Transcript Introduction to EM

UNIVERSITI MALAYSIA PERLIS
EKT 242/3:
ELECTROMAGNETIC
THEORY
CHAPTER 1 - INTRODUCTION
What is Electromagnetism?
• Electromagnetism - Magnetic forces
produced by electricity. Oscillating
electrical and magnetic fields.
• Electromagnetism - Magnetism arising
from electric charge in motion.
Electrostatic vs. Magnetostatic
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Electrostatic
Magnetostatic
Fields arise from a potential
difference or voltage gradient
Fields arise from the movement of
charge carriers, i.e flow of current
Field strength:
Volts per meter (V/m)
Field strength:
Amperes per meter (A/m)
Fields exist anywhere as long as
there was a potential difference
Fields exist as soon as current
flows
We will see how charged dielectric
produces an electrostatic fields
We will see how current flows
through conductor and produces
magnetostatic fields
Example of electrostatics:
vigorously rubbing a rubber rod
with a piece of fur and bring to a
piece of foil – foil will be attracted to
the charged rod
Example of magnetostatics:
Current passes through a coil
produces magnetic field about each
turn of coil – combined will produce
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two-pole field, south & north pole
Timeline for Electromagnetics
in the Classical Era
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 1785 Charles-Augustin de Coulomb (French)
demonstrates that the electrical force between
charges is proportional to the inverse of the
square of the distance between them.
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Timeline for Electromagnetics
in the Classical Era
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 1835 Carl Friedrich Gauss (German) formulates
Gauss’s law relating the electric flux flowing
through an enclosed surface to the enclosed
electric charge.
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Timeline for Electromagnetics
in the Classical Era
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 1873 James Clerk Maxwell (Scottish) publishes
his “Treatise on Electricity and Magnetism” in
which he unites the discoveries of Coulomb,
Oersted, Ampere, Faraday and others into four
elegantly constructed mathematical equations,
now known as Maxwell’s Equations.
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Units and Dimensions
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 SI Units
 French name ‘Systeme Internationale’
 Based on six fundamental dimensions
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Multiple & Sub-Multiple Prefixes
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Example:
4 x 10-12 F
becomes
4 pF
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The Nature of Electromagnetism
stop here
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Physical universe is governed by 4 forces:
1. nuclear force – strongest of the four but its range is limited to
submicroscopic systems, such as nuclei
2. weak-interaction force – strength is only 10-14 that of the
nuclear force. Interactions involving certain radioactive particles.
3. electromagnetic force – exists between all charged particles.
The dominant force in microscopic systems such as atoms and
molecules. Strength is of the order 10-2 of the nuclear force
4. gravitational force – weakest of all four forces. Strength is of
the order 10-41 that of the nuclear force. Dominant force in
macroscopic systems, e.g solar system
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The Electromagnetic Force
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Gravitational force – between two masses
Gm1m2
Fg 21   R̂ 12
R122
(N)
Where;
m2, m1 = masses
R12 = distance
G = gravitational constant
R̂ 12
= unit vector from 1 to 2
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Electric fields
• Electric fields exist whenever a positive or
negative electrical charge is present.
• The strength of the electric field is measured in
volts per meter (V/m).
• The field exists even when there is no current
flowing.
• E.g rubbing a rubber sphere with a piece of
fur.
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Electric Fields
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Electric field intensity, E
due to q
~
ER
q
(V/m) (in free space)
2
40 R
~
where R = radial unit vector
pointing away from charge
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Electric Fields
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Electric flux density, D
D  E (C/m )
2
where E = electric field intensity
ε = electric permittivity of the material
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Magnetic Fields
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• Magnetic field arise from the motion of electric
charges.
• Magnetic field strength (or intensity) is
measured in amperes per meter (A/m).
• Magnetic field only exist when a device is
switched on and current flows.
• The higher the current, the greater the strength
of the magnetic field.
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Magnetic Fields
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• Magnetic field lines are induced by current flow
through coil.
north
pole
south
pole
• Magnetic field strength or magnetic field intensity
is denoted as H, the unit is A/m.
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Magnetic Fields
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 Velocity of light in free space, c
c
1
0 0
 3 108 (m/s)
where µ0 = magnetic permeability of free space
= 4π x 10-7 H/m
 Magnetic flux density, B (unit: Tesla)
B  H
where H = magnetic field intensity
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Permittivity
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 Describes how an electric field affects and is
affected by a dielectric medium
 Relates to the ability of a material to transmit (or
“permit”) an electric field.
 Each material has a unique value of permittivity.
 Permittivity of free space;
 Relative permittivity;

r 
0
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Permeability
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 The degree of magnetization of a material that
responds linearly to an applied magnetic field.
 The constant value μ0 is known as the magnetic
constant, i.e permeability of free space;
 Most materials have permeability of  0 except
ferromagnetic materials such as iron, where  is larger
than  0 .
 Relative permeability;

