DIELECTRICS

Objectives
 At the end of lesson we shall understand about,
 Static dielectric constant.
 Types of Polarizations.
 Internal or local fields in solids and liquids.
 Lorentz field in cubic materials.
 Clausius Mossotti equation.
Introduction

A dielectric is a substance that is highly resistant to the flow of an electric current. In

other words, a dielectric is an electrically non conducting material that provides

electrical insulation between two media (conductors) which are at different potentials.

Eg - Glass, Wax paper, Ceramics, Mica, Porcelain. When a dielectric medium interacts

with an applied electric field, charges are redistributed within its atoms or molecules.

This redistribution alters the shape of an applied electrical field both inside the

dielectric medium and in the region nearby. When two electric charges move through a

dielectric medium, the interaction energies and forces between them are reduced.

Dielectric Constant

Faraday discovered that the capacitance of the condenser increases when the region

between the plates is filled with dielectric. If C 0 is the capacitance of the capacitor

without dielectric and C is the capacitance of the capacitor with dielectric then the

ratio C/C0 gives εr called as relative permittivity or Dielectric constant. Also for a given

isotropic material, the electric flux density is related to the applied field strength by

the equation D = εE, where ε is absolute permittivity. In SI system of units, the relative

permittivity is given by the ratio of absolute permittivity to the permittivity of free

space. ε = ε0εr, where ε0 is permittivity of the free space and εr is the relative

permittivity or dielectric constant. For an isotropic material, under static field

conditions, the relative permittivity is called as static dielectric constant. It depends

upon the structure of the atom of which the material is composed.

Dipole: A dipole is an entity consisting equal number of positive and negative charges

separated by a small distance. A dipole moment is a vector directed from positive field.

-q q

Polarization: The displacement of charges in the atoms or molecules of a dielectric

under the action of applied field leading to the development of dipole moment is called

as polarization.

Electrical polarization

The polarization of the dielectric is the process of formation of dipoles or alignment of

already existing dipoles by the application of an electric field on the dielectric material.

The ratio of induced dipole moment to the effective applied electric field is called as

polarizibility.

Polar and non-Polar dielectrics

In dielectrics there are no free electrons, the center of positive charges are centered or
concentrated at the center of the atom and center of negative charges are concentrated
in the electron cloud. When the center of gravity of positive charges coincides with
center gravity of negative charges, then it neutralizes each other; hence their dipole
moment is zero. Such dielectrics are called non-polar dielectrics.

In some other dielectrics like water, center of gravity of positive charges never
coincides with the center of gravity of negative charges even in the presence of applied
field. In such dielectrics, each molecule behaves as if it contains a pair of positive and
negative charges separated by a distance. Hence they have a permanent dipole
moment. They are known as polar dielectrics.

Consider a dielectric material placed between two plates of a parallel plate capacitor as

shown in figure 2.

Let a dc potential be applied between the plates; the atomic dipoles in the material will

align in the electric field. The mean position of electrons will align towards the positive

plate of the capacitor and mean position of positively charged nucleus will align

towards negative plate of capacitor. Inside the material the dipoles formed will align in

such a way that the positively charged particles are attracted towards the negatively

charged particles. In fact, at the surface of dielectric layer, negative charge is formed

near positively charged plate of capacitor; a layer of positive charge is formed adjacent

to the negatively charged plate of capacitor and these charges on the surface of

dielectric material is called as polarized charges.

+
Conducting plate (Metal)

- - - - - - - - -

+ + + + + + + + +
E0
- - - - - - - -

+ + + + + + + +

Conducting plate (Metal)

-
Fig 2: Polarization
 Different Types of Polarization Mechanisms

The polarization is the alignment of permanent or induced atomic or molecular dipoles

under the action of an applied field. Hence depending up on the dielectric material and
the manner of applied electric field, there are four types of polarization mechanisms:

1. Electronic polarization

2. Ionic polarization

3. Orientation or molecular polarization

4. Space charge polarization

Electronic Polarization

This is the most common type of polarization, which occurs in most of the dielectrics.

