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EMP 1
DIELECTRICS – MACROSCOPIC VIEW
Reference: Young & Freedman Chapter 24 Capacitance & Dielectrics
Dielectric are insulators – charges tend not to move easily in non-metallic solids
How does an object acquire a charge?
Why does a charged object attract a neutral object?
How do we investigate the electrical properties of an insulator or dielectric material?
How does a dielectric change the capacitance?
What is meant by potential difference (voltage)?
What produces an electric field?
BACKGROUND CAPACITORS
Capacitors – two conducting plates separated by an dielectric
Capacitors – 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
Capacitor – stores charge on conducting plates, stores electric potential energy due to the
work done is separating the charges.
Capacitance –“ability” to store charge
Q
C
V
For a parallel plate capacitor
C
r 0 A
d

 A capacitance only depends upon the
d geometry and dielectric
Capacitors in series (charge on
each plate is the same)
1
Ctotal 
1
1

 ...
C1 C2
ijcooper/physics/p2/em/emp_01.doc
Capacitors in parallel (voltage across each
capacitor is the same)
Ctotal  C1  C2  ...
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EMP1.1
CHARGED CAPACITOR NOT CONNECTED TO A BATTERY
(Q = Qfree = constant)
Parallel plate capacitor – no dielectric (vacuum or air)
How is the charge located on the surfaces of the conducting plates?
area of plates A
+ + + + + + + + + + + +
+Qfree on inner surface
plate separation d
-Qfree on inner surface
- - - - - - - - - - - -
Charge on plates  electric field between plates
E  V
 Ex  
E=V/d
V
x
f
V  V f  Vi    E  dl
i
V=Ed
Gauss’s Law – the total electric flux through any closed surface is equal to the net charge
enclosed within the surface.
E   E  dA 
E A
E
Qenclosed
o
V A Q free

d
0
 free
0
V

Q free
o
 C
d  free
0
Q free
V

0 A
d
q free   free A
Energy stored and energy density (energy stored by the field) Q  Qfree
1 Q2 1
1
1
1
U
 CV 2  QV   0 E 2 Ad  u   0 E 2
2 C 2
2
2
2
Proof
Work done to charge capacitor
q dq
1 Q
1 Q2
dW  v dq 
W   q dq 
C
C 0
2 C
The operation of assembling upon a conductor a group of charges that mutually repel one
another requires work and therefore results in the production of potential energy - this
potential energy is possessed by the charged conductor itself but it may be more correct to
picture the energy stored in the field surrounding the conductor.
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EMP1.2
Electric displacement or electric flux density D
is a useful quantity, since its value only depends upon the density of free charges and
in not changed by the introduction of the dielectric.
D  0 E
 D
Gauss’s Law
dA    free dv  Q free
 D   free
Parallel plate capacitor
– isotropic dielectric inserted that fills space between plates
Fixed charge on plates of capacitor (Q = Qfree same value without or with dielectric inserted)
Dielectric fills space between the capacitor plates.
dielectric constant (relative permeability) r
permeability of dielectric  = r 0
no dielectric – subscript 0
with dielectric – subscript
r  1
d
Capacitance only depends upon the geometry and dielectric
C0 
0 A
d
Cd 
r 0 A
d

A
d
Cd  C0
C=Q/V
Insertion of dielectric  C increases  reduction in V and E since Q = constant 
redistribution of the charges in the dielectric material (polarization).
r  K 
Vd 
V0
r
Cd
1
C0
Ed 
E0
r
D  Dd  D0   free   f
ijcooper/physics/p2/em/emp_01.doc
voltage & electric field decrease by the factor r
electric displacement remains constant
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EMP1.3
Dielectric increases the breakdown voltage (dielectric strength)  can use a larger voltage
before a disruptive discharge occurs.
Dielectric constant (  r  K) and dielectric strength
Dielectric Dielectric
constant
strength
V.m-1
Air
1.00058
~ 3106
Water
80
Mica
7-8
Polystyrene
2.5
Titanium dioxide ceramics
15 - 500
~ 2107
Barium titanate
500 - 6000 ~ 2106
How can we explain the reduction in the electric field between the capacitor plates?
Reduced electric field – net charges induced on the surface of the dielectric (dielectric still
neutral) – polarization.
P   bound   b
E  Ed 
Ed 
1
0
1
0
D   free   f
1
( D  P) 
0
( free   bound ) 
( free   bound )
E0
r


