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Electromagnetic Induction
The electromotive force (e.m.f.) induced in a closed contour:
=
=
S
ind
SdB
dt
dU
rr
Lenz (1804-1865):
- the induced current is in such a direction as to oppose the magnetic
General aspects
magnetic flux
flux variation causing it.
The Faradays law - a new physical phenomenon: a time varying
magnetic field generates an electric field.
- the electric field can be created not only by electric charges, but
by a varying magnetic field as well.
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Cu ring
Electromagnet
S
Source
if S closed Cu ring is moving to the... (right or left?)
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Cu ring
Electromagnet
S
Source
S closed Cu ring is moving to the... left
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Cu ring
Electromagnet
S
Source
S stays closed long enough time steady state currentthen
S opened Cu ring is moving to the... (right or left?)
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Cu ring
Electromagnet
S
Source
S opened Cu ring is moving to the... right!
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Differential Form of the Electromagnetic Induction Law
=L S
SdB
dt
dldE
rrrr
=L
ind ldEUrr
=S
SdBrr
Using Stokes formula ( )
== Sdt
BSdEldE
r
r
rrrr
because magnetic
flux
t
BE
=
r
r
- the differential form of Faradays law
The volume density of the magnetic field energy BHwrr
21=
= dVBHWrr
2
1For the entire space, the total energy of the field is
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The Displacement Current
1864 the English theoretical physicist Clerk Maxwell recognized the
dilemma posed by the application of Amperes circuital law to a system of
accelerated charges
B is unique determined byjBBrot
Bdivrrr
r
0
0
===
-r
(*)
tFrom (*) we have ( ) 0
1
0
= Bdivjdivrr
0
tfor a system of moving charges
The solution of this dilemma was posed by Maxwell
the displacement currentHe redefines the current density adding
But
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Ddivr
=
( ) 0=
+ Ddivt
jdivrr
0=
+
t
Djdiv
r
r
t
D
jjtot
+=
r
rr
- the total current density
jr
- the conduction current density
from electrostatics
agrees with: ( ) 01 = Bdivjdivrr
t- the displacement current density
Now we can write
t
DjH
+=
r
rr
t
EjH
+=
r
rr
0
- the displacement current creates a magnetic field like the conduction current.
t
E
H
=
r
r
00=jr
If
t
EjB
+=
r
rr
000 or
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Maxwells Equations
t
EjH
+= 00
r
rr
- Maxwell-Ampres circuital law
t
BE
=
r
r
- Faradays law of the electromagnetic induction
=E
r
=Dr
- Gauss law for the electric flux
0=Br
- Gauss law for the magnetic flux
HBHB
EPED
i
rrrr
rrrr
=+=
=+=
0
0
- the materials equations
In dielectrics: .. ctct == 00 == jr
t
EH
=
r
r
t
BE
=
r
r
0=Er
0=Hr
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Maxwells equations predict that electric and magnetic fields may
exist in regions where no electric charges or currents are present.
If the fields at one point of space vary with time, then some variation of
the fields must occur at every other point of space at some other time.Thus changes in the electric and magnetic fields should propagate
through space. The propagation of such a disturbance is called an
electromagnetic wave (experimental proof 1884 Heinrich Hertz).
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Fundamentals on e.m. waves
- the existence and features of e.m. waves were theoretically described and
predicted by James Maxwell, in 1864;- first experimental proof of this theory was given by Heinrich Hertz in 1888, ten
years after Maxwell's death.
- Hertz used an oscillatory circuit with a capacitor made of two bowls, K1 and K2
- the "coil" was made of two straight conductors- the bowls could be moved along the conductors the capacitance of the circuit
could be altered, and also its resonance frequency;
- with every interruption from the battery, a high voltage was produced at the output
of the inductor, creating a spark between the narrow placed balls k1 and k2
- whenever there was a spark in the oscillator between the balls k1 and k2, a sparkwould also be produced by the resonator, between balls k3 and k4.
