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Engineering Electromagnetics
Lecture 7
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
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Erwin Sitompul
EEM 7/1
Engineering Electromagnetics
Chapter 5
Current and Conductors
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EEM 7/2
Chapter 5
Current and Conductors
Current and Current Density
 Electric charges in motion constitute a current.
 The unit of current is the ampere (A), defined as a rate of
movement of charge passing a given reference point (or
crossing a given reference plane).
dQ
I
dt
 Current is defined as the motion of positive charges, although
conduction in metals takes place through the motion of
electrons.
 Current density J is defined, measured in amperes per square
meter (A/m2).
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Erwin Sitompul
EEM 7/3
Chapter 5
Current and Conductors
Current and Current Density
 The increment of current ΔI crossing an incremental surface
ΔS normal to the current density is:
I  J N S
 If the current density is not perpendicular to the surface,
I  J S
 Through integration, the total current is obtained:
I   J  dS
S
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Erwin Sitompul
EEM 7/4
Chapter 5
Current and Conductors
Current and Current Density
 Current density may be related to the velocity of volume
charge density at a point.
• An element of charge ΔQ = ρvΔSΔL moves
along the x axis
• In the time interval Δt, the element of charge
has moved a distance Δx
• The charge moving through a reference
plane perpendicular to the direction of
motion is ΔQ = ρvΔSΔx
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Erwin Sitompul
x
Q
 v S
I 
t
t
EEM 7/5
Chapter 5
Current and Conductors
Current and Current Density
 The limit of the moving charge with respect to time is:
I  v Svx
 In terms of current density, we find:
J x   v vx
J  v v
 This last result shows clearly that charge in motion constitutes
a current. We name it here convection current.
 J = ρvv is then called convection current density.
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Erwin Sitompul
EEM 7/6
Chapter 5
Current and Conductors
Continuity of Current
 The principle of conservation of charge:
“Charges can be neither created nor destroyed.”
 But, equal amounts of positive and negative charge (pair of
charges) may be simultaneously created (obtained) by
separation or destroyed (lost) by recombination.
I

S
J  dS
• The Continuity Equation in Closed Surface
 Any outward flow of positive charge must be balanced by a
decrease of positive charge (or perhaps an increase of
negative charge) within the closed surface.
 If the charge inside the closed surface is denoted by Qi, then
the rate of decrease is –dQi/dt and the principle of
conservation of charge requires:
dQi
I   J  dS  
• The Integral Form of the Continuity Equation
S
dt
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Erwin Sitompul
EEM 7/7
Chapter 5
Current and Conductors
Continuity of Current
 The differential form (or point form) of the continuity equation is
obtained by using the divergence theorem:

S
J  dS   ( J)dv
vol
 We next represent Qi by the volume integral of ρv:
d
vol (  J )dv   dt vol v dv
 If we keep the surface (and thus the enclosed volume)
constant, the derivative becomes a partial derivative,
v
vol (  J)dv  vol  t dv
v
(  J )v  
v
t
v
J  
• The Differential Form (Point Form)
of the Continuity Equation
t
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Erwin Sitompul
EEM 7/8
Chapter 5
Current and Conductors
Continuity of Current
 Example
1 t
The current density is given by J  e ar A m2 .
r
• Total outward current at time instant t = 1 s and r = 5 m.
I  J r Sr  ( 15 e1ar )(4 52 ar )  23.11 A
• Total outward current at time instant t = 1 s and r = 6 m.
I  J r Sr  ( 16 e1ar )(4 62 ar )  27.74 A
• Finding volume charge density:
v
1
1  2 1 t

 J  2
(r e )  2 e  t
t
r
r r
r
1 t
1 t
v    2 e dt  2 e  K (r )
r
r
t  , v  0
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 K (r )  0  v 
Jr
1 t
3

