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Engineering Electromagnetics
Lecture 8
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
President University
Erwin Sitompul
EEM 8/1
Chapter 5
Current and Conductors
The Method of Images
 One important characteristic of the dipole field developed in
Chapter 4 is the infinite plane at zero potential that exists
midway between the two charges.
 Such a plane may be represented by a thin infinite conducting
plane.
 The conductor is an equipotential surface at a potential V = 0.
The electric field intensity, as for a plane, is normal to the
surface.
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Erwin Sitompul
EEM 8/2
Chapter 5
Current and Conductors
The Method of Images
 Thus, we can replace the dipole configuration (left) with the
single charge and conducting plane (right), without affecting
the fields in the upper half of the figure.
 Now, we begin with a single charge above a conducting plane.
► The same fields above the plane can be maintained by
removing the plane and locating a negative charge at a
symmetrical location below the plane.
 This charge is called the image of the original charge, and it is
the negative of that value.
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Erwin Sitompul
EEM 8/3
Chapter 5
Current and Conductors
The Method of Images
 The same procedure can be done again and again.
 Any charge configuration above an infinite ground plane may
be replaced by an arrangement composed of the given charge
configuration, its image, and no conducting plane.
President University
Erwin Sitompul
EEM 8/4
Chapter 5
Current and Conductors
The Method of Images
 Example
Find the surface charge density at P(2,5,0) on the conducting
plane z = 0 if there is a line charge of 30 nC/m located at x = 0,
z = 3, as shown below.
• We remove the plane and
install an image line charge
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• The field at P may now be
obtained by superposition of
the known fields of the line
charges
Erwin Sitompul
EEM 8/5
Chapter 5
Current and Conductors
The Method of Images
R   2a x  3a z
R   2a x  3a z
30 109 2a x  3a z
L
E 
aR 
2 0 R
2 0 13
13
30 109 2a x  3a z
L
E 
aR 
2 0 R
2 0 13
13
E  E  E
180 109

az
2 0 (13)
x = 0, z = 3
P(2,5,0)
x = 0, z = –3
D   0E  2.20a z nC m 2
S  DN  2.20nC m2 at P
 249a z V m
• Normal to the plane
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Erwin Sitompul
EEM 8/6
Chapter 5
Current and Conductors
Semiconductors
 In an intrinsic semiconductor material, such as pure
germanium or silicon, two types of current carriers
are present: electrons and holes.
 The electrons are those from the top of the filled
valence band which have received sufficient energy
to cross the small forbidden band into conduction
band.
 The forbidden-band energy gap in typical semiconductors is of
the order of 1 eV.
 The vacancies left by the electrons represent unfilled energy
states in the valence band. They may also move from atom to
atom in the crystal.
 The vacancy is called a hole, and the properties of
semiconductor are described by treating the hole as a positive
charge of e, a mobility μh, and an effective mass comparable to
that of the electron.
President University
Erwin Sitompul
EEM 8/7
Chapter 5
Current and Conductors
Semiconductors
 The conductivity of a semiconductor is described as:
    e e   h  h
 As temperature increases, the mobilities decrease, but the
charge densities increase very rapidly.
 As a result, the conductivity of silicon increases by a factor of
100 as the temperature increases from about 275 K to 330 K.
President University
Erwin Sitompul
EEM 8/8
Chapter 5
Current and Conductors
Semiconductors
 The conductivity of the intrinsic semiconductor increases with
temperature, while that of a metallic conductor decreases with
temperature.
 The intrinsic semiconductors also satisfy the point form of
Ohm's law: the conductivity is reasonably constant with current
density and with the direction of the current density.
J E
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Erwin Sitompul
EEM 8/9
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 A dielectric material in an electric field can be viewed as a freespace arrangement of microscopic electric dipoles, a pair of
positive and negative charges whose centers do not quite
coincide.
 These charges are not free charges, not contributing to the
conduction process. They are called bound charges, can only
shift positions slightly in response to external fields.
 All dielectric materials have the ability to store electric energy.
This storage takes place by means of a shift (displacement) in
the relative positions of the bound charges against the normal
molecular and atomic forces.
President University
Erwin Sitompul
EEM 8/10
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 The mechanism of this charge displacement differs in various
dielectric materials.
 Polar molecules have a permanent displacement existing
between the centers of “gravity” of the positive and negative
charges, each pair of charges acts as a dipole.
 Dipoles are normally oriented randomly, and the action of the
external field is to align these molecules in the same direction.
 Nonpolar molecules does not have dipole arrangement until
after a field is applied.
 The negative and positive charges shift in opposite directions
against their mutual attraction and produce a dipole which is
aligned with the electric field.
President University
Erwin Sitompul
EEM 8/11
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 Either type of dipole may be described by its dipole moment p:
p  Qd
 If there are n dipoles per unit volume, then there are nΔv
dipoles in a volume Δv. The total dipole moment is:
nv
p total   pi
i 1
 We now define the polarization P as the dipole moment per
unit volume:
1 nv
P  lim
pi  np  nQd