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Wave Equations. General characteristics of waves
A disturbance that propagates in a given medium - wave
A transverse wave
-one-dimensional
- two-dimensional
- three dimensional
ong u na wave
A wave that is linearly polarized in the direction of the y-axis
A pulse traveling through a
string with fixed endpoints
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The Phase
A solitary wave pulse that propagates along a horizontal taut string
( )txfy ,= - wave function for the pulse
e o server n a coor na e rame a moves n ex rec on w e
same velocity, v
( )'' xfy= stationary pulse with a fixed shape
The connection''
yyvtxx
= += ( )vtxfy =
If the pulse is travelling in the opposite direction ( )vtxgy +=
vtxu = - the phase of the wave
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The phase velocity
( )vtxfy =
A wave pulse travels to the right with a velocity valong a taut string. The
location of the pulse is shown at times t1 and t2.
To give the same phase u0 at these instants 22110 vtxvtxu ==
12
12
ttxxv
= - phase u0 of the pulse peak to be a constant, independent on
time
- for all phases u we must have: 0=dt
du
t
u
t
x
x
u
dt
du
+
= vtxu = 1=
x
u vt
u=
v
dt
dx
dt
du== 10
dt
dxv= - any feature of the wave pulse has a coordinate location x that
moves with a velocity -phase velocity
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Phase velocity and group velocity
There are two velocities that are associated with waves, the phase velocity and
the group velocity.
Phase velocity
wavelength, A - amplitude
The wavelenght =vT(T is time period,
T=1/) is the shortest distance over which the
wave repeats itself.
A
kv
ph
= phase velocity
Group velocity
- a wave with the group velocity
and phase velocity going indifferent directions.
The group velocity depends upon the dispersion
relation connecting and k
dk
dvgr
=
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Superposition
- principle of superposition
( ) ( )vtxgvtxfy ++=
(a) Destructive interference. (b) Constructive interference.
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The Wave Equation
2
2
22
2 1
tvx
= wave equation - waves that propagate in one dimension
(x-direction)
2222 1 =
+
+
- in a three-dimensional medium
v wave velocity
tvzyx
(x,t) represents a generalized displacement from equilibrium (e.g. the
displacement of a string, a pressure variation, electric or magnetic field
variation, etc.).
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Plane Waves
( ) += kxttx sin),( 0 - solution of wave eq.; sinusoidal wave
k=2/ - the wave number
The wavelenght =vT(T is time period, T=1/) is the shortest
distance over which the wave repeats itself.
kv= - the angular velocity-v wave velocityphaseinitial
frequency
=
2The solution is periodic in x and t.
n
r
- unit vector direction of wavepropagation( ) nkrkttr
r
r
r
r
rrr
2sin),( 0 =+=
- a 3D plane wave; each color represents adifferent phase of the wave.
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( ) nkrktr
tr r
r
r
r
r
rr
2sin),( 0 ==
Spherical Waves from a Point Source
2
1~
r
I - the wave intensity
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Standing waves
A standing wave, orstationary wave, is a wave that remains in a constant
position. This phenomenon can occur because the medium is moving in theopposite direction to the wave, or it can arise in a stationary medium as a
result ofinterference between two waves traveling in opposite directions.
The sum of two counter-propagating waves (of equal amplitude andfrequency) creates a standing wave.
One-dimensional standing
waves; the fundamental mode
and the first 5 overtones.
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Waves on strings
Tv= T - tension of the string ; - linear mass density
Acoustic waves
E E Youngs modulus;
Acoustic or sound waves travel at speed given by
= -- in solid media
0
Bv=
B the adiabatic bulk modulus;
0 - volume mass density- in fluids
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Wave Equations for electric and magnetic fields
For dielectric media we have t
E
H
=
r
r
( )Ett
EH
r
r
r
=
= )(
( ) ( ) ( )BACCABCBAHHH
rrrrrrrrr
rrr
=
= 2)( 0=Hr
t
HE
=
r
r
02
2
=
t
H
H
r
r
r
wave equations !But
In a similar way one can obtain 02 =
tE
r
=
12v - the velocity of the wave propagation in the dielectric
00
2 1
=c 0=8.8510-12 F/m and 0=410-7 H/m
c=2.99792458108 m/s
rrnv
c
== - the index of refraction of the medium
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Plane Waves
( ))sin(
sin
0
0
kxtHH
kxtEE
==
rr
rr
- solutions of wave eq.
k=2/ - the wave number
The wavelenght =vTis the shortest distance over which the
wave repeats itself (T is time period, T=1/).
vk= - the angular velocity= 2
The solutions are periodic in x and t.
