Download Magnetic field

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Introductory EM theory
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Importance of electromagnetic theory
•
•
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EM theory helps understand how electrical signals propagate along
conductors as well as free space.
Predicts voltages and currents using the concept of electric and magnetic
field.
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Example : a simple DC circuit
Charge
distribution
E/H-field
distribution
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Generation of charges : battery
An amount of positive charges are generated such that the terminal voltages are
sustained.
2NH 4  2e   2 NH3  H 2
Electrons(-) are absorbed.
(+) charges are generated
Electrons(-) are generated.
(+) charges are absorbed.
Zn  Zn 2  2e 
Electrons are generated via
electro-chemical reaction.
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Electromagnetic theory
EM-theory
Material
Electric field (E)
Sources (q, J)
Magnetic field (H)
Electro-magnetic field (E,H )
Material (ε, μ)
Mathematics
Coordinate systems
Vector calculus
Green’s function
Wave equation
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Basic EM-theory
1.
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Electric field
① Coulomb’s law
② Gauss’s law (divergence)
③ Electric potential (gradient)
④ Capacitance
⑤ Ohm’s law
2.
Magnetic field
① Biot-Savart law
② Ampere’s law (curl)
③ Inductance
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3.
Electro-magnetic field
① Faraday’s law
② Displacement current
③ Maxwell’s equations
④ Plane wave
⑤ Reflection/transmission
4.
Transmission lines
① Impedance matching
② Smith chart
③ Waveguides
5.
Radiation
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Coulomb’s law
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•
This law is discovered by Coulomb experimentally.
•
In the free space, the force between two point charges is proportional to the charges of
them, and is inversely proportional to the square of the distance between those charges.
Fk
q 1q 2
r
2
k  9  109 [ Nm2C2 ] 
rˆ
1
40
ε0 : permittivity of vacuum.
If q1, q2 have the same
polarity, the force is
repulsive.
+q1
+q2
 

R  R 2  R1

R2

R1
Coulomb’s law only states that
the force between two charge is
related to the distance between
them and their charges. It does
not tells us how the interaction
occurs.
O
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Electric field distribution near charged plates
E
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If a charge is brought into the plates, it will be accelerated
along the direction of electric field.
F  ma  qE
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Electric potential energy (V)
•
•
•
•
As in the gravitational field, a potential energy
for the electric field can be introduced.
The potential energy in an electric field is
defined as the energy levels of the test charge
with +1C.
The unit of potential energy is “voltage”
named after the physicist Volta.
The position far away from the source charge
has zero potential energy.
+qt
Electric potential energy defined as the work to
move a test charge with (+1C)
+q
rB
rB
rA
rA
W   ( F)  dr   (  qt E)  dr  qtV
rB
V   ( E)  dr
rA
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Potential energy in a gravitational field
U (r2 )
r2
•
The scalar field quantity of potential energy is
introduced to represent energy levels inherent
to positions in the space.
•
Differences between the energy levels can be
obtained from the work that must applied for
the object to move from the initial position to
the final position.
•
The position that corresponds to zero energy
level is one that is located far away from the
earth.
U (r2 )
r2
r1
U  U (r2 )  U (r2 )  W
O
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Work in a gravitational field
r2
W   ( F)  dr
GMm
F   2 r̂12
r
r2
r2
mass : m
To move an object in the
gravitational field, an external
force must be applied that
compensates the force due to
gravity.
r2
r1
mass : M
r2
GMm
rˆ  rˆ dr
2
r2
r
W F
r
O
2
1 1
 1
 GMm    GMm  
 r  r2
 r2 r2 

GMm
( r2  r2 )  mgr.
r2

GM 
 g 

r
2 

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Magnetic field
A charged particle in
motion generates magnetic
field nearby.
In the same way, currents
generate magnetic field
nearby.
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Motion of a charge in a magnetic field
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F  qv  B
Charged particles in motion are influenced by
magnetic fields
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Biot-Savart law
ˆ
Ids  R
dH 
4R 2
Current
segment
Id s
r'
R  r  r'
r
Direction of
H-field
The generated magnetic field can be predicted
by Biot-Savart’s law
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Electromagnetic law – Maxwell equations
Maxwell equations
B
E  
t
D
H  J 
t
D  
B  0
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1. Electromagnetic phenomena are explained
by the four Maxwell equations.
2. Through the equations, electric field and
magnetic field are coupled to each other.
3. Quantities on the right hand side are the
source terms.
4. Quantities on the left side are the resulting
phenomena.
5. The independent variables are current
density vector J and charge density .
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Ampere’s law
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E
H  J 
t
Current or increase of
electric field strength
E,J
H
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Faraday’s law
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  E  
H
H
t
Increase of
magnetic field
E
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Faraday’s law
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The time-varying magnetic
field generates electric field
nearby.
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Gauss’ law
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E   /
E
+Q
-Q
Electric field lines emanate from positive
charges and sink into negative charges.
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 H  0
Magnetic field lines always form
closed loops
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Example – Hertzian dipole antenna
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spheres for storing electric charges
Heinrich Hertz (1857-1894)
arc monitoring
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Schematic diagram of Hertz experiment
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Transformer for high voltage generation
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Propagation of electromagnetic wave
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Electric field : red
Magnetic field : blue
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Radio communication
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Reception of EM wave
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current
E
Transmitting
antenna
V
Receiving
antenna
The charges on the receiving antenna move
toward the antenna terminal, which causes
voltage drop across them.
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Example – Signal propagation over a line trace
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V
H-field due to
moving charges
t
E
V


H


E
H  J 
t


ZL

H
  E  
t
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Electromagnetic problem
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Source distributions are known
over only a small area.
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