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PHYS2012
EMP10_04
DIELECTRICS – MACROSCOPIC VIEW
Reference: Young & Freedman Chapter 24 Capacitance & Dielectrics
REVIEW
Dielectrics are insulators – charges tend not to move easily in non-metallic solids
Gauss’s Law

 E dA 
qenclosed
0

q f  qb
0

qf
r 0
Electric Field and Potential

 V
V
V
E  V   
xˆ 
yˆ 
y
z
 x
V    E.dl


Parallel plate capacitor
V  E  dl  E d


zˆ 

V
E
d
Capacitors
 Two conducting plates separated by an dielectric
 Uses (basic component of most electronic circuits): timing circuits, filtering,
smoothing fluctuating voltages, transmission of ac signals, resonance circuits, flash
lights in cameras, pulsed lasers, air bag sensors, ac circuits, etc
 Stores charge on conducting plates, stores electric potential energy due to the work
done is separating the charges. Energy stored in the electric field.
 Capacitance –“ability” to store charge
Q
C
Q=CV
V
 For a parallel plate capacitor
C
r 0 A
d

 A capacitance only depends upon the
d geometry and dielectric
 Capacitors in series
|Q| on each plate
V = V1 + V2 +...
1
Ctotal 
1
1

 ...
C1 C2

Capacitors in parallel
Q = Q1 + Q2 + ...
V across each capacitor
Ctotal  C1  C2  ...
Energy
U
emp10_04.doc
1 Q2 1
1
 QV  CV 2
2 C 2
2
1
u  r 0 E2
2
14 sep 10
4. 1
Polarization
 P  e  0 E

Pn p

 b  P nˆ
e   r  1
(special case)
 Px Py Pz 




x

y
z  (more general)

b   P   
Electric displacement
1
DP
 E
0


E890
FIELDS INSIDE DIELECTRIC MATERIALS
E
D P
Dielectric materials consist effectively of a large number of electric dipoles.
An electric dipole consists of two equal and opposite charges +q and –q separated by a vector
distance d
dipole moment p  pe = q d
d
pe  qd
points from negative to positive
-q
+q
pe
We can consider the polarization of the dielectric in terms of the induced electric dipoles or
we can simply the description of dielectric behaviour by discussing different kinds of fields –
the electric field E , electric displacement field D and the polarization P to account for the
macroscopic properties of dielectrics.
If we insert a dielectric material between two charged plates, the voltage across the plates
decreases. When we remove it, the voltage goes back up again. The charge upon the plates
can’t be affected, what the dielectric does is to reduce the electric field E and hence the
voltage V ( V    E dl ). Why is the electric field reduced?
Electric displacement D
Historically, to account for behaviour of a dielectric material in an external electric field, the
concept of the electric displacement field was introduced. The free charges Qfree which might
consist of electrons on a conductor or ions embedded in the dielectric material give rise to the
electric displacement field D . Gauss’s Law can be expressed as
emp10_04.doc
14 sep 10
4. 2
 D
dA    free d  Q free
D  f
f 
Qf
A
Q f  Q free  f   free
 D  f
where f is the free charge surface density and f is the free charge volume density.
The field lines for D connect free charges (positive to negative).
Electric polarization P
The molecules within the dielectric material experience an electrostatic force due to an
electric field. The molecules are said to be polarized – each molecule becomes a tiny electric
dipole. A bound charge which means charge that can’t leave its “home” molecule is
produced by the polarization. The effect of the dielectric is due entirely to the bound charge.
We can smooth over the internal structure of the material and assign it an average dipole
moment pe per unit volume d and define this as the electric polarization P
dp
P  e  n pe
d
where n is the number density of the electric dipoles (number of dipoles per unit volume).
The lines of P connect bound charges (negative to positive). The polarization describes the
extent to which permanent or induced dipoles become aligned. The polarization gives rise to
a surface bound charge density  b   bound and a volume bound charge density b  bound
P   b nˆ
where n̂ is the normal outward pointing unit vector (special case).
Thus, the polarization equals the magnitude of the bound (induced) charge per unit area on
the surface of the dielectric material. Also, the polarization can be obtained through the
relationship
 b   P
(no proof, more general)
where  b is the volume density of the bound charges.
To develop a simple model of a dielectric material, we need to make a number of
assumptions. For an ideal dielectric material:
Homogeneous – properties don’t change with position
Isotropic – properties don’t depend upon direction
Linear – polarization is proportional to the electric field
Stationary – all charges are stationary
For ideal dielectrics, we can write
P  e  0 E  r  e  1
where  e is the dimensionless constant of proportionality, known as the electric
susceptibility and r is the dielectric constant or the relative permittivity (  r  1 ) of the
material.    0  r is the permittivity of the dielectric material [F.m-1 or C2.N.m-2]. For
anisotropic dielectric P and E are not in the same direction and  e is not a constant but a
tensor.
emp10_04.doc
14 sep 10
4. 3
Electric field E
The average electric field inside the dielectric material is due to contribution of both the free
and the bound charges.
How can we explain the reduction in the electric field between the capacitor plates?
The bound charges on the surface of the dielectric partly cancel the effect of the free charges,
hence reducing the resulting electric field. This can be seen by applying Gauss’s Law to find
the average electric field inside the dielectric E  Edielectric
 E dA 
qenclosed
0
0 E   f b