0Er
0Hr
- the amplitudes of the electric and magnetic field components
nEnH r
r
r
r
, - the electromagnetic waves are transversal wavesnr
- unit vector direction of wave propagation
=
H
E- the amplitudes of the fields are related
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( )
( )kxti
kxti
eHHeEE
==
0
0rr
rr
- complex notation; the imaginary part describes our wave
iikirr
)(==
r
= - for an arbitrar direction of wave ro a ation
- forxdirection
Applying this operator to Maxwells eqs.:
0
0
==
==
EnikD
HnikBr
r
r
r
r
r
0=Er
0=Br
We have nEnH r
r
r
r
,
The electromagnetic waves are transversal waves
in vacuum
or dielectrics
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Err
( ) HieiHt
Ht
B kxtirr
rr
===
0
Ei
t
D rr
=
From Maxwells equations in dielectricst
EH
=
r
r
t
BE
=
r
r
Now we show that:
HiEnikt
BE
EiHnikt
DH
rr
r
r
r
rr
r
r
r
=
=
==
( )
( ) HHHEn
EEEHn
rrrr
r
rrrr
r
===
===
v
v
1v
v
=
=k
EnHr
r
r
= rr
=
H
E
0
0
=
H
E- the amplitudes of the fields are related
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HErr
nEnH
r
r
r
r
,
,
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Electromagnetic Energy
The energy density of an electromagnetic wave
( ) ( )222
1
2
1HEHBEDw +=+= 22 HEw ==
The intensity of an electromagnetic wave
dWS
1= dtdAwdW = v wS = v
[J/m3]
=H
E
[Jm-2s-1]
HEHEHHSrr
==== 221
[W/m2]
HESrrr
= Poyntings vector
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The average intensity of this wave
Using Eqs. ( ))sin(
sin
0
0
kxtHH
kxtEE
==
rr
rr
rmsrmsHEHEx
T
tsinHES ==
= 00
2
002
12
- the time average intensity of the wave1
2sin1 2 =
T
dtt
because
because both E and H behave like sine functions0
2
2
1EEErms ==
The square root of the average square of the electric field strength is
called the rms field strength
( ) =A
dt
dWAdHErrr
the flux of the Poyntings vector through the surface
20
TT
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Electromagnetic Momentum and Radiation Pressure
- linear momentum density G
r
]/[,11 4
2 mJsnwcScG r
rr
==
The total wave momentum contained within a volumeActwill be absorbed bythe surface
- a force Fis exerted by the wave on an areaA of the surface
wcS =
tAcwc
tF = 1
The force per unit area is the radiation pressure,prad
[ ]2/, mwFprad == - radiation pressure
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COLOR Wavelengths Range (nm)
Violet 400-450
BlueGreen
Yellow
Orange
Red
450-500500-550
550-600
600-650
650-700
- there are no precisely defined boundaries
between the bands of the electromagnetic
spectrum; rather they fade into each other
like the bands in a rainbow
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Electromagnetic Radiation Spectrum (cont.)
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Boundaries
A discussion of the regions (or bands or types) of the electromagnetic spectrum
is given below. Note that there are no precisely defined boundaries between the
bands of the electromagnetic spectrum; rather they fade into each other like the
bands in a rainbow (which is the sub-spectrum of visible light). Radiation of eachfrequency and wavelength (or in each band) will have a mixture of properties of
two regions of the spectrum that bound it. For example, red light resembles
infrared radiation in that it can excite and add energy to some chemical bonds
photosynthesis and the working of the visual system.
Electromagnetic radiation and Matter
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Electromagnetic radiation and Matter
Electromagnetic radiation interacts with matter in different ways in different parts of the
spectrum. The types of interaction can be so different that it seems to be justified to referto different types of radiation. At the same time, there is a continuum containing all these
"different kinds" of electromagnetic radiation. Thus we refer to a spectrum, but divide it
up based on the different interactions with matter.
Region of the spectrum Main interactions with matter
RadioCollective oscillation of charge carriers in bulk material (plasmaoscillation). An example would be the oscillation of the electrons
in an antenna.
Microwave through far infrared Plasma oscillation, molecular rotation
,
VisibleMolecular electron excitation (including pigment molecules found
in the human retina), plasma oscillations (in metals only)
UltravioletExcitation of molecular and atomic valence electrons, including
ejection of the electrons (photoelectric effect)
X-rays
Excitation and ejection of core atomic electrons, Compton
scattering (for low atomic numbers)
Gamma rays
Energetic ejection of core electrons in heavy elements, Compton
scattering (for all atomic numbers), excitation of atomic nuclei,
including dissociation of nuclei
High energy gamma rays
Creation of particle-antiparticle pairs. At very high energies a
single photon can create a shower of high energy particles andantiparticles upon interaction with matter.
