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Chapter 20. Wave motion
Liew Sau Poh
2
Outline 20.1 Progressive Waves 20.2 Wave Intensity 20.3 Principle of Superposition 20.4 Standing Waves 20.5 Electromagnetic Waves
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Objectives a) interpret and use the progressive wave
equation, y = a sin ( t kx) or y = a cos ( t kx)
b) sketch and interpret the displacement-time graph and the displacement-distance graph
c) use the formula = 2 x/ d) derive and use the relationship v = f e) define intensity and use the relationship I
A2
4
Objectives g) describe the variation of intensity with
distance of a point source in space h) state the principle of superposition i) use the principle of superposition to explain
the formation of standing waves j) derive and interpret the standing wave
equation k) distinguish between progressive and
standing waves
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Objectives l) state that electromagnetic waves are made up of
electrical vibrations E = E0 sin ( t - kx) and magnetic vibrations B = B0 sin ( t - kx)
m) state the characteristics of electromagnetic waves n) compare electromagnetic waves with mechanical
waves o) state the formula c = ( 0 0)1/2 and explain its
significance p) state the orders of the magnitude of wavelengths
and frequencies for different types of electromagnetic waves.
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20.1 Progressive waves
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Wave motion
A wave travels along its medium, but the individual particles just move up and down.
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Wave motion
All types of traveling waves transport energy Study of a single wave
pulse shows that it is begun with a vibration and transmitted through internal forces in the medium.
Continuous waves start with vibrations too. If the vibration is SHM, then the wave will be sinusoidal.
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Wave characteristics
Amplitude, A Wavelength, Frequency f and period T Wave velocity, v = f
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Wave characteristics
between any two identical points on adjacent waves Period T: the time required for two identical points of adjacent waves to pass by a point.
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Wave characteristics Frequency f: the inverse of the period. Amplitude A: The maximum displacement
Wave velocity, v = f
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Wave characteristics The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal).
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Progressive waves is defined as the one in which the wave profile propagates. The progressive waves have a definite speed called the speed of propagation or wave speed. The direction of the wave speed is always in the same direction of the wave propagation .
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Progressive waves There are two types of progressive wave,
a. Transverse progressive waves b. Longitudinal progressive waves.
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Progressive waves
If the tension in the string is T, and its mass per unit length
The wave speed is :
stringTV
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Progressive waves The wave speed is :
= 0.002 kg/m
2
stringT mgV
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Progressive waves
Traveling in the positive x-direction
1Tf
22
V fT k k
2(wave number) k
22 fT
( , ) sin( )oD x t A kx t
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2( , 0) sin( )oD x A x
19
Longitudinal and Transverse Waves
The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal).
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Longitudinal and Transverse Waves
Sound waves are longitudinal waves:
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Longitudinal and Transverse Waves Earthquakes produce both longitudinal and transverse waves. Both types can travel through solid material, but only longitudinal waves can propagate through a fluid in the transverse direction, a fluid has no restoring force. Surface waves are waves that travel along the boundary between two media.
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Reflection and Transmission of Waves
A wave reaching the end of its medium, but where the medium is still free to move, will be reflected (b), and its reflection will be upright. A wave hitting an obstacle will be reflected (a), and its reflection will be inverted.
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Reflection and Transmission of Waves
A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter.
TT FLM
Fv/
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Reflection and Transmission of Waves
Two- or three-dimensional waves can be represented by wave fronts, which are curves of surfaces where all the waves have the same phase. Lines perpendicular to the wave fronts are called rays; they point in the direction of propagation of the wave.
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Reflection and Transmission of Waves
The law of reflection: the angle of incidence equals the angle of reflection.
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20.2 Wave Intensity
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Wave Power and Intensity
24P PI
Area r
2P vaAThe power carried by a wave is given by where a is a constant that depends on the kind of wave. The intensity of a wave is the power per unit area at the wavefront. spherical
wavefront
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20.2 Wave Intensity Definition: the amount of energy transferred by waves that passes through unit area per second of any plane surface normal to the direction of propagation of the waves. Unit: Js-1m-2 , or W m-2
Direction of wave
Area (sphere), A = 4 r2
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20.2 Wave Intensity A point source of disturbance generates waves that propagate outwards in three dimensions in a homogenous medium. It can radiate energy of P joules per second (P watts). Since the source of disturbance is a point source, the waves radiated from the source travel outwards in all possible directions, resulting in wavefronts being spherical in shape, with the source as the centre.
