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UNIT-II


A medium plays a significant role in
determining the response of an electric field.
Depending upon the behavior of different
materials in an electrostatic field, they are
divided in two main categories:
1. Conductors
2. Insulators or dielectrics.


Conductors are materials which contain
electrons that are free to move and an electric
field produces a steady drift of charge i.e.
current.
They “conduct” the current or allow the flow of
electric charges like electrons fairly easily.



Dielectric is basically an electrically insulating
material.
The electrons are strongly bound to the atoms
or molecules.
They cannot be separated by the application of
an electric field. A steady flow of electrons
cannot flow through it.

The electric moment of a system of charges
with zero net charge is generally called the
electric dipole moment of the system.
m  aQ

the vector m points from the negative to the
positive charge.
+Q
a
rp
- Q
rn

Dielectric materials are classified into
two broad groups:
1. Polar
2. Non-polar
Polar molecule




The molecules in which the
arrangement or geometry of
the atoms is such that one
end of the molecule has a
positive electrical charge and
the other side has a negative
charge.
The centers of gravity of the
positive and negative charge
distributions do not coincide.
They have a finite electric
dipole moment (permanent
dipole moment).
E.g. H2O, NH3, HCl, SO2, H2S,
CO.
+
-
Non-polar molecule





A non-polar molecule is that
in which the electrons are
distributed more
symmetrically
It does not have an excess
/abundance of charges at the
opposite sides.
The centers of gravity of the
positive and negative charge
distributions coincide.
They do not have any
permanent dipole moment
E.g. CO2, H2, N2, O2, CH4,
CCl4
+
-



In polar dielectric molecules, permanent
dipoles are present
however, due to thermal agitation, they are
generally in random orientations
Dielectric is in unpolarized condition i.e. the
net dipole moment within any small volume is
zero.



A dielectric possess no free electrons to provide
currents due to an applied external E.F.
Although there is no macroscopic migration of
charge when a dielectric is placed in an electric
field, microscopic displacements (on the order of
the size of atoms or molecules) of charge occur
resulting in the appearance of induced electric
dipoles.
There are two mechanisms of polarization:
…displacement of charge within the atom or
molecule
…orientation of a molecule with a permanent
dipole moment.




A dielectric is said to be polarized when
induced electric dipoles are present.
The presence of induced electric dipoles
within the dielectric causes the electric
field to be modified.
Polarizability is a measure of the ability of a
material to become polarized in the
presence of an applied electric field.
Polarization occurs in both polar and non
polar materials.
electron
cloud
nucleus


In the absence of an
applied electric field,
the positively charged
nucleus is surrounded
by a spherical electron
cloud with equal and
opposite charge.
The dipole moment is
zero.
Eapp

In the presence of
an applied electric
field, the electron
cloud is distorted
such that it is
displaced in a
direction (w.r.t.
the nucleus)
opposite to that of
the applied
electric field.
e
e
p  E
dipole
moment
(C-m)
polarizability
(F-m2)

The net effect is
that each atom
becomes a small
charge dipole
which affects the
total electric field
inside the
material.



In polar dielectric
molecules, permanent
dipoles are present
however, due to thermal
agitation, they are
generally in random
orientations
Dielectric is in unpolarized
condition i.e. the net
dipole moment within any
small volume is zero.
Eapp

In the presence
of an applied
electric field, the
dipoles tend to
align themselves
with the applied
electric field.


When an electric field is applied, the dipoles
are oriented by rotation and aligned in the
direction of the electric field
one type of bound charges (+ve or -ve)
appear on a surface and the opposite type on
opposite surface.
e
e
p  E
dipole
moment
(C-m)
polarizability
(F-m2)

The net effect is
that each polar
molecule is a small
charge dipole which
aligns with the
applied electric
field and influences
the total electric
field.
A neutral atom
+
-
R
Polarized
atom
-

E0



E
+
r
X-axis
Consider an atom of radius R having spherical
symmetric charge distribution. (non-polar atom
with zero dipole moment)
If atom is subjected to an external E.F., both
positive and negative charges will experience
force in opposite directions.


