2022-23- AP handout by Dr. T.Rajani, Asst.

Prof (Physics)

Unit-IV (Part-I) 1

Dielectric and Magnetic Materials

Contents
Dielectric Materials: Basic definitions- Types of polarizations (qualitative) -Frequency
dependence of polarization, Local field, Clausius-Mossotti relation, Ferroelectric, Piezoelectric,
and Pyroelectric materials – Applications.

Magnetic Materials: Basic definitions- Types of Magnetic materials, Antiferro and ferri
magnetic materials, Weiss-Domain theory of ferromagnetism, Hysteresis - Soft and hard
magnetic materials, Multiferroics – Applications.

• Introduction:
Dielectrics are insulating materials. There are no free charge carriers in a dielectric. In these
materials forbidden gap is greater than 3eV. Such large gap precludes thermal excitation of
electrons from valance band to conduction band. Dielectrics find extensive use in electrical and
electronics industries. They are used for insulation purpose. All capacitors use dielectric materials
between their plates.

• Basic terminology in Dielectrics

(a) Dipole:
Two equal and opposite charges (+q, -q) separated by a distance (r) called a dipole.

(b) Dipole moment (μ)

The multiplication of charge and distance between charges is called dipole moment.
𝜇 = 𝑞. 𝑟
(c) Polarization

The process of inducing dipole moment in dielectric by applying electric field is called
“Polarization”.
Dipole moment per unit volume is defined as polarization vector (𝑃̅).
Polarization can also be defined as induced charge per unit area.
𝑞𝑖
𝑃̅ =
𝐴
(d) Permittivity (ε)
It is the property of every material, which measures the opposition offered against the formation of
an electric field. Represented by the Greek alphabet ε. It tells the number of charges required to
generate one unit of electric flux in the given medium.

Unit-IV: Dielectric and magnetic materials

 2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

2
𝜀 = εo εr

ε𝑜 is permittivity of free space and its value is 8.85 X 10-12 farad/meter.

εr is relative permittivity and it is the ratio of absolute permittivity to the permittivity of free space.
It is considered as dielectric constant.
̅ ) and applied electric field (𝐸̅ ) is
The relation between displacement vector (𝐷

̅ = ε𝑜 ε𝑟 E
D ̅

(e) Relation between 𝑬̅ , 𝐏,

̅𝐃 ̅:
The three vectors E, D, P are related by
̅ = ε𝑜 𝐸̅ + ̅
D P

(f) Electric susceptibility:

It is a quantitative measure of the extent to which an electric field applied to a dielectric material
causes polarization. For most linear dielectric materials, the polarization P is directly proportional
to the average electric field strength E so that the ratio of the two, P/E, is a constant that expresses
an intrinsic property of the material.

̅
Pα𝐸
̅
P=χ𝐸
̅
P
χ=
𝐸
It can be defined as “Polarization produced per unit electric field”. Electric susceptibility is always
a dimensionless positive number.

(g) The Polarization:

We have seen that a dielectric material gets polarized by the application of an electric field
‘E’. the polarization P depends on applied field (E) and no. of particles present in that
material, temperature and nature of molecules etc.
Suppose a dielectric contain N molecules per unit volume, when E is applied then dipoles
were induced and it gets polarized. When E is switched off, P becomes zero slowly. Now,
polarization can be written as,
̅ = N α 𝐸̅
P
Where α is the proportionality constant called “Polarizability”
α is sum of four different polarizabilities namely, Electronic , Ionic, Orientational and Space
charge polarizabilities
α = αe + αi + αo + αs

Unit-IV: Dielectric and magnetic materials

 2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

3
• Electronic Polarization

Definition: In any dielectric material, when an electric field is applied then the displacement of
effective center of electron cloud from nucleus happens and forms the diploes is called “electric
polarization”

The electronic polarization

𝑃 = 𝑁𝛼𝑒 𝐸

Where electronic polarizability 𝛼𝑒 = 4 𝜋𝜀𝑜 𝑎3

‘𝛼𝑒 ’ is independent of temperature.

