1.
Dielectric
s
By:Mayank Pandey
VIT University, Vellore
2.
introDuction
Dielectrics are the materials having electric dipole moment permantly
Dipole: A dipole is an entity in which equal positive and negative
charges are separated by a small distance..
DIPOLE moment (µele ):
The product of magnitude of either of the charges and separation
distance b/w them is called Dipole moment.
µe = q . x coul – m
+q
-q
X
All dielectrics are electrical insulators and they are mainly
used to store electrical energy. It stores with minimum dissipation
power). Since, the e- are bound to their parent molecules & hence,
there is no free charge
Ex: Mica, glass, plastic, water & polar molecules…
3.
dipole
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+
Electric field
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+
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+
+
+
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+
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+
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Dielectric atom
4.
Important terms in dielectrics
1) Electric intensity or electric field strength
Def:- The force per unit charge “dq” is known as electric field strength
(E).
Where “dq” is point charge , E is electric field, F is force applied on point
charge “dq”.
E= F/dq = Q / 4πεr2
where “ε” is permittivity.
What is permittivity?
It is a measure of resistance that is encountered when forming an electric
field in a medium.
“ In simple words permittivity is a measure of how an electric field effects
and is effected by a dielectric medium”.
5.
a) ε (permittivity of medium):- How much electric field generated per
unit charge in that medium.
b) ε0 (permittivity of space) :- The electric field generated in vacuum.
It is constant value ε0=8.85 x 10-12 F/m.
Imp points:1) More electric flux exist in a medium with a high permittivity(because of
polarization).
2) Permittivity is directly related to “ Susceptibility” which is a measure of how
easily a dielectric polarize in a response of an electric field.
“ permittivity relates to a materials ability to transmit an electric field”
ε = εr . ε0 = (1+χ ) ε0
Relative permittivity
Susceptibility
6.
2) Electric Flux density or Electric displacement Vector:The electric flux density or electric displacement vector “D” is
the number of flux line’s crossing a surface normal to the lines, divided by
the surface area.
D = Q/ 4π r2
where , 4π r2 is the surface area of the sphere of radius “r”.
3) Dielectric Parameters :a) Dielectric constant(εr ):- It is defined as the ratio of permittivity of
medium(ε) to the permittivity of free space(ε0).
εr = ε / ε0
b) Electric dipole moment (μ):- The product of magnitude of charges &
distance of separation is known as electric dipole moment (μ ).
μ = Q.r
7.
c) Electric Polarization :- The process of producing electric dipoles by an
electric field is called polarization in dielectrics.
“ In simple words polarization P is defined as the dipole moment per unit
volume averaged over the volume of a cell”
P = μ / volume
d) Polarizability :- When a dielectric material is placed in an electric field, the
displacement of electric charge gives rise to the creation of dipole in the
material . The polarization P of an elementary particle is directly
proportional to the electric field strength E.
P∝E
P = αE
α → polarizability constant
The unit of “α” is Fm2
8.
Electric susceptibility:•
The polarization vector P is proportional to the total electric flux
density and direction of electric field.
Therefore the polarization vector can be written
P = ε 0 χe E
P
χe =
ε0E
ε 0 (ε r − 1) E
=
ε0E
χe = ε r −1
9.
Various polarization processes:
1. Electronic polarization
2. Ionic polarization
3. Orientation polarization
4. Space charge polarization
10.
1) Electronic Polarization:- When an EF is applied to an atom,
+vely charged nucleus displaces in the direction of field and ẽ could in
opposite direction. This kind of displacement will produce an electric
dipole with in the atom i.e, dipole moment is proportional to the
magnitude of field strength. This displacement between electron and
nucleus produces induced dipole moment & hence polarization. This is
called electronic polarization.
It increases with increase of volume of the atom.
This kind of polarization is mostly exhibited in Monatomic gases .
It occurs only at optical frequencies (1015Hz).
It is independent of temperature.
He
α e = ____ × 10-40 F − m 2
Ne
Ar
0.18 0.35 1.46
Kr
Xe
2.18 3.54
11.
Expression for Electronic Polarization :Consider a atom in an EF of intensity ‘E’ since the nucleus
(+ze) and electron cloud (-ze) of the atom have opposite charges and
acted upon by Lorentz force (FL).
