Electromagn Induction Waves

8/9/2019 Electromagn Induction Waves

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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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Electromagn Induction Waves