q f  qb
conductor
0
E 0
E
0
- dielectric
+
+
0 E  D  P
1
+
-
 free  bound
 D  P
+
-
E
+
The field lines for E connect net charges, free & bound (positive to negative)
+Qfree on inner surface
P
+ + + + + + + + + + + +
- - - - - - -qbound
Symmetry
– fields must be uniform
– field lines perpendicular to plates
+qbound
+ + + + + +
- - - - - - - - - - - -
D
-Qfree on inner surface
Interior points electric
field must be zero
E
E
1
0
( D  P)
The reduction in the electric field can be expressed in terms of the dielectric constant for the
material r
E
Edielectric  air
r
emp10_04.doc
14 sep 10
4. 4
Since P is parallel to E , the equation  0 E  D  P implies that D is also parallel to E and
this equation can be written as
1
E
DP
0
E
1
0


D    E
e
0
D   0 1   e  E   r  0 E   E
E
D
r 0

 r  1   e 
  r 0
D

D  r 0 E
b  r 0 (
 f b
)
0

1
b   f

 r 
Hence, the magnitude of the bound surface charge b of the dielectric is less than the
magnitude of the free charge density f on the conductor.
 b   f 1 
The above figure shows what happens to the electric field inside the dielectric. In this case,
2/3 electric field lines start on the positive plate are cancelled by negative charges in the
dielectric and reappear on the other side. The electric field induces a polarization within the
dielectric material – the negative charges move a little to the left and the positive to the right
under the influence of the applied electric field. For this figure, e  2  r  3 . D is the same
inside and outside the dielectric material and the value of E inside is only 1/3 of its value
outside. E is reduced by the factor  r inside the dielectric and 2/3 of the E field lines are
swallowed by the bound surface charge of the dielectric. If the dielectric were replaced by a
metal instead, all the electric field lines would disappear and E would be zero inside, the
metal behaves like a dielectric of infinite dielectric constant.
E095
E567
emp10_04.doc
E906
14 sep 10
4. 5
MAXWELL’S DISPLACEMENT CURRENT
When an uncharged capacitor is first connected to a battery, a current is established in the
conductors to charge the capacitor. Maxwell showed that it is necessary to assume a current
of the same value also flowed in the space between the capacitor plates.
Electric displacement current density
Jd 
d free
dt