Subsequently nucleus moves in the direction of field and
electron cloud in opposite direction.
When electron cloud and nucleus get shifted from their
normal positions, an attractive force b/w them is created and the
separation continuous until columbic force F C is balanced with
Lorentz force FL, Finally a new equilibriums state is established.
12.
E
x
+Ze
R
R
No field
fig(1)
In the presence of field fig (2)
fig(2) represents displacement of nucleus and electron
cloud and we assume that the –ve charge in the cloud
uniformly distributed over a sphere of radius R and the
spherical shape does not change for convenience.
13.
Let “σ” be the charge density of the sphere
− Ze
σ=
4 3
πR
3
- Ze represents the total
charge in the sphere.
Thus the charge inside the sphere is
4
q e ⇒ σ . π .x 3
3
− ze 4
⇒ 4
.π .x 3
.πR 3 3
3
(
=
− ze 3
x
3
R
)
- - - - - (1)
1 qe .q p
1 − ze.x 3
− z 2e 2 x
( ze ) =
Now Fc =
. 2 =
- - - - - (2)
2
3
3
4πε 0 x
4πε 0 x R
4πε 0 R
“Where Fc is a coulomb force acting between nucleus and
electron”
14.
Force experienced by displaced nucleus in EF of Strength E
is FL = Eq = ZeE -----(3)
Fc = Fl
− z 2e 2 x
4πε 0 R 3
= ZeE
- - - - - (4)
− zex
=E
4πε 0 R 3
− zex
− zex
=
4πε 0 R 3
αe
α e = 4πε 0 R 3
∴αe = 4πε0 R
3
Hence electronic Polarisability is directly proportional to cube of the
radius of the atom.
15.
2) Ionic polarization
The ionic polarization occurs, when atoms form molecules and
it is mainly due to a relative displacement of the atomic
components of the molecule in the presence of an electric field.
When a EF is applied to the molecule, the positive ions
displaced by X1 to the negative side electric field and negative
ions displaced by X2 to the positive side of field.
The resultant dipole moment µ = q ( X1 + X2)..
16.
+
+
+
Electric field
_
anion
cat ion
_
_
+
+
_
x1 x2
+
_
_
+
_
+
_
17.
Restoring force constant depend upon the mass of
the ion and natural frequency and is given by
Where, x1 & x2 are displacement of +ve and –ve ions.
“M” is the mass of -ve ion and “m” is the mass of
+ve ions.
ω0 is the angular frequency of respective ions
2
F = eE = m.w0 x
or
eE
x=
2
m.w0
∴ x1 + x2 =
eE 1 1
[ + M]
2 m
w0
18.
Therefore, the induced dipole moment is given by:2
∴ µionic
e E 1 1
= e( x1 + x2 ) = 2 [ m + M ]
w0
or α ionic
µionic
e 1 1
=
⇒ 2 [m + M]
E
w0
2
Hence, αionic is the ionic polarisation
a)This polarization occurs at frequency 1013 Hz (IR).
b)It is a slower process compared to electronic
polarization.
c)It is independent of temperature.
19.
3) Orientation Polarization
It is also called dipolar or molecular polarization. The molecules such
as H2,N2,O2,Cl2,CH4,CCl4 etc, does not carry any dipole because centre of
positive charge and centre of negative charge coincides. On the other hand
molecules like CH3Cl, H2O,HCl, ethyl acetate (polar molecules) carries dipoles
even in the absence of electric field.
How ever the net dipole moment is negligibly small since all the
molecular dipoles are oriented randomly when there is no EF. In the presence
of the electric field these all dipoles orient them selves in the direction of field
as a result the net dipole moment becomes enormous.
a) It occurs at a frequency 106 Hz to 1010Hz.
b) It is slow process compare to ionic polarization.
c) It greatly depends on temperature.
20.
Expression for orientation polarization
2
N .µ orie .E
Po = N .µ orie ⇒
= N .α o .E
3kT
2
µ orie
αo =
3kT
∴α = α elec + α ionic + α ori = 4πε o R 3 +
This is called Langevin
dielectrics.
e2
2
w0
[
1
M
+
1
m
]
2
µ ori
+
3kT
– Debye equation for total Polaris ability in
21.