The electronic polarization is due to the displacement of center of gravity of negatively

charged particles relative to center of gravity of positive charges. This is called as

electronic because of the dipole moment resulting due to the shift of the electron cloud

relative to the nucleus as shown in Fig 3. This type of polarization is due to the

induced dipole moments. The electrons have very high natural frequencies of the order

of 1015 Hz and hence a light of frequency 1015 Hz can cause an electronic

polarization. It is found that the electronic polarization is temperature independent.

which occurs in over a short interval of time of 10-15 sec.

Fig 3: Atom without Electric field Atom with Electric field

2. Ionic Polarization.
Since the induced dipole moment is directly proportional to the applied electric field strength E,

e is proportional to E.
Hence, e = αeE

Electronic Polarization P = Ne where N is the no of atoms per m3.

Therefore,

NαeE =  o ( r  1) E

αe =  o ( r  1) /N

Ionic Polarization

The ionic polarization occurs only in ionic materials like NaCl etc. In this type of

materials, under equilibrium conditions, the cations and anions remain at their mean

equilibrium conditions. When the field is applied, the cations and the anions get

displaced from their mean positions in opposite directions and give rise to a net dipole

moment as shown in Fig 4. As the dipole moment occurs only under an applied

electric field, the ionic polarization is due to the induced dipoles; and also the ions are

heavier than electrons. This type of polarization is a slow process and ionic

polarization is limited to frequencies up to 10 13 hertz and hence the light frequencies

of 1015 cannot cause ionic polarization.

Fig 4: Ionic polarization

3. Orientation Polarizations or Molecular Polarization

The orientation polarization occurs in polar dielectrics in which there are molecules

with a permanent dipole moment. The orientation of these molecules are random due

to their thermal agitation, because of the randomness in orientation, the material has

net zero dipole moment in the absence of electric field. When an electric field is

applied, each dipole undergoes a rotation so as to orient along the direction of the

field, which exerts a torque in them and thus the material itself develop the dielectric

polarization as shown in Fig 5. In the orientation polarization, the restoring forces do

not exist, however the dipole alignment is balanced by thermal agitation and this type

of polarization is strongly temperature dependent.

- + - - +
- - +
+
+ + -
+ -
+ - - + - +
+ + -
- qu
+ ote
- + - +
+ - + - from
+ the
Aligned dipoles in electric docu
Fig 5: Randomly oriented permanent dipoles field men
The orientation polarizibility is given by, α0 = µ2/3kT t or
the
The orientation polarization P o is given by, Po = Nµ2E/3kT sum
mary
4 Space Charge Polarization of an
inter
The space charge polarization occurs in multiphase
+ dielectric substances in which estin
g
there is a change of resistivity between different phases when an electric field is
poin
applied at a high temperature. The electric charges get accumulated at the interface t.
You
due to the sudden change in conductivity. This accumulation of charges with opposite
can
posit
ion
the
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box
polarities at opposite parts in low resistivity phase leads to the development of dipole

moment (Fig 6).

+ - + - + -

_ +
+ - + - + -

+ - + - + -

Fig 6: Space charge polarization

The space charge polarization is not an important factor in most common dielectrics.

The total polarization α of a material is thus given by the sum of electronic, ionic and

orientation polarizations,

ie. α = αe+ αi+ αo

Internal Field

When the electric field is applied to a dielectric material either liquid or solid, each
atom in the material develops dipole moment and acts like an electric dipole, since the
atoms either in liquids or solids are surrounded on all sides by polarized atoms, the
internal field at given point inside the material is equal to the electric field created by
the neighboring atoms and the applied field.

“The internal field is defined as the electric field that acts at the site of any given atom
of a solid or liquid. Dielectric field subjected to an external field and is resultant of the
applied field and the field due to all the effects of the surrounding atoms.”