v

rm s
e C m
r
2
v
r
Erwin Sitompul
EEM 7/9
Chapter 5
Current and Conductors
Reading Only
Metallic Conductors
 The energy-band structure of three types of materials at 0 K is
shown as follows:
 Energy in the form of heat, light, or an electric field may raise
the energy of the electrons of the valence band, and in
sufficient amount they will be excited and jump the energy gap
into the conduction band.
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Erwin Sitompul
EEM 7/10
Chapter 5
Current and Conductors
Reading Only
Metallic Conductors
 First let us consider the conductor.
 Here, the valence electrons (or free conductive electrons) move
under the influence of an electric field E.
 An electron having a charge Q = –e will experiences a force:
F  eE
 In the crystalline material, the progress of the electron is
impeded by collisions with the lattice structure, and a constant
average velocity is soon attained.
 This velocity vd is termed the drift velocity. It is linearly related
to the electric field intensity by the mobility of the electron μe:
v d   e E
J    e e E
J E
    e e
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• The Point Form of Ohm’s Law
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EEM 7/11
Chapter 5
Current and Conductors
Reading Only
Metallic Conductors
 The application of Ohm’s law in point form to a macroscopic
region leads to a more familiar form.
 Assuming J and E to be uniform, in a cylindrical region shown
below, we can write:
V  EL
I
V
J  E 
S
L
L
V 
I
S
V  IR
I   J  dS  JS
S
R
a
Vab   E  dL
b
L
S
a
 E dL
a
V
R  ab 
I
b
 E  Lba  E  Lab
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Erwin Sitompul
  E  dL
b
  E  dS
S
EEM 7/12
Chapter 5
Current and Conductors
Conductor Properties and Boundary Conditions
 Property 1:
The charge density within a conductor is zero (ρv = 0) and the
surface charge density resides on the exterior surface.
 Property 2:
In static conditions, no current may flow, thus the electric field
intensity within the conductor is zero (E = 0).
 Now our next concern is the fields external to the conductor.
 The external electric field intensity and electric flux density are
decomposed into the tangential component and the normal
component, with respect to the conductor surface.
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EEM 7/13
Chapter 5
Current and Conductors
Conductor Properties and Boundary Conditions
 The tangential component of the electric field intensity is seen
to be zero Et = 0  Dt = 0.
 If not, then a force will be applied to the surface charges,
resulting in their motion and thus it is no static conditions.
 The normal component of the electric flux density leaving the
surface is equal to the surface charge density in coulombs per
square meter (DN = ρS).
 According to Gauss’s law, the electric flux leaving an
incremental surface is equal to the charge residing on that
incremental surface.
 The flux cannot penetrate into the conductor since the
total field there (inside the conductor) is zero.
 It must leave the surface normally.
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Erwin Sitompul
EEM 7/14
Chapter 5
Current and Conductors
Conductor Properties and Boundary Conditions
 E  dL  0
    0
b
a
c
b
d
c
a
d
Et w  EN ,at b 12 h  EN ,at a 12 h  0


top

S
bottom
D  dS  Q

sides
Q
DN S  Q   S S
h  0, w   Et w  0
DN  S
Et  0
Dt  Et  0
DN   0 EN   S
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Erwin Sitompul
EEM 7/15
Chapter 5
Current and Conductors
Conductor Properties and Boundary Conditions
 Example
Given the potential V = 100(x2–y2) and a point P(2,–1,3) that is
predefined to lie on a conductor-to-free-space boundary, find V,
E, D, and ρS at P, and also the equation of the conductor
surface.
VP  100((2)2  (1) 2 )  300 V
Conductor surface is equipotential
 The surface equation is 300  100( x 2  y 2 )
x2  y 2  3
E  V  200 xa x  200 ya y
EP  400a x  200a y V m
2
DP =  0EP  3.542a x  1.771a y nC m
DN = DP = 3.96 nC m2
S , P  DN = 3.96 nC m2
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• Carefully examine
the surface
direction
Erwin Sitompul
EEM 7/16
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 7/17
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 7/18
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.
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Erwin Sitompul
EEM 7/19
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 7/20
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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EEM 7/21
Chapter 5
Current and Conductors
Reading Only
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.
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Erwin Sitompul
EEM 7/22
Chapter 5
Current and Conductors
Reading Only
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.
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Erwin Sitompul
EEM 7/23
Chapter 5
Current and Conductors
Reading Only
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 7/24
Chapter 5
Current and Conductors
Seminar on Embedded PC Technology
 Thursday, 30 May 2013.
 09:00-17:00,
 At PU Auditorium, 5th Floor.
 Collaboration of TDS Technology,
Indotama, and EE PU .
 7 sessions, 4 speakers.
 Your task: Visit 2 sessions and write
individual short summary of each
session (@ 1 A4 page).
 This task will be graded as Homework
of EEM.
 Originality (writing with your own words)
will be highly regarded. Plagiarism
means zero grade for giver and taker.
 Due: Tuesday, 04 June 2013.
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Erwin Sitompul
EEM 7/25
Chapter 5
Current and Conductors
Homework 6
 D5.1
 D5.2.
 D5.5.
 D5.6. (Extra Question, + 20 points if correctly made)
 All homework problems from Hayt and Buck, 7th Edition.
 Due: Monday, 03 June 2013.
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Erwin Sitompul
EEM 7/26