Transcript 9_waves

Electromagnetism
INEL 4152 CH 9
Sandra Cruz-Pol, Ph. D.
ECE UPRM
Mayagüez, PR
In summary
 Stationary
 Steady
Charges
currents
 Time-varying
currents
 Electrostatic
fields
 Magnetostatic
fields
 Electromagnetic
(waves!)
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Outline
 Faraday’s
Law & Origin of emag
 Maxwell Equations explain waves
 Phasors and Time Harmonic fields

Maxwell eqs for time-harmonic fields
9.2
Faraday’s Law
Electricity => Magnetism

In 1820 Oersted discovered that a steady
current produces a magnetic field while
teaching a physics class.
This is what Oersted
discovered accidentally:
ò H × dl = ò J × dS
L
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s
Would magnetism would produce
electricity?

Eleven years later,
and at the same time,
(Mike) Faraday in
London & (Joe) Henry
in New York
discovered that a
time-varying magnetic
field would produce
an electric current!
Vemf
d
 N
dt
¶
ò E × dl = -N ò ¶t B × dS
L
s
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Electromagnetics was born!

This is Faraday’s Law the principle of motors,
hydro-electric generators
and transformers
operation.
*Mention some examples of em waves
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Faraday’s Law
 For
N=1 and B=0
Vemf
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d
 N
dt
9.3
Transformer & Motional EMF
Three ways B can vary by
having…
1.
2.
3.
A stationary loop in a t-varying B field
A t-varying loop area in a static B field
A t-varying loop area in a t-varying B field
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1. Stationary loop in
a time-varying B field
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2. Time-varying loop area
in a static B field
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3. A t-varying loop area in
a t-varying B field
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Transformer Example
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9.4
Displacement Current, Jd

Maxwell noticed something
was missing…
And added Jd, the
displacement current
 H  dl   J  dS  I
L
enc
I
S1
 H  dl   J  dS  0
L
S1
I
L
S2
d
dQ
L H  dl  S J d  dS  dt S D  dS  dt  I
2
2
S2
At low frequencies J>>Jd, but at radio frequencies both
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terms are comparable in magnitude.
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9.4
Maxwell’s Equation
in Final Form
Summary of Terms
E
= electric field intensity [V/m]
 D = electric field density
 H = magnetic field intensity, [A/m]
 B = magnetic field density, [Teslas]
 J = current density [A/m2]
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Maxwell Equations
in General Form
Differential form Integral Form
  D  v
 D  dS   v dv
s
 B  0
v
 B  dS  0
s
B
 E  
t
D
 H  J 
t
Gauss’s Law for E
field.
Gauss’s Law for H
field. Nonexistence
of monopole

Faraday’s Law
L E  dl   t s B  dS
D 

H

dl

J


L
s  t   dS
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Ampere’s Circuit
Law
Maxwell’s Eqs.
 Also
the equation of continuity
 Maxwell
 v
J  
t
D
t
added the term
to Ampere’s
Law so that it not only works for static
conditions but also for time-varying
situations.

This added term is called the displacement
current density, while J is the conduction
current.
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Relations & B.C.
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9.6
Time Varying Potentials
We had defined
 Electric
Scalar & Magnetic Vector potentials:
Related to B as:
 To find out what happens for time-varying fields
Substitute into Faraday’s law:
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Electric & Magnetic potentials:
 If
we take the divergence of E:
 We
have:
 Taking
the curl of:
we get
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& add Ampere’s
Electric & Magnetic potentials:
 If
we apply this vector identity
 We
end up with
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Electric & Magnetic potentials:
 We
use the Lorentz condition:
To get:
Which are
both wave
equations.
and:
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9.7
Time Harmonic Fields
Phasors Review
Time Harmonic Fields
 Definition:
is a field that varies periodically
with time.

Ex. Sinusoid
 Let’s
review Phasors!
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Phasors & complex #’s
Working with harmonic fields is easier, but
requires knowledge of phasor, let’s review
 complex numbers and
 phasors
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COMPLEX NUMBERS:
 Given
a complex number z
z  x  jy  re
j
 r  r cos   jr sin 
where r | z |
x  y is the magnitude
2
  tan
1
2
y
is the angle
x
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Review:
 Addition,
1/ j =
j=
1/
 Subtraction,
 Multiplication,
 Division,
 Square
examples :
Root,
 Complex Conjugate
e
j 45o
/ j=
e
j 45o
- j=
3e
j 90 o
2e
2e
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+j=
+ j 45o
+ j 45o
=
+10e
+ j 90 o
=
For a Time-varying phase
f = wt + q
Real and imaginary parts are:
Re{re } = r cos(wt + q )
jf
Im{re }= rsin(wt + q )
jf
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PHASORS
a sinusoidal current I(t) = Io cos(wt + q )
equals the real part of I o e j e jt
 For
j
I
e
 The complex term o
which results from
dropping the time factor e jt is called the
phasor current, denoted by I s (s comes
from sinusoidal)
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Advantages of phasors
 Time
derivative in time is equivalent to
multiplying its phasor by j
A
 jAs
t
 Time integral is equivalent to dividing by
the same term.
As
 At  j
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How to change back from
Phasor to time domain
The phasor is
1. multiplied by the time factor, e jt,
2. and taken the real part.
A  Re{ As e
j t
}
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9.7
Time Harmonic Fields
Time-Harmonic fields
(sines and cosines)
 The
wave equation can be derived from
Maxwell equations, indicating that the
changes in the fields behave as a wave,
called an electromagnetic wave or field.
 Since
any periodic wave can be
represented as a sum of sines and
cosines (using Fourier), then we can deal
only with harmonic fields to simplify the
equations.
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Maxwell Equations
for Harmonic fields (phasors)
Differential form*

 DE
 v  v
Gauss’s Law for E field.
 BH
0

Gauss’s Law for H field.
No monopole
0
  E   jH
 E  
B
t
Ñ ´ H = s E + jwe E
D
 H  J 
t
* (substituting
Faraday’s Law
Ampere’s Circuit Law
D   E andCruz-Pol,
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Ex. Given E, find H
E
= Eo cos(t-bz) ax
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Ex. 9.23
 In

free space,  50
8
E  cos(10 t  kz)  V / m

 Find
k, Jd and H using phasors and
Maxwells eqs. Recall:
é
ˆ
r̂
r
f
ê
1ê ¶
¶
Ñ´ A = ê
r ê ¶r ¶f
ê Ar r Af
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