• Ionic Polarizability (𝜶𝒊 )

Definition: The polarization produced due to relative displacement of ions is called “Ionic
polarization”.

When electric field is applied to an ionic dielectric then positive ion move in the direction of the
field and negative ion move in the direction opposite to the field as shown in below figure.

Then, ionic polarizability can be written as,

𝑒2 1 1
𝛼𝑖 = 2
[ + ]
𝜔𝑜 𝑀 𝑚

𝛼𝑖 is independent of temperature.

Unit-IV: Dielectric and magnetic materials

2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

• Orientational polarization: 4
When an external field is applied to polar dielectrics, they tend to align themselves in the
direction of external applied field. The polarization due to such alignment is called Orientational
polarization.
“The polarization produced due to orientation of already existing diploes in polar substances is
called orientational polarization”.

The Orientational polarization is strongly

temperature dependent. With increase of
temperature, the thermal energy tends to randomize
the alignment. An expression for Orientational
polarizability given below and it can be obtained by
following the same procedure adopted in Langevin
theory of paramagnetic. Here, instead of magnetic
field, electric field is used.
2
𝜇𝑚
𝛼𝑂 =
3𝐾𝑇

• Space charge polarization:

Space charge polarization occurs due to accumulation of charges at the electrodes or at the
interfaces in multiphase dielectrics. As shown in figure, the ions diffuse over appreciable distance
in response to the applied field. This gives rise to redistribution of charges in dielectric medium.
The space charge polarization is not an important factor in most common dielectrics. It is found in
ferrites and semiconductors.

Unit-IV: Dielectric and magnetic materials

2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

• Frequency dependance of polarization 5

On application of an alternating field across the material, the polarization process occurs as a function of
time. Electronic polarization is extremely rapid and is complete at any instant of time even when the
frequency of the voltage is very high in the optical range. Thus it occurs at all frequencies. But ionic
polarization is slower and the ions do not respond at all when the voltage is corresponding to optical
frequencies. That is, the electric field here is changing too rapidly for the ions to reorient themselves in
response to the field. Therefore, ionic polarization does not occur at optical frequencies where as it
occurs at infrared frequencies. The Orientation polarization is slower than the ionic polarization and
occurs only at microwave frequencies
which are smaller than the infrared
frequencies. Space-charge polarization is
slowest process and occurs at power
frequencies (50-60 per sec). Thus at low
frequencies, the value of total polarization
is very high and at high frequencies
(optical frequencies) the value of total
polarization is very small. Above 1015 Hz,
None of the polarization mechanisms are
able to switch rapidly enough to remain in
step with the field. The material no longer
possesses the ability to polarize, and the dielectric constant drops to 1, the same as that of vacuum.

• Local field (or) Internal field

In dielectric solids, the atoms or molecules experienced not only the external applied electric
field but also the electric field produced by the dipoles. Thus the resultant electric field acting on
the atoms or molecules of dielectric substance is called the “Local field or an internal field.”

Derivation: Consider a dielectric material placed in an External field ‘E1’, placed between the
parallel plates of a capacitor. As a result opposite type of charges are induced on the surface of
dielectric.

Unit-IV: Dielectric and magnetic materials

2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

Imagine a small spherical cavity of radius ‘r’. In this sphere inside dipoles are present. Consider a 6
dipole at the center of spherical cavity. This dipole experiences the following fields, in addition
to the externally applied field ‘E1’. The total internal field experienced by the dipole

𝐸𝐿𝑜𝑐𝑎𝑙 = 𝐸𝑖 = 𝐸1 + 𝐸2 + 𝐸3 + 𝐸4 − − − −(1)

Field E1:

E1 is the field intensity at ‘A’ due to charge distribution on plates, from field theory

𝑃
𝐸1 = 𝐸 + − − − −(2)
𝜀0

Field E2:

E2 is the field intensity at ‘A’ due to the charge density induced on the two sides of the dielectric

𝑃
𝐸2 = − − − − −(3)
𝜀0

Field E3:

E3 is the field intensity at ‘A’ due to other atoms contained in the cavity. We are assuming that
the dielectric is a cubic then field is ‘0’ due to symmetry.

