Tài liệu Chapter XII Magnetic materials pdf

b, the dipoles tend to align with the applied field, and the vector sum of all atomic magnetic moments becomes non-zero vector, then we denote it by B’ :... 4/1/2008 132.3 Permeability a

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GENERAL PHYSICS II

Electromagnetism

&

Thermal Physics

Chapter XII Magnetic materials

§1 Atomic magnetic moment - Bohr magneton

§2 Magnetization, paramagnetism and diamagnetism

§3 Ferromagnetism

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 Investigation of the magnetic properties of materials is very important, because magnetic phenomena have various scientific and technical

applications

 The macroscopic properties of matter are a manifestation of the

microscopic properties of the atoms of which it is composed

 The magnetic properties of materials may be very different for types of material, depending on their nature and structure

§1 Atomic magnetic moment – Bohr magneton:

In order to understand the magnetic properties of matter we must

know the magnetic properties of atoms

1.1 The magnetic moment of an orbiting charge :

 Moving electrons, protons, neutrons create currents they have magnetic dipole moments

 The motions of these particles can be decomposed into orbital

motion and spinning motion

The current created by the orbiting particle is

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• Its magnetic moment is

The vector form:

where L0 is angular momentum of the orbital motion

It is known that for an electron in the ground (non-excited) state of

the hydrogen atom the angular momentum equals to 1.05 x 10 -34 J.s

(we will learn later in quantum physics), so we have for the orbital

magnetic moment of electron:

This quantity is the fundamental unit of magnetic moment, it is called

Bohr magneton = 9.22 x 10-24 A.m2

2.3 The magnetic moment of a spinning charge:

Consider a spinning charge:

A ring dq of the spinning charge creates the current:

The corresponding magnetic moment of the ring is

Summing over all the rings:

Assume that the charge is distributed in the same way as the mass,

we can write

I = L

The spin magnetic moment vector

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The particular case of an electron:

An electron is known to have spin angular momentum of

0.527 x 10-34 J.s So, its spin magnetic moment is

However, experiments give the result of twice bigger Why?

The problem lies with the assumption about the charge distribution

that we have used

To correct this mistake one introduces the factor g called

“gyromagnetic ratio“, and writes

For electrons g = 2.

§2 Magnetization, paramagnetism and diamagnetisme:

2.1 Some general view on the magnetic properties of materials:

 The material which has the most striking magnetic properties is

iron Similar magnetic properties are shared also by nickel, cobalt,… That kind of magnetic properties is called

ferromagnetism

 All other ordinary substances do show some magnetic effects,

but very small ones – a thousand → million times less than the effects in ferromagnetic materials

This small magnetism is of two kinds In other words, there are two signs to the magnetic effects: paramagnetism and

diamagnetisme.

Strong magnetic effects Weak magnetic effects

in ferromagnetic materials in paramagnetic

materials

in diamagnetic materials

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Two signs to the magnetic effect:

• If the small cylinder is of bismuth → it is repelled by the sharp pole

• If the small cylinder is of aliminium → it is attracted by the sharp pole

b)

Applying the formula of magnetic field for a ring

current on its axis, we have for each atomic current:

2.2 Magnetization:

First we consider a piece of paramagnetic

material

When no external magnetic field is present, the

atomic magnetic dipoles are randomly aligned

(pic a) The total magnetic field due to all the

dipoles cancels to zero

If we apply an external magnetic field B 0 (pic b),

the dipoles tend to align with the applied field, and

the vector sum of all atomic magnetic moments

becomes non-zero vector, then we denote it by B’ :

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Since atomic current ring is very small, we have

where we have introduced the quantity M , considered as a type

of average dipole moment per unit volume We can write

The quantity M is called the magnetization of the material

and we obtain the total magnetic field is

 Units:

 The units of magnetic moment are (current) x (area), that is A.m 2

 The units of magnetization M is (A / m2) / m 3 = A / m

 The units of M is the same as the units of B (as it must be):

( T.m / A )( A / m ) = T

 It is natural to think that the sum of the atomic magnetic moments tend to align with the external magnetic field, then

B > B0 and M > 0

However, by experiments one observed that this is true not for all

materials, but only for most common materials Such materials are called

paramagnetic For these materials M > 0 but fairly small.

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2.3 Permeability and susceptibility:

 We have said that for paramagnetic materials the total magnetic field

inside the material B is greater than the external field B 0 So we can write

B = K m B 0

where K m is a dimensionless factor, called relative permeability.

The value of K m is typically ranges from 1.0001 → 1.003 (see the table

In the page 1089 of the textbook)

 The expression of the magnetic field in materials relates to that in

vacuum by the replacement by

= Km 

which is called the permeability of the material.

 Since for paramagnetic materials the value of K m is a small deviation from unity, it is convenient to introduce the quantity

χ m = K m – 1

which is called the magnetic susceptibility.

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2.4 Diamagnetism:

There are a type of materials for which the

magnetization vector is opposite to the external

magnetic field, that is M < 0.

Why? We can explain this phenomenon in

a simplified version as follows:

 In the absence of external fields the

electrons move randomly (pic a)

 When the external magetic field is applied (pic b),

the electrons begin move in circular orbits This

orbiting electrons create a field which is opposite

to the external field

This type of materials is called diamagnetic. For it

K m is typically of the order of 0.9999 → 0.99999

( see the table)

a)

b)

§3 Ferromagnetism:

3.1 Strong magnetization of ferromagnetic materials:

 The third type of materials is called

ferromagnetic materials, which includes

iron, nickel, cobalt, … These materials

manifest strong magnetic effects

 The magnetic field inside them is much

larger than the applied external field, the

relative permeability K m is of the order of

1.000 to 10.000.

The properties of ferromagnetic materials

are explained by their microscopic structure:

In these materials the atomic magnetic moments are extremely easy to align together, due to strong interactions between them

Inside the material there exist regions called magnetic domains, even when no external field is applied In each domain the atomic magnetic moments are parallel to each other

In the absence of external magnetic field the magnetic moments

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When an external field B0 is applied, the domain magnetic moments

tend to orient themselves paralell to the field that leads to the shift of

domain boundaries: domains that have magnetic moments parallel to

the external field will grow, other domains will shrink

The magnetic moment of each domain have the order of thousands

of Bohr magnetons, the torques that tend to align the domains with the

external field are much stronger than in paramagnetic materials After

rearrangment of domain magnetic moments the magnitude of the

magnetization vector of the material is much larger than the external field

Two important features of ferromagnetism are

• the saturation of magnetization

• the hysteresis

3.2 Saturation of magnetization and hysteresis:

Consider the dependence of the magnitude of the magnetization vector

on the magnitude of external magnetic field

M

B 0

M sat

Increase B 0 from zero → the magnitude

of magetization increases

When B 0 reachs to some enough large value, further increase of the external field causes no increase in magnetization This phenomenon is called the saturation of magnetization.

The saturation of magnetization is explained as follows: When the

external field is enough large, all the domain magnetic moments are

aligned parallel to it, and the magnetization can’t increase further

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Hysteresis loop

M

B0

When the material is magnetized to

saturation we reduce the external field

to zero, the magnetization decreases

(the curve b), but some magnetization

remains when B 0 = 0.

The material becomes then a

permanent magnet It has own

magnetic moment when the external

field is removed

To reduce the magnetization to zero

We must apply an external field in the

inverse dirction

The variation of the magnetization with

the change of the applied magnetic

field is described by the hysteresis loop