As positive charge is rigidly fixed in nucleus,
its position remains fixed whereas negative
charge is pushed along negative x-axis.
Therefore atom acquires an induced dipole
moment.
p=Qe=Ze r

Also net E.F. inside dielectric:
Enet  E0  E

The force on nucleus due to applied electric
field is


F  ZeE0  ZeE0iˆ



There will be another force on nucleus due to
negative charge cloud which will tend to restore
original configuration.
The charge on negative cloud is Q   Ze
The E.F. due to this cloud at the nucleus will be:

E


Qr
 Zer ˆ

i
3
3
40 R
40 R
The force on nucleus due to the charged cloud
is:
2 2
'
 Z e r
ˆ
F  ZeE 
i
40 R 3

At equilibrium the total force on nucleus
must be zero.  
F  F'  0
2 2
Z
er ˆ
ˆ
ZeE0i 
i 0
3
40 R
40 R 3 E0
r
Ze
but p  Zer
 p  40 R 3 E0
p  E0

This shows that the induced dipole moment
is proportional to external E.F.
Atomic polarizability:
As p  40 R 3 E0
Let   40 R 3
 p  E0
where is  called
Atomic polarizabi lity.
It is defined as induced dipole moment per unit
E.F.
p

E0
For a given value of E0, larger the value of  ,
larger will be induced dipole moment.

Also, direction of induced dipole moment is
same as external E.F.



 p  E0


-

E0
E

p
+
X-axis
If a sample of dielectric contain a N atoms per
unit volume of the sample. Let p be the
dipole moment induced in each atom.
Then polarization produced in the sample is



P  Np  NE0

N is the number of dipoles per unit volume.

p is the average dipole moment of the


(m-3)
dipoles in the medium: Coulomb-m (C-m)
 is the atomic polarizability of the dipoles
in the medium: Farad-m2 (F-m2)
P is the polarization per unit volume:
Coulomb- m-2 (C/m2)





The charges which are developed on the
surfaces of dielectric during polarization are
called bound charges.
As these charges are attached with atoms of
material, they can’t take part in conduction.
The bound charges are also called polarization
charges.
These charges are denoted by qb or q p
All charges except produced by polarization
are called free charges.

A parallel-plate capacitor consists of
two oppositely charged, conducting
parallel plates separated by a finite
distance.

The field between the plates is
uniform in direction (perpendicular
to the plates) and magnitude.

Charge density

E.F.
  q/ A
E0  q /  o A   /  o

If a dielectric slab is placed within
capacitor plates, then dielectric gets
polarized.

As a result of polarization, the E.F.
inside dielectric become
E  E0  EP
where E P is E.F. produced by
polarization charges.

The dielectric constant of a material is defined
as the ratio of E.F. inside the capacitor without
dielectric to E.F. inside same capacitor with
dielectric.
k  E0 / E
 E0 /( E0  EP )
but E P   P /  0

k
  P


For vacuum,  P =0.
Therefore k=1


The polarization of the dielectric is caused by
the E.F. which causes alignment of dipoles.
Therefore polarization should depend on net
E.F. i.e. polarization is directly proportional to
the net E.F.
 
PE


 P   0 e E

electron
susceptibility
(dimensionless)
For vacuum, there is no polarization. Thus
susceptibility of vacuum is zero.
Under the influence of this field, the positive and negative
charges in the particle are moved apart: the particle is
polarized. In general, these induced dipoles can be treated as
ideal; permanent dipoles, however, may generally not be
treated as ideal when the field at molecular distances is to be
calculated.
The values of molecular dipole moments are usually expressed
in Debye units. The Debye unit, abbreviated as D, equals 10-18
electrostatic units (e.s.u.).
The permanent dipole moments of non-symmetrical molecules
generally lie between 0.5 and 5D. It is come from the value of
the elementary charge eo that is 4.410-10 e.s.u. and the
distance s of the charge centers in the molecules amount to
about 10-9-10-8 cm.
In the case of polymers and biopolymers one can meet much
higher values of dipole moments ~ hundreds or even
thousands of Debye units. To transfer these units to CI system
one have to take into account that 1D=3.3310-10 coulombsm.
A volume distribution of dipoles may
be represented as an equivalent
volume (qevb) and surface (qesb)
distribution of bound charge.
 These charge distributions are related
to the dipole moment distribution:

