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Engineering Electromagnetics
Lecture 7
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
President University
Erwin Sitompul
EEM 7/1
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).
President University
Erwin Sitompul
EEM 7/2
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
President University
Erwin Sitompul
EEM 7/3
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
President University
Erwin Sitompul
x
Q
 v S
I 
t
t
EEM 7/4
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.
President University
Erwin Sitompul
EEM 7/5
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, destroyed, or 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
President University
Erwin Sitompul
EEM 7/6
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 constant, the derivative becomes a
partial derivative. Writing it within the integral,
v
vol (  J)dv  vol  t dv
v
(  J )v  
v
t
v
J  
• The Differential Form (Point Form)
of the Continuity Equation
t
President University
Erwin Sitompul
EEM 7/7
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 7/8
Chapter 5
Current and Conductors
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.
President University
Erwin Sitompul
EEM 7/9
Chapter 5
Current and Conductors
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
Erwin Sitompul
EEM 7/10
Chapter 5
Current and Conductors
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
President University
Erwin Sitompul
  E  dL
b
  E  dS
S
EEM 7/11
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 components and the normal
components.
President University
Erwin Sitompul
EEM 7/12
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 is zero.
 It must leave the surface normally.
President University
Erwin Sitompul
EEM 7/13
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
President University
Erwin Sitompul
EEM 7/14
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 7/15
Chapter 5
Current and Conductors
Homework 6
 D5.1
 D5.2.
 D5.4.
 D5.5. (Bonus Question, + 20 points if correctly made)
 All homework problems from Hayt and Buck, 7th Edition.
 Deadline: 05 June 2012, at 08:00.
President University
Erwin Sitompul
EEM 7/16