dD
dE dP
 0

dt
dt dt
dP/dt rate of change of polarization – associated with the actual motion of charges in the
dielectric: rotation of permanent dipoles or induced dipoles – displacement of charges
– posses a current character.
0
dE
current associated with change in electric field strength even when a vacuum is
dt
between the plates.
E040
E851
emp10_04.doc
14 sep 10
4. 6
FREQUENCY RESPONSE OF THE DIELECTRIC CONSTANT
The capacitance of any capacitor is directly proportional to the dielectric constant of the
material between the capacitor plates. Hence, the dielectric constants of two materials can be
readily compared by introducing the materials, in turn, into a given capacitor and determining
the resulting capacitances. For a given material, the change in dielectric constant as a function
of pressure, temperature, or some other variable can be measured with high precision by
employing the material-filled capacitor as the capacitive element in a tuned circuit.
Resonance frequency LC tuned circuit
f0 
1
LC
If the circuit is sharply resonant, a small change in the capacitance of the capacitor results in
a significant change in the resonant frequency of the circuit. By this means, for example, even
the small changes in the dielectric constants of gases which occur when the temperature is
altered have been accurately studied.
When a DC voltage is applied to a capacitor, the polar molecules in the dielectric orient
themselves under the action of the electric field. When the applied voltage is an alternating
one, the polar molecules again attempt to line up with the field and are, in fact, equally
successful if the frequency of the AC voltage is low. As the polarity of the voltage changes,
the polar molecules obligingly change their direction. When the frequency of the applied field
is high, however, the polar molecules may not
dielectric
have time to orient themselves to the same extent
Constant
before the polarity changes. For this reason, in a
(polar
material that possesses permanent polar
molecules)
molecules, the dielectric constant decreases with
increasing frequency. If, on the other hand, the
polar molecules in the dielectric are induced
ones, resulting from a displacement of the
planetary electron systems there is no observed
frequency
decrease with increasing frequency, because this
displacement is practically instantaneous.
In most materials, both permanent and induced polar molecules contribute to the polarization.
The dielectric constant of water falls from its low frequency value of 80 to less than 2 at
optical frequencies (~1014 Hz).
E588
emp10_04.doc
14 sep 10
4. 7
REFRACTIVE INDEX
Maxwell  prediction of electromagnetic waves
Electromagnetic waves  time-varying electric and magnetic fields whose directions are
mutually perpendicular. In unbounded dielectric media the waves are transverse.
Velocity of propagation of electromagnetic waves depends upon the electric (permittivity )
and magnetic (permeability ) properties of the medium. For an unbounded medium
1
v

For non-magnetic materials
1
1
v

 0
 r  0 0
For a vacuum
c
1
 0 0
The change in the velocity of the electromagnetic wave as it passes from one medium to
another is responsible for refraction. The refractive index n of non-magnetic materials is
n
  