1
  bound   free  1  
 r 
 D   r  0 Ed   Ed
Lines of D connect
free charges (positive to
negative)
Lines of P connect
bound charges
(negative to positive)
Lines of E connect net
charges = free & bound
(positive to negative)
In isotropic materials:
D , E and P all have
the same direction
 free  free

r 0

Vd 
 free d
r 0
Cd 
r 0 A
d
 bound   free
+Qfree on inner surface
P
+ + + + + + + + + + + +
- - - - - - -qbound
Symmetry
– fields must be uniform
– field lines perpendicular to plates
+qbound
+ + + + + +
- - - - - - - - - - - -
D
-Qfree on inner surface
ijcooper/physics/p2/em/emp_01.doc
Interior points electric
field must be zero
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E
E
1
0
( D  P)
EMP1.4
conductor
+
- dielectric
Gauss’s Law
+
E 0
+
-
 free  bound
+
-
E

free
  bound 
0
E
+
Isotropic materials – assume polarization P proportional to the electric field
P  e  0 Ed
  r  1  e
Maxwell’s displacement current
Switched closed – current through conductors to charge capacitor. Maxwell showed that it is
necessary to assume a current of the same value also flowed in the space between the
capacitor plates.
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.
Parallel plate capacitor with compound dielectric (Q = Qfree = constant)
Portion of the inter-plate region occupied by a dielectric slab of thickness t, the rest of the
region being air filled
C
0 A
d  t  
t
r
How do you prove this?
ijcooper/physics/p2/em/emp_01.doc
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EMP1.5
Forces between parallel plates / energy in electric field (Q = constant)
The charges on the plates are all on the inner
surfaces  unbalanced electrical attractive force F
between the plates.
Suppose the top plate is raised a distance dy by an
external force so that there is no increase in kinetic
energy  |F| = |external force| = |attractive force
between the plates|. The change in |potential energy|
is equal to |work done| by the force necessary to
raise the plate
|dU| = |F dy|
+ + + + + + + + +
dy
+ + + + + + + + +
F
- - - - - - - - -
|F| = |dU/dy|
1 Q2 1
  r  0 AQ 2 y
2 C 2
dU  r  0 AQ 2
F 

dx
2 y2
U
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EMP1.6
DIELECTRIC BEHAVIOR
Insulator (dielectric) – will not permit a current (passage of charge) although local
microscopic displacements of charge may take place.
All atoms or molecules can be temporarily polarized under the action of an electric field.
Permanent polar molecules (H+Cl-) may exist in a dielectric.
Separation of charge  electric dipole
A dipole consists of two equal and opposite charges + and –q separated by a vector distance d
dipole moment p = pe = q d
pe  qd
d
points from negative to positive
-q
+q
Induced dipole moment – helium atom
E
-e
+2e
Zero electric field –
helium atom
symmetric  zero
dipole moment
-e
-e
+2e
A
-e
d
B
effectively charge +2e at A and -2e at B
dipole moment
p = 2ed
p
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EMP1.7
Potential and electric field from an electric dipole
V ( P) 
1 q q
q  r2  r1 
  


4   r1 r2  4   r 1r2 
Er
E
r  d
V ( P) 

P
q d cos 


d2
4   r 2  cos2  
4


r2  r + (d/2)cos
q d cos  p cos 
p r


2
2
4  r
4  r
4  r 3
r1  r – (d/2)cos
r
r extends from the centre of the
dipole to the point P
(d/2)cos
The radial and tangential components
of the field at point P are

V 2 p cos 

r
4  r 3
1 V p sin 
E  

r  4  r 3
-q
Er  
along the axis of the dipole
along the right bisector of the dipole
d
+q
 = 0  E = 0
 = /2  Er = 0
Electric field approaches zero much more quickly than a point charge. ??? Why ?
See Matlab plot page 10
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EMP1.8
Electric polarization
The extent to which permanent or induced dipoles become aligned is described by the
electric polarization P
dpe
 n pe
dv
where n is the number density for the electric dipoles (number of dipoles per unit volume)
electric dipole moment per unit volume
P
Consider, polarization of the dielectric between the plates of a charged parallel plate capacitor
dA
+f
+ + + + + + + + +



d

Throughout the body of the dielectric, the charges on
adjacent ends of the polar molecules neutralize one
another. At both the top and bottom of the dielectric
the charges do not neutralize each other  bound
surface charges b .
For a cylinder of the dielectric of cross-sectional area
dA extending from one plate to the other 
-f
-b
+b
- - - - - - - - -
electric dipole moment pe = q d
dpe  ( b dA)d
P
dpe dpe ( b dA)d


dv dAd
dAd
P  b
P   b nˆ
where n̂ is the normal outward pointing unit vector.
Polarization equals the magnitude of the bound (induced) charge per unit area on the surface
of the dielectric.
Also, the polarization can be obtained through the relationship
 b   P
where  b is the volume density of the bound charges.
Number density
The number density for a gas is obtained from the ideal gas equation
pV  N k B T
p
N
kB T  n kB T
V
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n
p
kB T
EMP1.9
For a gas at atmospheric pressure and 20 oC, the number density n is
p = 1 atm = 1.013105 Pa T = 20 oC = 293 K
kB = 1.381023 J.K-1
n = 2.51023 molecules.m-3
For a solid or liquid (density , molecular mass m, number of molecules N,mass of sample
msample) the number density n is obtained as follows