𝐸3 = 0 − − − −(4)

Field E4:

E4 is the field intensity at ‘A’ due to polarization of charges on the surface of cavity and was
calculated by Lorentz as given below

If the ‘dA’ is the surface area of the sphere

of radius ‘r’ lying between ‘θ’ and ‘θ+dθ’,
where θ is the direction of the applied force,
then

𝑑𝐴 = 2π (PQ) (QR)

But, from triangle POQ,

𝑃𝑄
sin θ = = 𝑃𝑄 = 𝑟 sin θ
𝑟

𝑄𝑅
dθ = = 𝑄𝑅 = 𝑟 d θ
𝑟
𝑑𝐴 = 2π (𝑟 sin θ) (𝑟 d θ) = 2πr 2 sin θ d θ

Unit-IV: Dielectric and magnetic materials

2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

The charge ‘dq’ on the surface dA is equal to normal component of the polarization multiplied by 7
the surface area

𝑑𝑞 = P cos θ X dA

= 𝑃 cos θ 2πr 2 sin θ d θ

The field due to this charge at ‘A’ is

dq x cos θ
𝑑𝐸4 =
4𝜋𝜀0 𝑟 2

𝑃 cos θ 2πr 2 sin θ d θ cos θ

=
4𝜋𝜀0 𝑟 2

𝑃cos 2 θ sin θ d θ
=
2𝜀0

The total field ‘E4’ due to the charges on the surface of the entire cavity is obtained by
integrating it.

𝑃 𝜋 2
𝐸4 = ∫ 𝑑𝐸4 = ∫ cos θ sin θ d θ
2𝜀0 0

𝑃 𝜋 2
= ∫ cos θ 𝑑 (−𝑐𝑜𝑠 θ)
2𝜀0 0
𝜋
−𝑃 cos3 θ
= [ ]
2𝜀0 3 0

−𝑃
= [−1 − 1]
6𝜀0

𝑃
𝐸4 = − − − −(5)
3𝜀0

Substitute the equations (2), (3), (4), (5) in equation (1)

Total internal field is

𝐸𝑖 = 𝐸1 + 𝐸2 + 𝐸3 + 𝐸4

𝑃 𝑃 𝑃
𝐸𝑖 = 𝐸 + − +0+
𝜀0 𝜀0 3𝜀0

𝑃
𝐸𝑖 = 𝐸 +
3𝜀0

Unit-IV: Dielectric and magnetic materials

2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

• Clausius - Mossotti Equation.

It is the relation between microscopic polarizability and macroscopic dielectric constant (relative
permittivity) of material.

Consider a dielectric material with density N and E be the external electric field. Then
Polarization vector is given by

𝑃 = 𝑁 𝛼 𝐸𝑖 − − − −(1)

Relation between E, D, P is given by, 𝐷 = 𝜀𝑜 𝐸 + 𝑃 by solving we get,

𝑃 = 𝜀𝑜 (𝜀𝑟 − 1)𝐸 − − − (2)

𝑃
Local field or Internal field is given by, 𝐸𝑖 = 𝐸 + − − − − − (3)
3𝜀𝑜

Substitute (3) in (1)

𝑃
= 𝑁𝛼 (𝐸 + )
3𝜀𝑜

Substitute (2) in above equation

𝜀𝑜 (𝜀𝑟 − 1)𝐸
= 𝑁𝛼 (𝐸 + )
3𝜀𝑜

(𝜀𝑟 − 1)
𝑃 = 𝑁𝛼 𝐸 (1 + ) − − − − − (4)
3

Compare (2) and (4)