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Elements of Photometry
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Blackbody Radiation
For a shiny metallic surface, the light isn't absorbed, it gets reflected.
For a black material like soot, light and heat are almost completely absorbed,
and the material gets warm.
- good absorbers of radiation are also good emitters.
Observing the Black Body Spectrum
A cavity approximates a blackbody
HESRTrrr
== || Poynting vector
- find how much radiant energy, RT, is being emittedin each frequency range, say the energy between
frequency and + d is RT()d.
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Black body thermal emission intensity as a
function of wavelength for various absolute
temperatures.
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Spectral density of a blackbody at 2000, 3000, 4000 and 5000 K versus frequency
42-84 /5.67x10, KmwattsTP ==
Stefan Boltzmann's Law of Radiation
[ ] [ ]HzTkTh
anm
T=
= 10max
6
max 10879.5;103
Wien's Displacement Law
The ultraviolet catastro he
- as we go to higher frequencies, there aremore and more possible degrees of freedom.
-the oven should be radiating huge amounts
of energy in the blue and ultraviolet
- the equipartition of energy isn't working!
dS )(0
will be infinitely large
- the total power radiated
EM radiation exhibits both wave properties and particle properties at the same time
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EM radiation exhibits both wave properties and particle properties at the same time
(wave-particle duality). Both wave and particle characteristics have been confirmed in
a large number of experiments. Wave characteristics are more apparent when EM
radiation is measured over relatively large timescales and over large distances while
particle characteristics are more evident when measuring small timescales and
distances.
- when electromagnetic radiation is absorbed by matter, particle-like properties will be
more obvious.- a contradiction between the wave theory of light on the one hand, and on the other,
observers' actual measurements of the electromagnetic spectrum that was being
emitted by thermal radiators known as black bodies ultraviolet catastrophe
- -,
the observed spectrum.- Planck's theory was based on the idea that black bodies emit light (and other
electromagnetic radiation) only as discrete bundles or packets of energy: quanta
( ) 118
/3
3
= kThec
hu
hE =0Js/m3
Plancks law
J/m4
sJh = 341062618.6 Plancks constant
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Dependence of photocurrent on
accelerating potential and on frequency.
Dependence of maximum energy of
photoelectron on light frequency
-1905 - Einstein gave a very simple interpretation of Lenard's results:
The Photon
- the radiation itself is quantized: an electromagnetic wave of frequency carries its
energy in packets of size h namedphotons
- particle-wave duality: - de Broglie wavelength:p
h=
A.E. proposed that the quanta of light might be regarded as real particles, and
the particle of light was given the name photon, to correspond with otherparticles being described around this time, such as the electron and proton. A
photon has an energy, E, proportional to its frequency, f, by:cphEph ==
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The Compton Effect - Optional
1923 - Compton gives the most direct confirmation of the photon hypothesis
Experimental setup for observing the Compton scattering of X rays
hcos=
mc
The Bohr Model
The spectrum of electromagnetic radiation from an excited hydrogen gas
- discreet energy levels En
,...2,1with,8 2220
4
0 == nnh
qmEn
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As a photon is absorbed by an atom, it excites the atom, elevating an electron to a higher
energy level (on average, one that is farther from the nucleus).
When an electron in an excited molecule or atom descends to a lower energy level, it emits
a photon of light equal to the energy difference. Since the energy levels of electrons in
atoms are discrete, each element and each molecule emits and absorbs its own characteristic
frequencies.
When the emission of the photon is immediate, this phenomenon is called fluorescence, a
type ofphotoluminescence. An example is visible light emitted from fluorescent paints, in
response to ultraviolet (blacklight). Many other fluorescent emissions are known in spectral
an s ot er t an v s e g t. en t e em ss on o t e p oton s e aye , t e p enomenon
is calledphosphorescence.
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The radiant power
( ) == A dtdWAdHEP
rrr
[W]
Luminous power
= PKV [lm]
= m an s ca e p o ome r c ac or
V- spectral sensitivity of normal human eyes
V=1 for =555 nm
A typical 100 watt incandescent bulb has a luminous power of about 1700 lumens.
