Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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294 Basics of Magnetostatics

induction (1.67), which is referred to as Lenz’s law. Macroscopic, induced currents

generally decay over time because of Ohmic resistance which any medium usually

has. Superconductivity is an exception. The microscopic currents that cause

magnetization (Ampere's molecular currents) do not decay. These flow without

resistance and persist for as long as the external field remains. This is the reason

why there is a unique relation between magnetization and magnetic field for which

we may write

(5.77)

is called the magnetic susceptibility. Because of Lenz’s law is negative.

The specific formulation of (5.77) makes dimensionless. The magnetization M

is oriented anti-parallel to B. This allows to identify a diamagnetic material by the

fact that in an inhomogeneous magnetic field, it experiences a force in the direction

where the field decreases.

The molecules of a so-called paramagnetic material exhibit net magnetic

moments even without an applied field. However, there is no preferred direction

without an applied field and therefore no magnetization. The individual dipoles are

statistically oriented in all directions and cancel each other on average (spatial and

time average). An applied field causes a torque, which then tries to align the

individual dipoles parallel to the external field. Still, temperature acts to mis-align

the individual dipoles. This mis-alignment is more successful the higher the

temperature. Nevertheless, partial orientation along the external field is achieved.

The diamagnetic effect of induction applies simultaneously, in an attempt to

decrease the magnetization. If paramagnetism is present in that material, then it

prevails over diamagnetism and the same Ansatz as (5.77) can be made, now with

positive . This is the reason why paramagnetic material is drawn toward the side

where an inhomogeneous field increases. for paramagnetic materials is

temperature dependent, while for diamagnetic materials does not depend on

temperature.

There are many more magnetic phenomena. Particularly important is

ferromagnetism, especially for electrical engineering. It is related to the spin of

electrons and like paramagnetism, can qualitatively be described by the orientation

of the related magnetic dipole moments. In contrast to paramagnetism, the

Fig. 5.35

t∂

∂B

B B

t∂

∂B

t∂

∂B

t∂

∂B

field changes → induced E-field → magnetization →

weakened field

M µ

5.5 B and H in Magnetizable Media 295

magnetization due to ferromagnetism is several orders of magnitude higher and the

relation between applied field and magnetization is not linear, not even unique (i.e.,

it is not a single-valued function). This means that magnetization is not only a

function of the external field but also of its previous states (i.e., it depends on its

history). Because of its non-linearity, a description of ferromagnetic materials by

susceptibility depends on the current state which, from a theoretical perspective,

does not necessarily make it useful. The susceptibility of diamagnetic and

paramagnetic materials is very small ( ), while the numbers for

ferromagnetism may reach some . Examples:

The reader is alerted about the non-standardized definitions of when

comparing these numbers with other sources.

The fact that ferromagnetism can not be described by a linear relation has

formal, far reaching consequences. The methods we have developed apply only to

linear problems. There are hardly any mathematical methods available to treat non-

linear problems analytically. Then, numerical methods need to be applied.

In the presence of magnetized media, it is possible to calculate the magnetic

field as in vacuum, if one explicitly considers all currents, including those bound

currents that originate from magnetization. We write

where

Therefore

(5.78)

We define the magnetic field strength for magnetizable media

(5.79)

This results in our previous relation between H and B for vacuum if .

Conversely

Oxygen (O

) at 18°C (64°F)

Palladium at 18°C (64°F)

Nitrogen (N

)

Bismuth

1«

1.8 10

6–

⋅=

782 10

6–

⋅=







paramagnetic

0.07–10

6–

⋅=

160–10

6–

⋅=







diamagnetic

∇ B×µ

free

bound

+ g

free

------

∇ M×+==

∇ B×µ

∇ M×+=

∇

BM–

---------------





× g

BM–

---------------

M 0=

296 Basics of Magnetostatics

(5.80)

a relation which holds for any medium, for example, even for a permanent magnet

which has a magnetization that persists even without an applied field. From (5.78)

and (5.79) follows that

(5.81)

This means that H is irrotational as long as there are only bound currents. However,

B is not irrotational in this case, but rather

For “linear media” follows from (5.80) that

Defining the relative permeability

(5.82)

and the (absolute) permeability

(5.83)

allows to write

(5.84)

When calculating H, only free currents are relevant, while the bound currents are

hidden in the relation between B and H. This approach is similar to the one in

electrostatics, where only the free charges are relevant when calculating D and the

impact of bound charges is hidden in the relation between D and E.