Frequency Dependence of Dielectric Properties
Polarisation : When an ac field is applied to a dielectric material is a
function of time, it follows the equation
Where,
a) P is the maximum polarisation attained due to prolong application of
the electric field, and
b) τr the relaxation time for the particular polarisation process, is the
average time between molecular collision (in case of liquid), during the
application of the electric field.
The relaxation time is a measure of the polarisation process and is
the time taken for a polarisation process to reach 67% of the maximum
value.
22.
Dielectric loss
When an ac field is applied to a dielectric material, some amount of electrical
energy is absorbed by the dielectric material and is wasted in the form of heat.
This loss is known as dielectric loss.
The dielectric loss is the major engineering problem.
a) In an ideal dielectric, the current leads the voltage by an angle of 90
degree.
b) But in the case of a commercial dielectric, the current does not exactly
leads the voltage by 90 degree. It leads by some other angle q less than 90
degree.
The angle f = 90 – q, is known as the dielectric loss angle.
23.
Relation between current and voltage in dielectrics
ω ε0 ε r ’ E0
I
I
ϕ
ϕ
J
ϕ
θ
90
0
ω ε0 εr’’ E0
V
V
E0
For a dielectric having capacitance C and voltage V applied to it at a
frequency f Hz, the dielectric power loss is given by,
P = VI cos θ
Since, I = V/ Xc where Xc is the capacitive reactance and is equal to 1/jωC.
Therefore,
24.
Since θ is very small, sin ɸ = tan ɸ and P= j V2 ɷ Ctan ɸ, where tan ɸ is
said to be the power factor of the dielectric.
The power loss depends only on the power factor of the dielectric as long
as the applied voltage, frequency and capacitance are kept constant.
The dielectric loss is increased by the following factors :
1) High frequency of the applied voltage.
2) High value of the applied voltage.
3) High temperature.
4) Humidity
A) The dielectric losses in the radio frequency region are usually due to dipole
rotation.
B) The dielectric losses at lower frequencies are mainly due to dc resistivity.
C) The dielectric losses in the optical region are associated with electron and
they are known as optical obsorption.
25.
Conductance: Electrical conductance measures how easily electricity
flows along a certain path through an electrical element. The SI derived unit of
conductance is the siemen. Because it is the reciprocal of electrical resistance
(measured in ohms), historically, this unit was referred to as the mho.
Resistance: The electrical resistance of an electrical conductor is the
opposition to the passage of an electric current through that conductor; the
inverse quantity is electrical conductance, the ease at which an electric
current passes. The SI unit of electrical resistance is the ohm (Ω).
Impedance : Electrical impedance, or simply impedance, describes a
measure of opposition to alternating current (AC). Electrical impedance
extends the concept of resistance to AC circuits, describing not only the
relative amplitudes of the voltage and current, but also the relative phases.
When the circuit is driven with direct current (DC) there is no distinction
between impedance and resistance; the latter can be thought of as impedance
with zero phase angle.
26.
Susceptance (B) : It is the imaginary part of admittance. The inverse of
admittance is impedance and the real part of admittance is conductance.
In SI units, susceptance is measured in siemens.
The general equation defining admittance is given by
Y= G + jB
where,
Y is the admittance, measured in siemens (a.k.a. mho, the inverse of
ohm).G is the conductance, measured in siemens. j is the imaginary unit, and B is
the susceptance, measured in siemens. The admittance (Y) is the inverse of the
impedance (Z).
or
27.
Where,
Z = R + jX
Z is the impedance, measured in ohms
R is the resistance, measured in ohms
X is the reactance, measured in ohms.
Note: The susceptance is the imaginary part of the admittance.
The magnitude of admittance is given by:
28.
Dielectric permittivity
If a varying field V(t) is applied to a material, then the polarization and
induced charge Q are related as
where ϵ* is the complex dielectric constant. The frequency dependent complex
dielectric permittivity is given by
Where ϵ s and ϵ α are the low and high frequency dielectric constants
respectively.
ɷ =2πf is the angular frequency, τ is the time constant. The ε* is given by
where ϵ ’ is the relative permittivity or dielectric constant, ϵ" is the dielectric
loss
29.
The dielectric permittivity ϵ * is related to complex impedance Z* by
ɷ is the angular frequency and C0 = t/Aɷϵ 0 is the capacitance of the free space
The real and imaginary parts of the dielectric permittivity are calculated
using the above equations, pellet dimensions and the measured impedance
data.