A typical dependence of the human eyes sensitivity
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Pointance or Intensity of Light
= I [cd] is the solid angle
Luminance (luminous sterance )
S
IB=
(- for an extended source)
[cd/m2]
Illumination
SE inc= [lx] inc is the flux of light striking the surface S
= Iinc 22 ricosS
r
Sn ==
Sn is the surface normal to the light direction
2
cos
r
iIE
=
-r is the distance from the source of light
- i is the incident angle
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Luminous efficiency
P= [lm/W]
The incandescent bulbs with nominal power P=25-1000 W have =718 lm/W
The fluorescent lamps have 50 lm/W.
The efficiency in visible
- the radiated power (P)
[%]100=Pviz
viz- the radiated power in visible (Pvis)
viz =34 % for incandescent bulbs
viz =20 % for fluorescent lamps
vis increases with the temperature increasing of the incandescent lamp.
Incandescent Light Bulb
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Tungsten bulbs
On 13 December 1904, Hungarian SndorJust and Croatian Franjo Hanaman were
granted a Hungarian patent for a tungsten
(W) filament lamp that lasted longer and
gave brighter light than the carbon
filament. Tungsten filament lamps werefirst marketed by the Hungarian company
Tungsram in 1904.
Original carbon-filament
bulb from Thomas Edison;
time life: 13.5 hours
Early carbon filaments had
a negative temperature
coefficient of resistance: as
they got hotter, their
electrical resistance
decreased
the lampsensitive to fluctuations Xenon halogen lamp
The bulb is filled with an inert gas
such as argon (93%) and nitrogen
(7%) to reduce evaporation of the
filament and prevent its oxidation at a
pressure of about 70 kPa (0.7 atm)
An electric current heats the filament totypically 2000 to 3300 K, well below
tungsten's melting point of 3695 K.
Fluorescent Lamp Operation
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Schematic for Ballast
Starter
Typical low pressure fluorescent tube I/V characteristic
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Near fields and far fields
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Three-dimensional perspective of the
radiation pattern of an elementary doublet.
Radiation pattern of an elementary doublet,shown in profile.
Microwave sources
The magnetron the microwave radiation of microwave ovens and some radar
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The magnetron - the microwave radiation of microwave ovens and some radar
applications is produced by a device called a magnetron.
- a "crossed-field" device
Electrons are released at the center hot cathode by the process of thermionic emission
The axial magnetic field exerts a magnetic force on these charges they tend to be
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The axial magnetic field exerts a magnetic force on these charges - they tend to be
swept around the circle.
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The electron path under the influence of
different strength of the magnetic field
The high-frequency electrical field
Rotating space charge wheel in an twelve
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Rotating space-charge wheel in an twelve-
cavity magnetron
Interaction between a cavity resonator and
the rotating Space-Charge Wheel
Tunnel diode
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A tunnel diode orEsaki diode is a type of semiconductor that is capable of very fast
operation, well into the microwave frequency region, made possible by the use ofthe quantum mechanical effect called tunneling.
It was invented in August 1958 by Leo Esaki when he was with Tokyo Tsushin
Kogyo, now known as Sony. In 1973 he received the Nobel Prize in Physics, jointly
with Brian Josephson, for discovering the electron tunneling effect used in these
diodes.
IVcurve similar to a tunnel diode
characteristic curve. It has negative
resistance in the shaded voltage
region, between v1 and v2.
The negative resistance region of
the tunnel diode makes oscillator action
possible. The unijunction transistor has a
similar oscillator application.
Tunnel Diode Oscillator
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Tunnel Diode Oscillator
Resonant Tunneling Diode
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The spin-torque oscillator
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Geometry of a spin-torque oscillatorconsisting of a 'fixed' magnetic layer, a
non-magnetic spacer and a 'free' magnetic
layer.
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Guided waves
1. Ionospheric reflection
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Ionospheric reflection is a bending, through a complex process involving reflection
and refraction, of electromagnetic waves propagating in the ionosphere back towardthe Earth.
The amount of bending depends on the extent of penetration (which is a function of
frequency), the angle of incidence, polarization of the wave, and ionospheric
conditions, such as the ionization density. It is negatively affected by incidents of
ionospheric absorption.
Effects of ionospheric density
on radio waves
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Different incident angles of radio
waves
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