c
 r 0 0
v
 0 0
n  r
Refraction  dispersion of light through a glass prism
When the frequency is comparable to the orbital frequency of the electrons in the material,
absorption and emission can take place – the index of refraction can display appreciable
frequency dependence, e.g., dispersion of visible light in passage through a glass prism. The
frequency for light is f ~ 1014 Hz.
emp10_04.doc
14 sep 10
4. 8
This prediction of Maxwell’s electromagnetic theory originally served as a basis for
criticizing the theory, for example DC values for water and air
water n = 1.3
 r  81  9
 n  r
It was not known at the time that water contained permanent polar molecules and as a result
the value of  r decreases with increasing frequency.
air
n = 1.000294
 r = 1.000295
 n  r
The polarization of the air molecules is entirely due to the displacement under the action of
the applied electric field of the electron clouds of their constituent atoms – since this
displacement occurs with great rapidity, r displays no frequency dependence.
emp10_04.doc
14 sep 10
4. 9
FORCES and ENERGY
Just as conductor is attracted into an electric field, so too is a dielectric. The bound charges
tend to accumulate near the free charge of opposite sign. But the calculation of forces on
dielectrics can be very tricky. For example, we assume that the electric field inside a parallel
plate capacitor is uniform and zero outside and the direction of the electric field is always
perpendicular to the plates. But a dielectric slab is drawn into the field region between the
plates, because in reality there is a fringe field around the edges and it is this non-uniform
electric field that pulls the dielectric into the capacitor.
Although it can be difficult, if not impossible, to calculate the forces directly using Newton’s
Laws, it is often a simple matter to derive expressions for the forces using the principle of
conservation of energy. The first step in this approach is to clearly identify the system and
secondly, how the work done by forces produces changes in the kinetic and potential energy
of the system. Assume that an applied external force Fme (the subscript me emphasizes the
application of the external force acting on the system) acts on the system to change only the
potential energy of the system without any change in the kinetic energy of the system. Then,
by the law of conservation of energy, the work done Wme on the system by an external force
Fme equals the change in potential energy Usystem of the system. For an incremental
displacement dy, we can write
dWme  Fme dy  dU system
Fme 
dU system
dy
The net force on the system must be zero because there is zero change in the kinetic energy of
the system. The internal force F that acts in the opposite direction to the external force Fme is
Fme   F
Force between the plates of a parallel plate capacitor
The charge on a conductor resides in a thin surface
Fme
layer. This is due to the mutual repulsion between
charges of like sign, so that the charges on the
conductor are trying to get as far away as possible
+ + + + + + + + +
from each other. Then, for a charged parallel plate
capacitor, there is an attractive force between the
+ + + + + + + + +
plates due to the thin layers of opposite charge on
each plate.
F
dy
- - - - - - - - -
emp10_04.doc
14 sep 10
4. 10
Consider a parallel plate capacitor with a medium of dielectric constant  r between the plates.
Q is kept constant (capacitor charged and disconnected from the battery) and the plate
separation y is increased by the application of an external force
Charge Q
Plate separation y  d
Capacitance C
constant
increases
decreases
Energy storage U
increases
Potential difference V
increases
Electric field E
constant
Force between plates
constant
C
r 0 A
y
y C
Q2
C  U 
2C
Q
Q
C
V
C  V 
V
C
Gauss’s Law Q = constant  E =
constant
F independent of y
F is in the direction of decreasing y
U
The system is the capacitor and the external force Fme acts on one plate to pull the plates apart
(increase the separation distance y) with zero change in the kinetic energy of the system. The
rate of change of the energy stored and the external force are
  A
Q2
U
C r 0
2C
y
dU dU dC

dy dC dy
dU
Q2

dC
2C2
  A
dC
C
  r 20  
dy
y
y

dU  Q 2  C   Q 2
y
Q2
 





 
dy  2 C 2   y   2  r  0 A y  2  r  0 A
dU/dy > 0 as expected. The external work done increases the stored potential energy. The
force between the plates is
dU
Q2
Q 2
Fme 

F   Fme 
dy 2 r  0 A
2 r  0 A
The minus sign for F denotes that the force acts in the opposite direction to the movement of
one plate, i.e. in the direction of decreasing y. The magnitude of the force is independent of
the plate separation y.
emp10_04.doc
14 sep 10
4. 11
V is kept constant (capacitor connected to the battery) and the plate separation y is
increased by the application of an external force
Potential difference V
Plate separation y  d
Capacitance C
constant
increases
decreases
Charge Q
decreases
Electric field E
decreases
Energy storage U
decreases
Force between plates
decreases
C
r 0 A
y
y C
Q
Q  CV
C Q
V
V
E
y E 
y
1
U  CV2
C  U 
2
F  1/y2
C
The system is the capacitor and the battery. The external force Fme acts on one plate to pull
the plates apart (increase the separation distance y) with zero change in the kinetic energy of
the system. For the capacitor, Q  C V and if the capacitance changes by dC then the charge
changes by dQ  V dC (V = constant)
  A
  A
  AV
dC
C r 0
  r 20
dQ   r 0 2
dy
y
dy
y
y
dQ is negative indicating that charge is transferred to the battery from the capacitor.
Therefore, energy is transferred to the battery to increase its potential energy
 r  0 AV 2
dU battery  V  dQ  
dy
y2
and the potential energy of the capacitor is decreased
 r  0 AV 2
1
 1    r  0 AV 
dU capacitor  V dQ   V   
dy   
dy
2
y2
2 y2
 2 