30.
Electric modulus
The complex electric modulus is defined by the reciprocal of the complex
Permittivity.
where M* is the complex modulus, ϵ* is the dielectric permittivity, M' is the real
and M" is the imaginary parts of modulus.
a) The complex electric modulus spectrum represents the measure of the
distribution of Ion energies or configuration in the structure and it also describes
the electrical relaxation and microscopic properties of Ionic glasses
b) The modulus formalism has been adopted as it suppresses the polarization
effects at the electrode/electrolyte interface Hence, the complex electric modulus
M (ω) spectra reflects the dynamic properties of the sample alone.
31.
For parallel combination of RC element, the real and imaginary parts of
the modulus are given by
32.
Applicability of electric modulus
1)
The applicability of the electric modulus formalism is investigated on a
Debye-type relaxation process, the interfacial polarization or MaxwellWagner-Sillars effect.
Debye-type relaxation:- Debye relaxation is the dielectric relaxation response of an
ideal, non interacting population of dipoles to an alternating external electric field. It is
usually expressed in the complex permittivity of a medium as a function of the
field's frequency:
Variants of the Debye equation
a) Cole–Cole equation
b) Cole–Davidson equation
c) Havriliak–Negami relaxation
d) Kohlrausch–Williams–Watts function (Fourier transform of stretched
exponential function)
33.
Maxwell-Wagner-Sillars effect : In dielectric spectroscopy, large frequency
dependent contributions to the dielectric response, especially at low frequencies,
may come from build-ups of charge. This, so-called Maxwell–Wagner–Sillars
polarization (or often just Maxwell-Wagner polarization),
It occurs either at inner dielectric boundary layers on a mesoscopic scale, or at
the external electrode-sample interface on a macroscopic scale.
In both cases this leads to a separation of charges (such as through a depletion
layer).
The charges are often separated over a considerable distance (relative to the
atomic and molecular sizes), and the contribution to dielectric loss can therefore
be orders of magnitude larger than the dielectric response due to molecular
fluctuations.
34.
2) Electric modulus, which has been proposed for the description
of systems with ionic conductivity and related relaxation
processes, presents advantages in comparison to the classical
approach of the real and imaginary part of dielectric permittivity.
3) In composite polymeric materials, relaxation phenomena in the
low-frequency region are attributed to the heterogeneity of the
systems.
4) For the investigation of these processes through electric
modulus formalism, hybrid composite systems consisting of epoxy
resin-metal powder-aramid fibers were prepared with various
filler contents and their dielectric spectra were recorded in the
frequency range 10 Hz-10 MHz in the temperature interval 30150 oC.
35.
5) The Debye, Cole-Cole, Davidson-Cole and HavriliakNegami equations of dielectric relaxation are expressed in the
electric modulus form.
6) Correlation between experimental data and the various
expressions produced, shows that interfacial polarization in
the systems examined is, mostly, better described by the
Davidson-Cole approach and only in the system with the
higher heterogeneity must be used.
“Heterogeneity is a problem that can arise when attempting to undertake
a meta-analysis. Ideally, the studies whose results are being combined in the
meta-analysis should all be undertaken in the same way and to the same
experimental protocols: study heterogeneity is a term used to indicate that this
ideal is not fully met.”
36.
Internal fields or local fields
Local field or internal field in a dielectric is the space and time average
of the electric field intensity acting on a particular molecule in the dielectric
material.
In other words, the field acting at the location of an atom is known as
local or internal field “E”.
The internal field Ei must be equal to the sum of applied field plus the
field produced at the location of the atom by the dipoles of all other atoms.
Ei = E + the field due to all other dipoles
37.
Evaluation of internal field
Consider a dielectric be placed between the
plates of a parallel plate capacitor and let there
be an imaginary spherical cavity around the
atom A inside the dielectric.
The internal field at the atom site ‘A’ can be
made up of four components E1 ,E2, E3 & E4.
39.
Field E1:
E1 is the field intensity at A due to the charge density
on the plates
E1 =
D
ε0
D = ε0 E + P
ε0 E + P
E1 =
ε0
E1 = E +
P
ε0
..........(1)
40.
Field E2:
E2 is the field intensity at A due to the charge
density induced on the two sides of the dielectric.