v  0 v
i 1
 The immediate goal is to show that the bound-volume charge
density acts like the free-volume charge density in producing
an external field ► We shall obtain a result similar to Gauss’s
law.
President University
Erwin Sitompul
EEM 8/12
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 Take a dielectric containing nonpolar molecules. No molecules
has p, and P = 0 throughout the material.
 Somewhere in the interior of the dielectric we select an
incremental surface element ΔS, and apply an electric field E.
 The electric field produces a moment p = Qd in each molecule,
such that p and d make an angle θ with ΔS.
 Due to E, any positive charges initially lying below the surface
ΔS and within ½dcosθ must have crossed ΔS going upward.
 Any negative charges initially lying above the surface ΔS and
within ½dcosθ must have crossed ΔS going downward.
President University
Erwin Sitompul
EEM 8/13
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 For n molecules/m3, the net total charge (positive and
negative) which crosses the elemental surface in upward
direction is:
Qb  nQd cos  S
Qb  nQd  S
 The notation Qb means the bound charge. In terms of the
polarization, we have:
Qb  P  S
 If we interpret ΔS as an element of a closed
surface, then the direction of ΔS is outward.
 The net increase in the bound charge within
the closed surface is:
Qb    P  dS
S
President University
Erwin Sitompul
EEM 8/14
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 Seeing some similarity to Gauss’s law, we may now generalize
the definition of electric flux density so that it applies to media
other than free space.
 We write Gauss’s law in terms of ε0E and QT, the total enclosed
charge (bound charge plus free charge):
QT 

S
 0E  dS
QT  Qb  Q
 Combining the last three equations:
Q  QT  Qb 

S
( 0E  P)  dS
 We may now define D in more general terms:
D   0E  P
• There is an added term to D when a
material is polarized
President University
Erwin Sitompul
EEM 8/15
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 For equations with volume charge densities, we now have:
Qb   b dv
v
Q   v dv
v
QT   T dv
v
 With the help of the divergence theorem, we may transform the
equations into equivalent divergence relationships:
  P   b
  D  v
   0E  T
President University
Erwin Sitompul
EEM 8/16
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 To utilize the new concepts, it is necessary to know the
relationship between E and P.
 This relationship will be a function of the type of material. We
will limit the discussion to isotropic materials for which E and P
are linearly related.
 In an isotropic material, the vectors E and P are always
parallel, regardless of the orientation of the field.
 The linear relationship between P and E can be described as:
P   e 0 E
D   0E  e 0E  ( e  1) 0E
 We now define:
 r  e  1
D   0 r E   E
χe : electric susceptibility,
a measure of how easily
a dielectric polarizes in
response to an electric field
   0 r
εr : relative permittivity
President University
Erwin Sitompul
EEM 8/17
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 In summary, we now have a relationship between D and E
which depends on the dielectric material present:
D  E
   0 r
  D  v

S
D  dS  Q
President University
Erwin Sitompul
EEM 8/18
Chapter 6
Dielectrics and Capacitance
The Nature of Dielectric Materials
 Example
We locate a slab of Teflon in the region 0 ≤ x ≤ a, and assume
free space where x < 0 and x > a. Outside the Teflon there is a
uniform field Eout = E0ax V/m. Find the values for D, E, and P
everywhere.
Eout  E0a x
Dout   0 E0a x
Pout  0 • No dielectric materials outside 0 ≤ x ≤ a
 r ,teflon  2.1  e  2.1  1  1.1
Din  2.1 0Ein
Pin  1.1 0Ein
• No relations yet established over the boundary
• This will be discussed in the next section
President University
Erwin Sitompul
EEM 8/19
Chapter 6
Dielectrics and Capacitance
Boundary Conditions for Perfect Dielectric Materials
 Consider the interface between two dielectrics having
permittivities ε1 and ε2, as shown below.
 We first examine the tangential components around the small
closed path on the left, with Δw<< and Δh<<< :
 E  dL  0
Etan1w  Etan 2 w  0
Etan1  Etan 2
President University
Erwin Sitompul
EEM 8/20
Chapter 6
Dielectrics and Capacitance
Boundary Conditions for Perfect Dielectric Materials
 The tangential electric flux density is discontinuous,
Dtan1
Dtan 2
 Etan1  Etan 2 
1
Dtan1 1