Expression for Internal or local fields in solids and liquids

Consider a dielectric material solid or liquid under the action of an electric field of
intensity ‘E’. In a dielectric, imagine an infinite string of similar equidistant atomic
dipoles parallel to the external applied field,
 Fig 7: Internal field

The components of the electric field at ‘P’ due to an atomic dipole in polar form
are given by,

μ cosθ μ sin θ
Er  Eθ 
2π 0 r 3 4π 0 r 3
(1)

Dipole at A1:

The distance of X from A1 is d.

i.e., r = d and  = 0

μ
Er  Eθ  0 (2)
2π 0 d 3
μ
Field at X due to A1: Er+ Eθ =
2π 0 d 3
Dipole at A2:

Since it is situated symmetrically on the other side of X, its field at X will also
be μ
2π 0 d 3
Field at X due to A2: μ (3)
2π 0 d 3
Therefore field at X due to both dipoles A1 and A2

Field at X due to A1 & A2: μ

E1 
π 0 d 3

μ
Field at X due to B1 & B2, located at a distance of 2d: E 2 
π 0 (2d) 3

The Total Field E' at X due to all dipoles:

E'  E1  E2  E3  ....
μ μ μ
    ....
π 0 (d) 3
π 0 (2d) 3
π 0 (3d) 3
μ 1 1
 [1  3  3  ....]
π 0 d 3
2 3

μ 1

π 0 d 3
n
n 1
3

where n = 1, 2, 3,….∞

1
But we know that, the summation of infinite series  3
=1.2
n 1 n

μ
 E' = 1.2 (5)
π 0 d 3

The total field at X which is the internal field Ei is the sum of the applied field E and

the field due to all the dipoles, i.e. E' .

Ei = E + E' (6)

If αe is the electronic polarizability for the dipoles, then

µ = αeEi (7)

1.2  e Ei
Ei = E +
0 d 3
By rearranging the terms in the above equation we have,

E
Ei =
1  1.2 e
π ε 0d 3 (8)

This is the expression for internal field in case of one–dimensional array of atoms in a

dielectric which is a solid or liquid.

Lorentz Field for a cubic lattice:

In 3D, the general equation for internal field is expressed as,

Ei = E + (γP/ε0), where P is the polarization and γ is a proportionality constant called

internal field constant.

In the 3D if it is a cubic lattice then, γ =1/3 and the internal field is named as Lorentz

Field given by

E Lorentz = E + P/3ε0

The above equation is known as Lorentz relation. One of the important results that

follow from this relation is Clausius-Mossotti relation.

CLAUSIUS-MOSSOTTI RELATION
Consider an element solid dielectric of a dielectric constant εr. If N is the
number of atoms/unit volume of the material,  is the atomic dipole moment,
then we have,
Dipole moment,
e = αeEi ………………….. (1)
Polarization of the medium is given by,
P = NαeEi …………………… (2)
Therefore,
Ei = P/Nαe …………………….. (3)

For a medium with a dielectric isotropy,

P = o(r-1)E (4)
Therefore,

E = P/o(r -1)………………. (5)

In a 3D dielectric material,

E i = E + (γP/ε0) ........................ (6)

Using (3), (5) and (6)

P P P ……………….. (7)
 
N  0 ( r  1)  0

1 1 1 1
   
N  0  ( r  1) 3 

as  = 1/3 for cubic crystals. Therefore,

0  r  2 
 
N  3( r  1) 

Rearranging we get,

  r  1  N
  
  r  2  3 0 (8)

This is Clausius-Mossotti equation.

Dielectric losses:

It is the loss of energy in the form of heat by a dielectric medium due to the internal
friction that is developed as consequences of switching action of molecular dipoles
under certain ac conditions.
Dipolar Relaxation:

Relaxation time is the time required for the dipole to reach the equilibrium orientation
from the disturbed position in alternating field conditions.

Summary:

1. Dielectrics are insulators and posses high electrical resistivity. Dielectric constant

is characteristic of materials and it measures polarization ability of dielectric

subjected to electric field

2. Dielectrics are broadly divided into polar and non-polar dielectrics.

3. The polarization phenomenon accounts for the ability of materials to increase

storage capability of capacitors.

4. The total polarization of materials is sum of the electronic, ionic and orientation

polarizations.

5. The Clausius-Mossotti equation holds good for crystals of high degree of

symmetry and also for non polar dielectric materials.