The B field is always source free and it is therefore always

The consequence is that H is not necessarily source free, namely if , then:

We have introduced as the fictitious magnetic charges eq. (5.73)

which gives

(5.85)

The implication is that the H field originates from, or ends at bound magnetic

charges. Because of

we can also write

B µ

HM +=

∇ H× g

∇ B×∇M×=

B µ

H µ

H+ µ

1 χ

+()H==

1 χ

µµ

B µH =

B∇• 0=

M∇• 0≠

H∇•

BM–

---------------

∇•

------

M∇•–==

M∇•–

mag

M∇•–=

H∇•

------

mag

H ψ∇–=

ψ∇–()∇• ∇

ψ–+

------

mag

5.5 B and H in Magnetizable Media 297

i.e.,

(5.86)

This represents the magnetic Poisson equation.

These properties of B and H can be confusing and are therefore summarized

in Table 5.2 for three cases.

The field of a cylindrical, uniformly magnetized permanent magnet shall

serve as an example. The H field can be calculated like the electric field of two

circular plates with a surface charge. This gives

The B field can be calculated like the field of a coil of finite length with the surface

current

or also as shown in Fig. 5.36 and Fig. 5.37, by calculating

The field of H in Fig. 5.37a) is irrotational but has sources and sinks in the form of

bound magnetic charges at the top and bottom surfaces, respectively. The field of B

in Fig. 5.37b) is source free but has curl in the form of bound azimuthal currents in

the cylinder wall. M in Fig. 5.37c) is neither source free nor curl free. Its curl is in

the cylinder wall (as for B) and its sources are at the top and bottom surfaces (as for

H). It shall be noted that Fig. 5.30 is not representing a magnetized material, but

rather the vacuum field created by given dipoles, which can only be calculated

outside of the cylinder. However, Fig. 5.37 is based on the now generalized

Table 5.2

Fields of free currents

Fields of magnetized matter

Fields of free currents and

fields of magnetized matter

H, B both

source free, but

not curl free

H not source free, but

curl free

B source free, but

not curl free

H neither source free

nor curl free

B source free, but

not curl free

∇

------

mag

–=

B∇• 0=

H∇• 0=

B∇× µ

H∇× g

B∇• 0=

H∇• ρ

mag

⁄=

B∇× M∇×=

H∇× 0=

B∇• 0=

H∇• ρ

mag

⁄=

B∇× µ

M∇×+=

H∇× g

mag

Mn•=

k k

------

Mn×==

B µ

HM+=

298 Basics of Magnetostatics

definition of H as of eq. (5.79). Only this new definition defines H inside a

magnetizable medium.

There are also anisotropic linear media for which and become a tensor.

The relation in this case reads:

(5.87)

or in short

(5.88)

The tensor is symmetric, i.e.,

(5.89)

Fig. 5.36

+ + + +

+ + +

+ + + + +

+ + + +

+ + +

- - - -

- - -

- - - - -

- - - -

- - -

Fig. 5.37

a) b) c)

+ + + + +

- - - - -

++=











B m H⋅=

5.6 Ferromagnetism 299

5.6 Ferromagnetism

Ferromagnetism is a property of iron, cobalt, nickel, and certain alloys. The

relationship between M and H or B and H is very complicated for these materials.

As previously mentioned, it is neither linear nor unique. It depends on the history

of the medium and is also rather different for the various kinds of iron. The exact

relation has to be established by measurements. In principle, the measurement

could be carried out as illustrated in Fig. 5.38. For simplicity reasons, it shall be

assumed that the cross section of the coil is very small compared to its radius r. The

primary current in the coil creates the field

where N is the total number of turns the current path (primary loop) takes around

the toroid. This creates an EMF in the induction loop (secondary loop) of

magnitude

Integrating the EMF over time gives

Measuring the corresponding values of I and V

provides the corresponding values

of B and H. Charting these values gives the so-called hysteresis loop (Fig. 5.39).