Dtan 2  2
2
 The boundary conditions on the normal components are found
by applying Gauss’s law to the small cylinder shown at the right
of the previous figure (net tangential flux is zero).
DN 1S  DN 2 S  Q  S S
DN 1  DN 2  S
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• ρS cannot be a bound surface charge
density because the polarization
already counted in by using dielectric
constant different from unity
• ρS cannot be a free surface charge
density, for no free charge available in
the perfect dielectrics we are
considering
• ρS exists only in special cases where
it is deliberately placed there
Erwin Sitompul
EEM 8/21
Chapter 6
Dielectrics and Capacitance
Boundary Conditions for Perfect Dielectric Materials
 Except for this special case, we may assume ρS is zero on the
interface:
DN 1  DN 2
 The normal component of electric flux density is continuous.
 It follows that:
 1 EN 1   2 EN 2
President University
Erwin Sitompul
EEM 8/22
Chapter 6
Dielectrics and Capacitance
Boundary Conditions for Perfect Dielectric Materials
 Combining the normal and the tangential
components of D,
DN 1  D1 cos 1  D2 cos2  DN 2
Dtan1 D1 sin 1 1


Dtan 2 D2 sin  2  2
 2 D1 sin 1  1D2 sin 2
 After one division,
tan 1 1

tan  2  2
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1   2  1  2
Erwin Sitompul
EEM 8/23
Chapter 6
Dielectrics and Capacitance
Boundary Conditions for Perfect Dielectric Materials
 The direction of E on each side of
the boundary is identical with the
direction of D, because D = εE.
E1
 1 EN 1   2 EN 2
Etan1  Etan 2
1   2  1  2
E2
President University
Erwin Sitompul
EEM 8/24
Chapter 6
Dielectrics and Capacitance
Boundary Conditions for Perfect Dielectric Materials
 The relationship between D1 and D2 may be derived as:
2
D2  D1 cos 1 
1
2
 
2
sin 2 1
 The relationship between E1 and E2 may be derived as:
1
2
E2  E1 sin 1 
2
 
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2
cos 2 1
Erwin Sitompul
EEM 8/25
Chapter 6
Dielectrics and Capacitance
Boundary Conditions for Perfect Dielectric Materials
 Example
Complete the previous example by finding the fields within the
Teflon.
Eout  E0a x
Dout   0 E0a x
Pout  0
• E only has normal
component
Din  Dout   0 E0a x
 0 E0 a x
Din

Ein 
 0.476 E0a x
 r 0
 r 0
 0 E0a x
Pin  1.1 0Ein  1.1 0
 0.524 0 E0a x
 r 0
President University
Erwin Sitompul
EEM 8/26
Chapter 6
Dielectrics and Capacitance
Boundary Conditions Between a Conductor and a Dielectric
 The boundary conditions existing at the interface between a
conductor and a dielectric are much simpler than those
previously discussed.
 First, we know that D and E are both zero inside the conductor.
 Second, the tangential E and D components must both be zero
to satisfy:
 E  dL  0
D  E
 Finally, the application of Gauss’s law shows once more that
both D and E are normal to the conductor surface and that
DN = ρS and EN = ρS/ε.
 The boundary conditions for conductor–free space are valid
also for conductor–dielectric boundary, with ε0 replaced by ε.
Dt  Et  0
DN   EN  S
President University
Erwin Sitompul
EEM 8/27
Chapter 6
Dielectrics and Capacitance
Boundary Conditions Between a Conductor and a Dielectric
 We will now spend a moment to examine one phenomena:
“Any charge that is introduced internally within a conducting
material will arrive at the surface as a surface charge.”
 Given Ohm’s law and the continuity equation (free charges
only):
J E
v
J  
t
 We have:
v
  E  
t
 v

 D  

t
President University
Erwin Sitompul
EEM 8/28
Chapter 6
Dielectrics and Capacitance
Boundary Conditions Between a Conductor and a Dielectric
 If we assume that the medium is homogenous, so that σ and ε
are not functions of position, we will have:
 v
D  
 t
 Using Maxwell’s first equation, we obtain;
 v
v  
 t
 Making the rough assumption that σ is not a function of ρv, it
leads to an easy solution that at least permits us to compare
different conductors.
 The solution of the above equation is:
v  0e (  )t • ρ0 is the charge density at t = 0
• Exponential decay with time constant of ε/σ
President University
Erwin Sitompul
EEM 8/29
Chapter 6
Dielectrics and Capacitance
Boundary Conditions Between a Conductor and a Dielectric
 Good conductors have low time constant. This means that the
charge density within a good conductors will decay rapidly.
 We may then safely consider the charge density to be zero
within a good conductor.
 In reality, no dielectric material is without some few free
electrons (the charge density is thus not completely zero), but
the charge introduced internally in any of them will eventually
reach the surface.
ρv
ρ0
v  0e (  )t
ρ0/e
ε/σ
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Erwin Sitompul
t
EEM 8/30
Chapter 6
Dielectrics and Capacitance
Homework 7
 D5.6.
 D5.7.
 D6.1.
 D6.2. (Bonus Question, + 20 points if correctly made)
 Deadline: 12.06.2012, at 08:00 am.
President University
Erwin Sitompul
EEM 8/31