r 
0
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The Electromagnetic Spectrum
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Atmosphere Opaque
Ionosphere Opaque
100%
0
X-rays
V
Medical diagnosis
i
Gamma rays
Ultraviolet s Infrared
i Heating,
Sterilization
Cancer therapy
b Night vision
l
e
1 fm
1 pm
10-15
10-12
1 nm
10-10
10-9
Radio Spectrum
Communication, radar, radio and TV broadcasting,
radio astronomy
1 μm
1 mm
1m
10-6
10-3
1
1 km
1 Mm
103
106
Wavelength (m)
108
Frequency (Hz)
1 EHz
1023
1021
1018
1 PHz
1 THz
1015
1012
1 GHz
109
1 MHz
1 kHz
1 Hz
106
103
1
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Electromagnetic Applications
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Review of Complex Numbers
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• You can use calculator .
j  1
• A complex number z is written in the
rectangular form Z = x ± jy
• x is the real ( Re ) part of Z
• y is the imaginary ( Im ) part of Z
• Value of
• Hence, x =Re (z) , y =Im (z)
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Forms of Complex Numbers
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• Using Trigonometry, convert from rectangular
to polar form,
z  x  jy
 z cos   j z sin 
 z (cos   j sin  )
• Alternative polar form,
z  ze
j
 z 
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Forms of complex numbers
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• Relations between rectangular and polar
representations of complex numbers.
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Forms of complex numbers
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Applying Euler’s identity
e  cos θ  j sin θ
jθ
thus
z  z e j  z cos  j z sin 
which leads to the relations
x  z cos  ,
NB: θ in degrees
z

x 2  y2 ,
y  z sin  ,
  tan 1 ( y / x ) ,
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Complex conjugate
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• Complex conjugate, z*
• Opposite sign (+ or -) & with * superscript
(asterisk)
z*  ( x  jy)*  x  jy
• Product of a complex number z with its complex
conjugate is always a real number.
• Important in division of complex number.
z  z*  real number
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Equality
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omplex numbers z1 and z 2 are given by
1 =
z1  x1  jy1  z1 e
j1
z 2  x 2  jy 2  z 2 e j2
x 1  x 2 and y1  y 2 or, equivalent
• z1 = z2 if and only if x1=x2 AND y1=y2
z 2 if and only if
 2 .
• Or equivalently, z1  z 2
AND
1   2
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Addition & Subtraction
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If two complex numbers z1 and z 2 are given by
z1  a  jb
z 2  x  jy
Hence
z 1  z 2  (a  x )  j ( b  y )
and
z 1  z 2  (a  x )  j ( b  y )
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Multiplication in Rectangular Form
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plex •numbers
and
are
given
by
z
z
bers
and
are
given
by
z
z
1
2
1
2
Given two complex numbers z1 and z2;
j1 j1
z1  xz11 
 xjy
jy1z1 e
z1 e
1 
1 
j2
z 2  xz2 
 xjy2 jy
 z2 ze e j2
2
2
2
2
• Multiplication gives;
z1z 2  (x1  jy1 )( x 2  jy 2 )
z1z 2  (x1  jy1 )( x 2  jy 2 )
 (x1x 2  y1y2 )  j(x1y2  x 2 y1 )
 (x1x 2  y1y2 )  j(x1y2  x 2 y1)
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Multiplication in Polar Form
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• In polar form,
form
z1 z 2  z1 e
j1
 z2 e
 z1 z 2 e
j2
j( 1  2 )
 z1 z2 cos (1 2)  j sin (1 2)
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Division in Polar Form
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z2  0
• For z  0
z1
z2


x1  jy1
x2  jy2
x1  jy1
x 2  jy2

x 2  jy2
x 2  jy2

x(1xxx2x2 yy1 1yy22) jj(xx22yy11 xx11yy2 2) 

x2 2  y 2 2
x
2
 y22
2

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Division in Polar Form
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z1
z2
z1 e j1

z2 e

z1
z2

z1
z2
j2
e
j(θ1  θ2)
cos (θ1 θ2 )  j sin (θ1  θ2 )
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Powers
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sitive
• integer
For anyn,positive integer n,
z  (z e )
n
j n
 z e
n
jn
 z (cos n  j sin n )
n
• And,
1
2
1
2
z  z
e
j 2
1
2
  z (cos 2  j sin 2 )
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Useful Relations:
Powers
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• Useful relations 1  e
j
je
e
j 2
 j  e
j  (e
 j

1 180 ,
 1 90 ,
j 2
j 2
 j e
)
e
1
2
 j 4
 j 2
 e

 1   90
j 4

 ( 1  j)
2
 ( 1  j)
2
,
,
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