−P
E2 =
...........(2)
ε0
Field E3:
E3 is the field intensity at A due to the atoms
contained in the cavity, we are assuming a cubic
structure, so E3 = 0.
41.
+ +
+
+
+
+
+
+
_
_
E
dA
+
+
+
+
A
θ dθ
r
_
_
p
_
_
_
_
_
r
_
_
_
q
R
42.
Field E4:
1) This is due to polarized charges on the surface of
the spherical cavity.
dA = 2π . pq.qR
dA = 2π .r sin θ .rdθ
dA = 2π .r sin θdθ
2
Where dA is Surface area between θ & θ+dθ…
43.
2) The total charge present on the surface area dA is…
dq = ( normal component of polarization ) X ( surface
area )
dq = p cos θ × dA
dq = 2πr p cos θ . sin θ .dθ
2
44.
3) The field due to this charge at A, denoted by dE4 is given by
1
dq
dE4 =
4πε0 r 2
The field in θ = 0 direction
1 dq cos θ
dE4 =
4πε 0
r2
1
dE4 =
(2πr 2 p cos θ . sin θ .dθ ) cos θ
2
4πε 0 r
P
dE4 =
cos 2 θ . sin θ .dθ
2ε 0
45.
π
4) Thus the total field E4
due to the charges on the
surface of the entire
cavity is
E4 = ∫ dE4
0
π
P
=∫
cos 2 θ. sin θ.dθ
2ε0
0
P
=
2ε0
π
cos 2 θ. sin θ.dθ
∫
0
let..x = cos θ →dx = −sin θ θ
d
P
=
2ε0
−
1
x 2 .dx
∫
1
− P x 3 −1
− P −1 −1
=
( )1 ⇒
(
)
2ε0 3
2ε0
3
P
E4 =
3ε0
46.
The internal field or Lorentz field can be written as
Ei = E1 + E2 + E3 + E4
p
p
p
Ei = ( E + ) − + 0 +
εo εo
3ε o
p
Ei = E +
3ε o
47.
Classius – Mosotti relation :
Consider a dielectric material having cubic structure , and
assume ionic Polarizability & Orientational polarizability are
zero..
αi = α0 = 0
polarization..P = Nµ
P = Nα e Ei ......where., µ = α e Ei
P
where., Ei = E +
3ε 0
48.
P = Nαe Ei
P
P = Nα e ( E +
)
3ε 0
P
P = Nα e E + Nα e
3ε 0
P
P − Nα e
= Nα e E
3ε 0
Nα e
P (1 −
) = Nα e E
3ε 0
Nα e E
P=
...................(1)
Nα e
(1 −
)
3ε 0
49.
We known that the polarization vector
P = ε 0 E (ε r − 1)............(2)
from eq n s (1) & (2)
Nα e E
= ε 0 E (ε r − 1)
Nα e
(1 −
)
3ε 0
1−
Nα e
Nα e E
=
3ε 0
ε 0 E (ε r −1)
1=
Nα e
Nα e E
+
3ε 0 ε 0 E (ε r − 1)
1=
Nα e
Nα e
+
3ε 0 ε 0 (ε r − 1)
1=
Nα e
3
(1 +
)
3ε 0
ε r −1
Nα e
1
=
3
3ε 0
(1 +
)
ε r −1
Nα e ε r − 1
=
...... → Classius Mosotti relation
3ε 0
εr + 2
Where “N” is the number
of dipoles per unit volume
50.
Ferro electric materials or Ferro electricity
Ferro electric crystals exhibit spontaneous
polarization i.e. electric polarization with out electric
field.
Ferro electric crystals possess high dielectric
constant.
Each unit cell of a Ferro electric crystal carries
a reversible electric dipole moment.
Examples: Barium Titanate (BaTiO3) , Sodium
nitrate (NaNO3) ,Rochelle salt etc..
51.
Piezo- electricity
The process of creating electric polarization by
mechanical stress is called as piezo electric effect.
This process is used in conversion of mechanical energy
into electrical energy and also electrical energy into
mechanical energy.
According to inverse piezo electric effect, when an electric
stress is applied, the material becomes strained. This strain is
directly proportional to the applied field.
Examples: quartz crystal , Rochelle salt etc.,
“Piezo electric materials or peizo electric semiconductors such
as Gas, Zno and CdS are finding applications in ultrasonic
amplifiers.”
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