Chapter 7: Polarization

Transcription

Chapter 7:
Polarization
Joaquín Bernal Méndez
Group 4
Campos Electromagnéticos
Class year 2008/2009
Ingeniería Industrial
Dpto. Física Aplicada III
1
Index

Introduction

Polarization Vector

The Electric Displacement Vector

Constitutive Laws: Linear Dielectrics

Energy in Dielectric Systems

Forces on Dielectrics
2
Introduction


Conductors: contain a great amount of free charge
Dielectrics: all charges are attached to specific
atoms or molecules


Examples: wood, plastic, stone...
Then ¿How does a dielectric substance respond to an
external electrostatic field?

Charges attached to molecules or atoms undergo
microscopic displacements
3
Induced Dipoles


An atom has a positively charged core (the nucleus)
and a negatively charged electron cloud surrounding it
The nucleus is pushed in the direction of the field and
the electrons the opposite way:

The atom gets polarized

Induced dipole moment:

Polarizability
If the electric field is too strong this relationship can
become nonlinear and the atom can even be ionized
4
Alignment of Polar Molecules


Some molecules have permanent dipole moments that
are not due to the action of an external electric field
Example: water molecule
Polar Molecules

This polar molecules tends to rotate to line up its
dipole moment parallel to the external electric field
5
Polarization

Dielectrics with neutral atoms or nonpolar molecules:


A dipole moment parallel to the field is induced in each
atom or molecule by the applied electric field
Dielectrics with polar molecules:

The external electric field exerts a torque on each
molecule that tends to line it up along the field direction

This will not be a complete alignment due to the effect of
random thermal motion
In both cases we obtain a polarized dielectric: a lot of
little dipoles aligned with the external field
6
Index

Introduction

Polarization Vector




7
Polarization Vector

We are going to study the field due to a piece of
polarized material



We will forget for a moment about the cause of the
polarization
Each molecule has a dipole moment:
From a macroscopic point of view we define the
polarization vector:

Dipole moment per unit volume
8
Electric Field Due to a Polarized
Material


Let's suppose that we know the
polarization vector. ¿Can we calculate
the the electric field created by the polarized material?
Idea: the total field can be obtained as a superposition
of the fields of all the tiny dipoles inside the material

Potential due to a dipole at the origin:

If the dipole is located at an arbitrary point
9
Material



Dipole moment due to a volume
element :
Potential created by this volume
element:
Integrating over the volume of the polarized material:
10
Material

This potential can be expressed in a different way

By using:

We can write down the integrand as:
11
Material

And we arrive to:

By applying the Divergence Theorem:
: Volume of the polarized material
: Surface boundary of the
polarized material
12
Polarization Charges


Potential created by a volume and a surface charge
densities:
By analogy we van define:
Surface density of polarization charges
Volume density of polarization charges
13

We can calculate the electric field produced by the
polarized material by finding the polarization charges
and calculating the field that they produce



We get to a problem of electrostatics (chapter 3)
We must know the polarization vector to apply this
technique
Questions about polarization charges:


¿Are they actual charges or just a mathematical tool?
If they are true charges, ¿How does polarization lead to
such accumulation of charge in a neutral material?
14
Physical Meaning of the

Uniformly polarized material:

The head of a dipole cancels the tail of its neighbor

But at the ends are two layers of charges left over:
15
Physical Meaning of the

Piece of material with nonuniform polarization

There is not complete compensation between adjacent
positive and negative layers
net bound charge
within the material
Polarization charges are real accumulations of charge
16
Total Polarization Charge

The total charge can be calculated by summing the
surface and volume polarization charges:
Divergence Theorem
There is no total polarization charge in a polarized
material (unless free charge has been deposited)
17
Index

Introduction

Polarization Vector




18


We have already calculated the field crated by a
polarized material: polarization charges
The total electric field is that produced by both the
polarization charges and the free charges. This field
obeys the Gauss's Law:
with:
Electric displacement vector
19

Gauss's Law can be written in terms of the electric
displacement vector:
Differential form
Integral form

This is an auxiliary field: it can not be measured

Units: C/m2 (same as

Its scalar sources are only the free charges

Boundary condition:
)
20
Vector Sources of the Electric
Displacement Vector


A vector field is determined by its divergence (scalar
sources) and its curl (vector sources)
From the definition:
(Electrostatics)

We get to:

The curl of the polarization vector is the vector source
of the electric displacement vector
21
Usefulness of the Electric
Displacement Vector