Application of Dielectric Materials:

1. Insulating Materials

a. The electrically insulating material should have high resistivity to reduce the
leakage current and high dielectric strength to enable to withstand higher
voltage without being breaking down.

b. The insulating dielectric materials are required to have low dielectric

constant, low dielectric loss and a high resistance.

c. They should possess an adequate chemical stability, high moisture

resistance and suitable mechanical properties.

d. polymers and ceramics are widely used as solid insulators.

e. In Aluminium or copper conductors, plastic or rubber insulators are used.

2. Dielectric medium in Capacitors

a. Dielectrics should have a high dielectric constant, a high specific resistance,

a high dielectric strength and a low dielectric loss.

b. Several layers of thin paper are also used as a capacitor dielectric.

c. Mica is used in discrete capacitors with very small capacitance values.

d. Polypropylene films and tissue paper impregnating with dielectrol-II are used
in power capacitor applications.

e. An electrolytic solution of ammonium borate or Sodium phosphate is used in

wet type capacitors.

3.Application of dielectric in Transformers:

a. The dielectric material in a transformer is used as an insulator and as a

cooling agent.

b. Dielectric liquids are used as electrical insulators in high voltage

applications, e.g. transformers.

c. In electrical transformers, mineral oils are used as a liquid dielectric and they
assist in the cooling process. Castor oil is used in high-voltage applications.

Solved examples:

1. Find the polarization produced in a dielectric medium of relative permittivity 15 in

the presence of an electric field of 500 V/m.

Solution:

Given: εr = 15, we know that, ε0 = 8.854 * 10-12 F/m

Ε =500 V/m

P=?

P = ε0(εr -1)E

=8.854 * 10-12 (15-1) 500

 = 6.195 * 10-8 C/m2

2. A parallel plate capacitor of area 650 mm2 and a plate separation of 4mm have a
charge of 2 * 10-10 C on it. What should be the resultant voltage across the capacitor
when a material of dielectric constant 3.5 is introduced between the plates?

Solution:

Given

Area of the capacitor, A= 650 mm2 = 650 * 10-6 m2

Distance of separation between the plates, d = 4 mm = 4 *10-3 m

Charge on the capacitor, Q = 2*10 -10 C

Dielectric constant εr = 3.5

We know that,

C = ε0εrA/d

Also, C = Q/V

Equating the above relations,

Q/V = = ε0εrA/d

Or V= Qd/ε0 εr A

= 2*10 -10 *4*10-3/8.85 *10-12*3.5 *650*10-6

= 39.73V

3. The dielectric constant of sulphur is 3.4. Assuming a cubic lattice for its
structure calculate the electronic polarizability of sulphur.

Solution:
Since the crystal structure of sulphur is cubic we can apply Clausius –Mossotti
equation,   r  1   N
   2  3
 r  0

   1  3 0
Hence αe =  r 

 r  2  N
Now, N the number of atoms/unit volume can be written as,

N A * 10 3 * D
N
atomicnumb er

6.025 * 10 26 * 2.07 * 10 3
N
32.07

N = 3.89*1028 /m3

   1  3 0
Hence αe =  r 

 r  2  N

12
αe =  3.4  1  3 * 8.854 * 10
 
 3.4  2  3.89 * 10
28

αe = 3.035 *10-40 Fm2

5. A elemental solid dielectric material has polarizability 7*10-40 Fm2. Assuming

that internal field to be Lorentz, calculate the dielectric constant for the
material if the material has 3*1028 atoms/m3.

Solution:

αe = 7* 10-40 Fm2

No of atoms/m3 =3*1028

The internal field is Lorentz field

Since the internal field is Lorentz field, we can apply Clausius –Mossotti
equation,
  r  1  N
  

 r  2  3 0
 r 1   3 * 10 28 * 7 * 10 40 
    12 


 r  2   3 * 8.854 * 10 
= 0.7906

(  r -1) = (  r +2) *0.7906

 r -1 = 0.7906  r + 1.5812

 r (1-0.7906) = 2.5812

 r =2.5812/0.2094

 r = 12.33