qevb    P
qesb  P  nˆ
33

Gauss’s law in differential form in free
space:
 0  E  qev

Gauss’s law in differential form in
dielectric:
 0  E  qev  qevb
34
 0  E  qev  qevb  qev    P
   0 E  P   qev
• Hence, the displacement flux density vector
is given by
D  0 E  P
35

Gauss’s law in differential form:

Gauss’s law in integral form:
  D  qev
D

d
s

Q
encl

S
36
 C 
D    3 
m 
   Ddv   dv
v
v

 D  ds  Q
s
(C )
 E  0
Gauss’s Law: The total outward
the dielectric displacement (or
the outward flux) over any
surface is equal to the total free
enclosed in the surface
flux of
simply
closed
charge
D   0 E  P   0 E   0 e E 
 C 
  0 (1   e ) E   0 r E   E  2 
m 
Where- ε is the absolute permittivity (F/m)
-εr is the relative permittivity or the dielectric cons
of the medium
-ε0 is the permittivity of free space
-χe is the electric susceptibility (dimensionless)

Assuming that
we have

P   0 e E





Dis the
 e permittivity
E  E or the
The parameter
0 1electric
dielectric constant of the material.
39



The concepts of polarizability and dipole
moment distribution are introduced to
relate microscopic phenomena to the
macroscopic fields.
The introduction of permittivity eliminates
the need for us to explicitly consider
microscopic effects.
Knowing the permittivity of a dielectric tells
us all we need to know from the point of
view of macroscopic electromagnetics.
40

For the most part in macroscopic
electromagnetics, we specify the
permittivity of the material and if
necessary calculate the dipole moment
distribution within the medium by
using
P  D   0 E     0 E
41

The relative permittivity of a dielectric is the
ratio of the permittivity of the dielectric to the
permittivity of free space

r 
0
42
3-6.2 Dielectric in static electric field


Bound charges (electric dipole) : External electric field causes a force
to be exerted on each charged particle and results in a small
displacements of positive and negative charges in opposite
directions. (fig. 3-14)
The induced electric dipoles modify the electric field both inside and
outside the dielectric material.

Polar molecules possess permanent electric dipole moment.

Polarization vector: the volume density of electric dipole moment.
n
P  lim
 0
p
k 1

k
(C/m 2 ).
3-7 Electric flux density and dielectric constant

Divergence modified to include the effect of equivalent polarization
volume charge density :
 EE 



1
(    p ) 
1
(  
 P
P) or
0
0

 (( 0 E  P )   .
Electric flux density (electric displacement) :
D   0E  P
New equation :
(C/m 2 ).

 D   (C/m3 ), where  : volume density of free charges.
Generalized Gauss's law:
  DDd    d , or
D ds  Q (C).
 D
 ds  Q (C )
V
V'
C
C

The total outward flux of the electric displacement over any closed
surface is equal to the total free charge enclosed in the surface.
3-7 Electric flux density and dielectric constant

For linear and isotropic medium,
P   0 eE, where e : electric susceptibility.

Electric displacement :
D   0 E  P   0 E   0  e E   0 (1   e )E   0 r E   E
where  r  1   e 
(C/m 2 ),

: relative permittivity (dielectric constant),
0
   0 r : absolute permittivity (permittivity).


A simple medium: a linear, homogeneous, and isotropic medium.
Anisotropic materials: the dielectric constant is different for different
directions of the electric field.
Dielectric constants and dielectric strengths of
some common materials
3-7.1 Dielectric strength



Dielectric strength: the maximum electric field intensity that a
dielectric material can withhold without breakdown.
Dielectric strength of air: 3 kV/mm.
Electric field intensity at a charged conductor surface is higher at
points of larger curvature.