The total change in the potential energy of the system is
dU system  dU battery  dU cap 
 r  0 AV 2
 r  0 AV 2
 r  0 AV 2
dy 
dy 
dy
y2
2 y2
2 y2
The energy transferred to the battery is twice the energy lost by the capacitor. The application
of the external force Fme increases the total energy of the system. The external force is
dU system  r  0 AV 2
Fme 

dy
2 y2
and the attractive force between the plates is
F   Fme  
 r  0 AV 2
2 y2
Historically, the force to separate the plates of the capacitor was used to measure the potential
difference V.
emp10_04.doc
14 sep 10
4. 12
Removing or inserting a dielectric in a parallel plate capacitor
Consider the case of a slab of dielectric material with dielectric constant  r inserted between
the square plates of a parallel plate capacitor with an area A = L2. An external force Fme acts
to remove the dielectric without increasing its kinetic energy.
Q is kept constant (capacitor charged and disconnected from the battery) and the
dielectric is pulled out of the capacitor by the application of an external force
Charge Q
Capacitance C
constant
decreases
Energy storage U
increases
Potential difference V
increases
C
r 0 A
y
r   C 
Q2
U
C  U 
2C
Q
Q
C
V
C  V 
V
C
Dielectric is attracted to the
charged plates of the capacitor
Force between plates &
dielectric
The system is the capacitor. When the dielectric has been displaced by a distance x, the
capacitance C is equivalent to two capacitors in parallel
  L  L  x
 Lx
 L
C1  0
C2  r 0
C  C1  C2   0   x   r  L  x  
d
d
 d 
The change in capacitance for an incremental distance dx is
dC   0 L 
 L

dC   0  1   r  dx  0  r  1
 1   r   1
dx  d 
 d 
The capacitance decreases as the slab is withdrawn. The change in the potential energy of the
system (capacitor) is
1 Q2
U system 
2 C
dU system
dx
dU system
dx
2
  0L 
 Q 2  dC  Q 2  
d


 
 
 1   r 
2 
 2 C  dx  2     0 L   x   r  L  x     d 

Q 2 d 1   r 
2  0 L   x   r  L  x  
2
0
The potential energy of the capacitor increases as the dielectric is removed. The work done
by the external force in withdrawing the dielectric increases the potential energy of the
capacitor and the external force is given by
dU system
Q2 d 1   r 
Fme 

0
2
dx
2  0 L   x   r  L  x  
The direction of the external force Fme is in the direction of increasing x. The force F on the
dielectric due to the charge on the capacitor plates is attractive and opposes the withdrawal of
the dielectric (F = - Fme).
emp10_04.doc
14 sep 10
4. 13
V is kept constant (capacitor charged and connected to the battery) and the dielectric is
pulled out of the capacitor by the application of an external force
Potential difference V
Capacitance C
constant
decreases
Charge Q
Energy storage Ucap
decreases
Energy storage Ubattery
Energy storage Usystem
Force between plates &
dielectric
increases
increases
C
r 0 A
y
r   C 
Q  CV C   Q 
1
U  CV 2 C   U 
2
Charge transferred to battery
|dUbattery| > |dUcap|
Dielectric is attracted to the
charged plates of the capacitor
The system is the capacitor and the battery. When the dielectric has been displaced by a
distance x, the capacitance C is equivalent to two capacitors in parallel
  L  L  x
 Lx
 L
C1  0
C2  r 0
C  C1  C2   0   x   r  L  x  
d
d
 d 
The change in capacitance for an incremental distance dx is
dC   0 L 
 L

dC   0  1   r  dx  0  r  1
 1   r   1
dx  d 
 d 
The capacitance decreases as the slab is withdrawn.
The change in the potential energy of the capacitor is
1
U cap  CV 2
2
dU cap  V 2  dC   0 LV 2 
 

 1   r   0
dx
2
dx
2
d
 


When the dielectric slab is withdrawn, the capacitance and the potential energy of the
capacitor decrease. Therefore, charge must be transferred to the battery increasing its
potential energy
Q
  LV 
C
Q V C   0
  x   r  L  x 
V
 d 
dQ   0 LV 