(𝜀𝑟 − 1)
𝜀𝑜 (𝜀𝑟 − 1)𝐸 = 𝑁𝛼 𝐸 (1 + )
3

𝑁𝛼
𝜀𝑜 (𝜀𝑟 − 1) = ( 2 + 𝜀𝑟 )
3

By re arranging terms

(𝜀𝑟 − 1) 𝑁𝛼
=
(𝜀𝑟 + 2) 3𝜀𝑜

Unit-IV: Dielectric and magnetic materials

 2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

• Piezo-electricity 9

When some crystals subjected to mechanical stress, electrical charges will induced on the surfaces
of the crystals. This phenomenon is called “piezo electricity”. Quartz crystals are showing this
property. Consider x-cut quartz crystal, used in feedback oscillator for producing ultrasonics.
Now, let us consider some simple lattice structures of charge distribution in crystals. The below
fig. explains this.

The field E displaces the opposite charges there by slightly altering the dimensions of the crystals.
Thus effect is the electrostriction charge arrays. On other hand, stress will produce a net dipole
moment in a crystal whose charges do not possess the center of symmetry as shown below.

If crystal shows center of symmetry, then induced dipole moments are equal and opposite hence
there is no net dipole moment.

Examples: Quartz, Lead Zirconate Titanate (PZT), Polyvinylidene Fluoride (PVDF), Lithium
Niobate, Rochelle Salt etc.,

• Ferro electricity:

The dielectric become spontaneously polarized, when temperature of dielectric is equal to certain
temperature (Curie temperature) is called” Ferro electricity” since its behavior is analogous to
ferromagnetism.

A large no. of small regions are formed in sample at their critical temperatures. The diploes within
any one of the domains are coupled together and point in the same direction, thus forming large
net dipole moment. These large diploes present in separate domains are randomly oriented, so the
sample as a whole in neutral. When electric field is applied, the sample exhibits high degree of
polarization.

Unit-IV: Dielectric and magnetic materials

2022-23- AP handout by Dr. T.Rajani, Asst.Prof (Physics)

Above Tc (ferro electric curie temperature), the material is normal polar dielectric as its 10
polarization P is directly proportional to E, so that P vs E plot is straight line passing through origin.
If T ≤ Tc, the material becomes ferroelectric and its polarization is non-linear with E. In this case
the plot exhibits Hysteresis loop, similar to ferromagnetism as shown below.

First, polarization increases with E rapidly and reaches to

saturation. If the field is decreased at this stage, P also
decreases but rate of decrease is less than that required.
When E is zero P does not become zero is called “residual
polarization”. If E increased on negative side, p reduces to
zero and the field is called “coercive field”. After that P also
becomes negative and reaches to negative saturation.
Further as increasing E positive side P also increases but
takes a new path and finally reaches to saturation and
completes a loop. The ferro electric materials are used in capacitors to store very large electric
field in a small region. These materials can be used as temperature sensors.

Examples: Barium Titanate, Lead Titanate, Strontium Titanate, Lithium Tantalate, Lead Zirconate
Titanate (PZT)

• Pyro electricity:

It is a property of certain materials to generate an electric charge in response to a change in

temperature. When a pyroelectric material is heated or cooled, it generates an electric charge due
to the movement of its positive and negative ions.

Pyroelectric materials can be used in a variety of applications. For example, they can be used as
sensors to detect temperature changes, as well as in energy harvesting devices to convert thermal
energy into electrical energy.
Some examples of pyroelectric materials include lithium tantalate, quartz, and tourmaline.

Tutorial Questions

(Short)

1. What is electric susceptibility. Write its units.

2. Define Electronic and Ionic polarizations.
3. List the applications of Piezo and pyro electric materials.
4. Write a short note on Ferro electricity.

(Long)

1. What is internal field in dielectrics? Derive expression for it

2. Derive expression for Clausius Mossotti relation.
3. Discuss the frequency dependance of polarization.

Unit-IV: Dielectric and magnetic materials