The parallel between


and
is subtle:
The electric displacement vector can NOT be obtained
in the same way as the electric field but forgetting about
the polarization charges
However for highly symmetric situations we usually
have:
and then the electric displacement
vector can be calculated in terms of the free charge
from the Gauss's Law:
22
Example

Parallel-plate capacitor filled with a dielectric slab
Plane symmetry
By applying Gauss's Law:
23
Example

Infinite straight line with a uniform line charge λ
surrounded by a dielectric cylinder



can be expressed in terms of the free charges:
If we knew
we could calculate
by using:
BUT USUALLY:
we need to know the
functional form of this relationship (constitutive equation)
24
Index

Introduction

Polarization Vector




25
Constitutive Laws


A dielectric is usually polarized due to an external
electric field
For many substances the polarization is proportional
to the field:



: electric susceptibility (dimensionless)
In vacuum:
is the total electric field (due to free and polarization
charges), not the externally applied electric field
Substances verifying this constitutive
equation are referred to as linear dielectrics
26
Linear Dielectrics



Homogeneous: its susceptibility is independent of
position
Isotropic: its susceptibility is a scalar magnitude (in
instead of a tensor)
Linear: polarization is proportional to the field


This is true as long as the field is not too strong
There exit substances not obeying this law:


Ferroelectric materials: the polarization depends on the
history of the particular chunk of material
Electrets: materials which are able to hold a permanent
electric polarization in absence of an external field
27
Linear Dielectrics



Relationship between the electric displacement and
the electric field:
We define:

Permittivity of the material:

Relative permittivity:
Therefore:
;(F/m)
and:
28
Dielectric Constants for Some
Common Substances
Material
Air
1.0006
Glass
4-10
Paper
2-4
Wood
2.5-8.0
Porcelain
6-8
Rubber
2.3-4.0
Ethyl Alcohol
28.4
sodium chloride
6.1
Sea water
72
Distilled water
80
29
Example

Parallel-plate capacitor filled with a linear dielectric
We have already obtained:
The capacitance is increased by a factor of
30
Example

Parallel-plate capacitor: polarization charges

In general, for linear dielectrics:
31
Capacitor filled with Insulating
Material

For a given free charge, the potential difference is
smaller when the capacitor is filled with a dielectric
material:


Because the electric field between the plates is partially
shielded by the polarization charges and hence its
magnitude is smaller than in the vacuum case
For a given difference of potential, the accumulated
free charge is larger when the capacitor is filled with a
dielectric material:

Because an extra amount of free charge is needed to
counteract the effect of the polarization charges in order
to attain the same electric field between the plates
32
Example

Parallel plate capacitor partially filled with a dielectric
Applying Gauss's Law:
Capacitance of two
capacitors connected
in series
33
Example

Parallel-plate capacitor partially filled with a dielectric
In both regions must be verified that:
Capacitance of two
capacitors connected
in parallel
34
Example

Conducting sphere carrying a charge q surrounded by
a dielectric sphere
Symmetry:
Gauss's Law:
35
Example

Conducting sphere carrying a charge q surrounded by
a dielectric sphere: polarization charges
Inside the material:
(linear dielectric)

Exercise: check that the total polarization charge is zero
36
Index

Introduction

Polarization Vector




37

We already know:


This equation give us the work that it takes to bring all
the charges from infinity to their final positions
When dealing with dielectric systems it is more
convenient to use this formula:

This equation give us the work that it takes to bring the
free charges from infinity to their final positions
Both formulas are correct, but they represent different things
38
Energy Stored in a Parallel-Plate
Capacitor

For a parallel-plate capacitor filled with a dielectric:

We have calculated:

Therefore:
39


The force exerted on the dielectric material in a
direction can be calculated by using the principle of
virtual work:
Example: dielectric slab partially inserted between the
plates of a parallel-plate capacitor:
40
Summary (I)

Polarization is the response of dielectric materials to
external electric fields:




The dipole moments of the molecules of the dielectric
tends to line up in the direction of the electric field.
The polarization vector describes the polarization of
the material from a macroscopic point of view.
Polarization charges account for the electric field
created by the polarized material.
The electric displacement vector is an auxiliary
vector field whose scalar sources are the free
charges.
41
Summary (II)


In highly symmetric distributions the electric
displacement vector can be calculated as a function of
the free charges by using Gauss's Law.
To calculate the total electric field we also need a
constitutive equation of the medium, which gives us
the relationship between the polarization vector and the
electric field.


For linear media the polarization vector is proportional to
the electric field.
To calculate the energy of dielectric systems we have
introduced an alternative formula of the energy that
does not include the work required to bring the
polarization charges from infinity.
42

Chapter 7: Polarization

Transcription

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