 1   r   0
dx  d 
dU battery
  LV 2 
 V (dQ)    0
 1   r   0
dx
 d 
emp10_04.doc
14 sep 10
4. 14
The total change in the potential energy of the system is
dU system dU battery dU cap
  LV 2 
  0 LV 2 


  0
1




 1   r 
r 
dx
dx
dx
 d 
 2d 
  LV 2 
  0
 1   r   0
dx
 2d 
Again, the energy transferred to the battery is twice the energy lost by the capacitor.
dU system
The application of the external force Fme increases the total energy of the system and the
external force is
dU system
  LV 2 
Fme 
  0
 1   r   0
dx
 2d 
and the attractive force between the charged plates and dielectric is
  LV 2 
F   Fme   0
 1   r 
 2d 
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PARALLEL PLATE CAPACITOR WITH COMPOUND DIELECTRIC
When there is more than one dielectric between the capacitor plates (compound dielectric),
we often replace this capacitor by a number of capacitors (single dielectric) connected in a
series and/or parallel combination.
Consider a parallel plate capacitor with a thin dielectric slab of thickness t (t < d) midway
between the plates of the charged capacitor that is not connected to a battery. Compare the
following parameters for the capacitor without and without the dielectric slab: capacitance,
charge, potential difference and energy stored. Derive an expression for the thickness of the
dielectric slab.
Air filled capacitor
(subscript 0 refers to when the dielectric is not inserted)
0 A
Capacitance
C0 
Charge
Q0
Potential difference
V0 
Energy stored
Q2
U0 
2C
emp10_04.doc
d
Q0
C0
14 sep 10
4. 15
Capacitor with thin dielectric slab
Charge Q1 = Q0 = constant
(subscript 1 refers to when the dielectric is inserted)
Capacitor charged and disconnected from battery
The capacitance is
Q
C
V
To determine the capacitance we need to find the potential difference between the plates. We
can do this by integrating the electric field along a contour from the negative plate to the
positive plate V1   E dl    E0  d  t   E1 t   E0  d  t   E1 t


The effect of the dielectric is to reduce the electric field inside the dielectric by the factor r
E
E1  0
r
So the potential and capacitance are
t 

d t  


r
t 
 V0  V0
V1  E0  d  t    
r  
d






t 

d t  

Q
r

V1  0 
C0 
d





Q0
d

C0  C0
V1 
t 
d t  
r 

The energy stored is

t
d t 
2
2 
r
1 Q0
1 Q0  
U1 

2 C1 2 C0 
d


C1 


t
 d t 

r
  U  
0


d






  U
0



The capacitance values C0 and C1 can be measured to estimate the thickness t of the dielectric
 C  

t  1  0  r  1 d
 C1   r 
Alternatively, we can consider the system to be two capacitors in series (which gives the
same result as before):
emp10_04.doc
14 sep 10
4. 16
0 A
Cair 
C1 
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E761
d  t 
1
Cair
1
 1
E202
E797
Cdielectric 

