Figure 1.1: Typical hysteresis loops for a permanent magnet show the M(H)
and B(H)
loops after saturation of magnetization. These loops are linked through equation (1.3). The shaded area highlights the maximum energy product (BH
).
The divergenceless field B
, often referred to as flux density or induction, is defined by:
B = \mu_0(H + M).
(1.5)
Here, \mu_0 = 4 \times 10^{-7} \text{N A}^{-2}
, is the magnetic constant, commonly called the permeability of free space. It is involved whenever the H
field applies force to a magnet or a current-carrying conductor.
Both H
and B
fields can vary locally. They follow the magnetostatic field equations derived from Maxwell’s equations, assuming no free currents:
\text{curl} \, H = 0 \quad \text{and} \quad \text{div} \, B = 0.
(1.6a, b)
In free space, M = 0
and B = \mu_0 H
. However, inside materials, the vector relationship and nonlinear dependence of M
on H
result in B
and H
being neither parallel nor proportional (see Figure 1.2). The unit of B
is \text{Vs m}^{-2}
, also known as the Tesla (T).
The Tesla is ideal for discussing ferromagnetism. Engineers often multiply H
and M
by \mu_0
and represent them in Tesla rather than \text{A m}^{-1}
. A related quantity, \mu_0 M
, denoted by J
, is called polarization. Thus, equation (1.5) can be written as:
B = \mu_0 H + J.
A high-quality permanent magnet typically has a remanent polarization around 1 \text{T} [10 \text{kG}]
, though ferrite magnets are limited to 0.6 \text{T} [6 \text{kG}]
. For pure iron, the magnetization is 1.76 \text{MA m}^{-1} [1760 \text{emu cm}^{-3}]
, equating to a polarization of 2.15 \text{T} [21.5 \text{kG}]
.
Permanent magnets in electrotechnical applications must generate a uniform magnetic flux density of about 1 \text{T}
. This is used to deliver flux into an air gap. In contrast, iron-core electromagnets and superconducting magnets can achieve flux densities of 2 \text{T}
and 15 \text{T}
, respectively. Pulsed currents in coils can momentarily create fields exceeding 100 \text{T}
. At the lower end, stray fields also play a role.