drop the suffix most of the time, believing in your intellectual power to keep the two apart.
|  | Of course, the four vectors H, B, J, M are all parallel in isotropic homogeneous media (i.e. in amorphous materials and poly-crystals).
|  | In anisotropic materials the situation is more complicated; c and µr then must be seen as tensors. That should be no surprise anymore.
| We are left with the question of the origin of the magnetic susceptibility. There are no magnetic monopolesthat could be separated into magnetic dipoles as in the case of the dielectric susceptibility, there are only magnetic dipoles to start from.
|  | Why there are no magnetic monopoles (at least none have been discovered so far despite extensive search) is one of the tougher questions that you can ask a physicist; the ultimate answer seems not yet to be in. So just take it as a fact of life.
|  | In the next paragraph we will give some thought to the the origin of magnetic dipoles.

© H. Föll (Advanced Materials B, part 1 - script)