If we expose a material that has not been magnetized to a H-field that

increases, starting from zero, then B goes through the so-called new curve or initial

curve until it reaches a region that is called saturation. Saturation is characterized

by the fact that the related magnetization does not increase any more. The

magnetization is shown in Fig. 5.40.

Now, letting H decrease, results not in a curve that is merely reversing

direction, but it takes a different path. B or M decrease less than what they had

increased when H was increased. The result is that even when H is zero, B still has

a finite value (remanence or remanent field). To bring B to zero requires a negative

Fig. 5.38

ferromagnetic ring

induction loop

exciting current

(secondary loop)

(primary

loop)

2πr

---------=

A==

∫

BA=

300 Basics of Magnetostatics

field (the so-called coercivity or coercive force). Sufficiently large negative fields

lead to saturation in the other direction. When increasing H thereafter, B will take

the path through the other part of the hysteresis loop, eventually arriving again at

the positive side of saturation. Smaller hysteresis loops are achieved when not

going all the way to saturation (Fig. 5.41). The shape of the hysteresis loop

depends on the material and can be wide or narrow, more or less tilted, or almost

rectangular. There are materials with specific hysteresis loops that are more or less

useful for any particular of the multitude of applications in electrical engineering.

An important feature is the area enclosed when passing through an entire hysteresis

loop, since this area represents the losses when changing magnetization. This is

remanence

coercivity

initial curve

Hysteresis loop

Fig. 5.39

MBµ

H–=

Fig. 5.40

Fig. 5.41

5.6 Ferromagnetism 301

plausible: The work done per unit of time, i.e., the power necessary to create the

field given in Fig. 5.38 is

where is the voltage in the primary turns

Therefore

is the volume of the ferromagnetic ring. The necessary power needed per unit

volume is

(5.90)

and

W is the work necessary to establish the field starting at . It corresponds per

unit volume to the shaded area of Fig. 5.42.

The area of the integral for an entire loop is shown in Fig. 5.43

(5.91)

This means that a wide hysteresis curve, which is characteristic of so-called “hard

materials”, leads to large losses during re-magnetization. Thus, hard materials are

not suited for transformers, for which “soft materials”, with a narrow hysteresis

loop are better suited.

-------- IV

Nφ

-------

-------- INA

-------

2πr

---------

A2πr

-------

Hτ

-------

== =

---

--------

-------

---

dW HdB=

----- HdB

∫

----- HdB

∫

H µ

dH dM+()

∫

HdM

∫

== =

302 Basics of Magnetostatics

Although from a purely formal perspective, the relation between B and H is

rather complicated, we can write the constitutive equation in this form

where or are now functions of H and its previous states. When subsequently

using this terminology, it shall not be understood to suggest any kind of linearity.

Rather a specific condition shall be characterized by a corresponding factor.

An electromagnet with a ferromagnetic core and a small air gap (Fig. 5.44)

shall serve as an example. As long as the toroid is relatively slim and the gap

sufficiently small, we may consider the fields inside the core and in the air-

gap approximately uniform and can also neglect that the different force lines have

different lengths. Then we may write

Fig. 5.43

HdB

∫

Fig. 5.42

HdB

∫

B µ

H µH==

Fig. 5.44

H ds•

∫

+ NI==

5.6 Ferromagnetism 303

and furthermore

The reason is that the perpendicular components of B must be continuous at the

boundary, which will be demonstrated in the next section. This leads to

and furthermore

(5.92)

This represents a very simple example of a so-called magnetic circuit. In

general, such a circuit may consist of several pieces of different length, different

cross section, and different permeability (Fig. 5.45). We approximate that

the flux is the same through each piece, which is equivalent to neglecting the flux

leakage (fringe effects). One writes:

B==

------

+ NI=

------

------+

-------------------=

Fig. 5.45

NI H

i 1=

∑

-----

----------==