Cdilectric
E288
E900
r 0 A
t
r 0 A
d  t   t 
r
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INTERESTING DIELECTRICS
Ferroic Materials cooperative solid state phenomena
Ferroics - is the generic name given to the study of ferromagnets, ferroelectrics, and
ferroelastics.
Ferromagnet - a material able to retain a permanent magnetic moment.
Ferroelectric - a material able to retain a permanent dipole moment.
Ferroelastic - a material able to retain a permanent shape after deformation (e.g. shape
memory alloys)
Ferroelectrics
Crystalline dielectric materials – permanent electric polarization
Rochelle salt, potassium dihydrogen phosphate and barium titanate
+
+
-
+
-
Ferroelectric material
-
+
+
-
+
-
-
+
+
-
+
+
+
-
+
+
-
+
+
+
-
+
+
-
+
+
+
-
+
-
+
+
Antif erroelectric material
Electrets Isotropic dielectric material - susceptibility is not
constant – e.g. electrets microphones. An electret is a stable
dielectric material with a permanently-embedded static
electric charge (which, due to the high resistance and
chemical stability of the material, will not decay for
hundreds of years). Electrets are commonly made by first
melting a suitable dielectric material such as a plastic or wax
that contains polar molecules, and then allowing it to resolidify in a powerful electrostatic field. The polar molecules
of the dielectric align themselves to the direction of the
electrostatic field, producing a permanent electrostatic
"bias". Modern electret microphones use PTFE plastic, either
in film or solute form, to form the electret. An electret
microphone is a type of condenser microphone, which
eliminates the need for a polarizing power supply by using a permanently-charged material.
emp10_04.doc
14 sep 10
4. 17
Piezoelectricity generate an electric charge in a material when subjecting
it to a mechanical stress, conversely, generate a mechanical strain in response to an applied
electric field.
Crystals that lack a centre of symmetry of ion distribution are electrically polarized (i.e. they
develop surface charges) when they are mechanically stressed.
Crystalline dielectric materials – mechanical pressure exerted upon the crystal results in
appearance of electric charge along its surface (polarization effect)
mechanical stress  emf
emf  mechanical stress – change in dimensions of crystal
+
Zero applied stress
Compressive stress
Induces a voltage
Applied voltage produces
An expansion
Quartz crystal – mechanically vibrating  ac voltage across its face – frequency of
mechanical vibration depends upon the dimensions and other parameters of the crystal  ac
voltage generated possesses a very constant frequency.
Quartz is piezoelectric but not ferroelectric. It has a piezoelectric constant of
2.2×10-12 C.N-1 or V.m-1. If you apply a force of 1 N perpendicular to the axis of the quartz
crystal, you get a charge of 2.2 pC; if you apply a volt, the length changes by 2.2×10 -12 m.
The piezoelectric constants for barium titanate and potassium sodium tartrate (Rochelle salt)
(ferroelectric) may be hundreds of times greater but are quite variable. These materials are
widely used in transducers. If you apply a voltage to a quartz crystal at a frequency
corresponding to a mode of mechanical oscillation, the resulting resonance is stable and
precise.
ac voltage applied to the faces of crystal – if frequency of the voltage is identical to natural
frequency of crystal’s mechanical vibrations  crystal vibrate “violently”  ultrasonic
waves
Transducer: electrical energy  mechanical energy.
emp10_04.doc
14 sep 10
4. 18
Pyroelectricity - migration of positive and negative charge (and therefore establishment of
electric polarization) to opposite ends of a crystal's polar axis as a result of a change in
temperature. Some of crystals that lack a centre of symmetry of ion distribution, (i.e.
piezoelectrics) can also spontaneously develop electric dipoles (polarize), with the degree of
polarization dependent on temperature. Crystal heated or cooled  changes in charge at the
surface – these are piezoelectric changes from the strain associated with thermal expansion or
contraction.
Reference http://en.wikipedia.org/wiki/Ferroelectricity
Polarization P  e  0 E   r  1  0 E
Most materials are polarized linearly with an external electric
field; nonlinearities are insignificant. This is called dielectric
polarization.
Some materials, known as paraelectric materials, demonstrate a more pronounced nonlinear
polarization. The electric permittivity, corresponding to the
slope of the polarization curve, is thereby a function of the
external electric field. In addition to being nonlinear,
ferroelectric materials demonstrate a spontaneous (zero field)
polarization. Such materials are generally called pyroelectrics.
The distinguishing feature of ferroelectrics is that the direction
of the spontaneous polarization can be reversed by an applied
electric field, yielding a hysteresis loop.
Typically, materials demonstrate ferroelectricity only below a
certain phase transition temperature, called the Curie
temperature TC, and are paraelectric above this temperature.
E893
E040
E600
E900
E095
E617
E906
emp10_04.doc
E105
E664
E202
E696
E232
E704
E288
E761
14 sep 10
E382
E797
E456
E851
E567
E890
E588
E893
4. 19