msample
n
V

Nm
V
M  NA m
m
M
NA
n
N
V
 n
M
NA
NA
M
Avogadro’s number NA = 6.021023 molecules.mol-1
Molar mass M (in kilograms)
For copper
 = 8.93103 kg.m-3
M = 63.5 g = 63.510-3 kg
n = 8.51028 atoms.m-3
Note: the number density of solids is much greater than that of gases.
Electric susceptibility  e
For isotropic dielectrics, (electrical properties identical in all directions) the polarization that
occurs due to the applied electric field has the same direction as the field. Also, the
magnitude of the polarization is proportional to the field
P  e  E
where  e is the constant of proportionality, known as the electric susceptibility.
For anisotropic dielectric P and E are not in the same direction and  e is not a constant but
a tensor.
Electrets Isotropic dielectric material - susceptibility is not constant – e.g. electrets
microphones
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EMP1.10
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. 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 have time to orient themselves to the same
extent before the polarity changes. For this reason, in a material that possesses permanent
polar 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 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).
dielectric
constant
frequency
ijcooper/physics/p2/em/emp_01.doc
2 May 2017
EMP1.11
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 em waves depends upon the electric and magnetic 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 its velocity as it passes from one medium to another is responsible to
refraction. Refractive index n (non-magnetic materials)
n
  
c
 r 0 0
v
 0 0
n  r
This prediction of Maxwell’s electromagnetic theory originally served as a basis for
criticizing the theory, for example DC values for air and water
air
n = 1.000294
water n = 1.3
 r = 1.000295
 n  r
 r = 8.9
 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. 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.
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.
ijcooper/physics/p2/em/emp_01.doc
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EMP1.12
INTERESTING DIELECTRICS
Ferroelectricity
crystalline dielectric materials – permanent electric polarization (analogy with ferro-magnetic
materials – it has nothing to do with iron)
Rochelle salt, potassium dihydrogen phosphate and barium titanate
Piezoelectricity
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
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.
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.
Pyroelectricity
crystal heated or cooled  changes in charge at the surface – these are piezoelectric changes
from the strain associated with thermal expansion or contraction.
ijcooper/physics/p2/em/emp_01.doc
2 May 2017
EMP1.13
ELECTRIC DIPOLE PLOT– MATLAB
Why is difficult to plot the potential in a plane passing through the axis of the dipole?
Potential: Electric Dipole
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
% electric_dipole.m
% Ian Cooper School of Physics,University of Sydney
close all
clear all
clc
% emconstants ------------------------------------------------------c = 3.00e-8;
% speed of light
e = 1.602e-19;
% elementary charge
eps0 = 8.85e-12;
% permittivity of free space
NA = 6.02e23;
% Avogadro constant
me = 9.11e-31;
% electron rest mass
mp = 1.673e-27;
% proton rest mass
mn = 1.675e-27;
% neutron rest mass
h = 6.626e-34;
% Planck's constant
kB = 1.38e-23;
% Boltzmann's constant
kC = 8.988e9;
% Coulomb constant
mu0 = 4*pi*1e-7;
% permeability of free space
amu = 1.66e-27;
% atomic mass unit
% Setup ------------------------------------------------------------q = e;
% dipole charge
d = 1.6795e-018;
% dipole separation distance
q1 = q; q2 = -q;
% separated charges
kc = 1/(4*pi*eps0);
% constant in Coulomb's Law
x1 = d/2; x2 = -d/2;
% position of dipole
y1 = 0; y2 = 0;
scale = 1.25;
% plotting region
xmax = scale * d;
ymax = xmax; xmin = -xmax; ymin = -ymax;
% plane above diople
num = 100;
x = linspace(xmin,xmax,num);
y = x;
[xx yy] = meshgrid(x,y);
r1 = sqrt((xx-x1).^2 + (yy-y1).^2);
ijcooper/physics/p2/em/emp_01.doc
% distance from charges
2 May 2017
EMP1.14
% to test point to calc. potential
r2 = sqrt((xx-x2).^2 + (yy-y2).^2);
V1 = kc .* q1 ./ (r1);
% potential from each charge
V2 = kc .* q2 ./ (r2);
Vtot = V1 + V2;
Vmax = max(max(Vtot));
sat = 0.5;
% saturate the potential
Vtot(Vtot > sat*Vmax) = sat * Vmax;
% potential near a charge
%
is extremely large
Vtot(Vtot < -0.5*Vmax) = -sat * Vmax;
Vtot = Vtot/(max(max(Vtot)));
figure(2);
% [3D] plot
surf(xx/d,yy/d,Vtot,'FaceColor','interp',...
'EdgeColor','none',...
'FaceLighting','phong')
daspect([1 1 1])
axis tight; view(-45,20)
camlight left; colormap(jet)
grid off; axis off
colorbar
title('Potential: Electric Dipole')
ijcooper/physics/p2/em/emp_01.doc
2 May 2017
EMP1.15