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20.2 Wave Intensity
Wavefront: a surface on which all particles vibrate in phase with one another.
Direction of the wave propagation
Spherical wavefront
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20.2 Wave Intensity At a particular instance, let the radius of one spherical wavefront be r. The surface area A of the wavefront is equal to the surface of a sphere given by A = 4 r2
The amount of energy crossing the entire spherical surface of this wavefront every second must be P joules (the power radiated by the source).
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20.2 Wave Intensity Therefore, the amount of energy crossing unit area is P/ 4 r2. By definition, this amount of energy is equal to the intensity I at distance r from the source. Thus, the intensity, I = P/A = P/4 r2. Hence,
I P I 1/r2
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Sound and the Human Ear The human ear is an amazing instrument. It can
respond to sound intensities that differ by a factor of a trillion!
The ear can do this because of its non-linear response to intensity. It turns out to be better to describe sound intensities in terms of decibels (dB):
where I is in W/m2 and I0 = 10-12 W/m2. An increase of 10
dB corresponds to an intensity increase of a factor or 10.
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0
10log II
Pain threshold ~ 130 dB 34
Wave Intensities
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Wave Intensity, W/m2
A whisper at 1m 10-10
TV signal, 5km from 50 kW transmitter
1.6 x 10-4
Sound, 4m from a loud rock band 1 Sound, 50 m from a jet aircraft 10
50 1368
Microwaves, inside a microwave oven 6000
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20.3 Principle of Superposition
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20.3 Superposition
The superposition principle says that when two waves pass through the same point, the displacement is the arithmetic sum of the individual displacements.
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20.3 Superposition
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20.3 Superposition
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20.3 Superposition
(a) exhibits destructive interference (b) exhibits constructive interference.
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20.3 Superposition These figures show the sum of two waves. In (a) they add constructively; in (b) they add destructively; and in (c) they add partially destructively.
41 In Phase 42 180o out of phase
43 43 Between in phase and 180o out of phase 44
20.4 Standing Waves
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20.4 Standing /Stationary waves
The frequencies of the standing waves on a particular string are called resonant frequencies. They are also referred to as the fundamental and harmonics.
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20.4 Standing /Stationary waves Standing waves occur when both ends of a string are fixed. In that case, only waves which are motionless at the ends of the string can persist. There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value.
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20.4 Standing /Stationary waves The wavelengths and frequencies of standing waves are:
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20.4 Standing /Stationary waves The characteristics of Stationary wave
Nodes and antinodes are appear at particular time that is determine by the equation of the stationary wave. The distance between adjacent nodes or antinodes is? The distance between a node and an adjacent antinode is?
= 2(the distance between adjacent nodes or antinodes)
N A N A N A N
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Node (N) is defined as a point at which the displacement is zero where the destructive interference occurred. Antinode (A) is defined as a point at which the displacement is maximum where the constructive interference occurred.
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20.4 Standing /Stationary waves The pattern of the stationary wave is fixed hence the amplitude of each points along the medium are different. Thus the nodes and antinodes appear at particular distance and determine by the equation of the stationary wave.
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20.4 Standing /Stationary waves The Equation of Stationary wave
By considering wave functions(equations) for two progressive sinusoidal waves having the same amplitude, frequency and wavelength but travelling in opposite directions in the same medium as shown below.
where y1 represents a wave travelling in the +x direction and y2 represents one travelling in the x direction. By applying the principle of superposition hence
)s i n (1 k xtay)s i n (2 k xtay
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20.4 Standing /Stationary waves The general equation of stationary wave is given by where
number wave the:frequencyangular the:
waveeprogressiv theof amplitude the: wavestationary theof amplitude the:
k
aA
21 yyyk xtak xtay s i ns i n
k xtk xtak xtk xtay s i nc o sc o ss i ns i nc o sc o ss i nk xtay c o ss i n2
aA 2tk xAy s i nc o s and
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20.4 Standing /Stationary waves Description of the equation of stationary wave
determine the amplitude for any point along the stationary wave. It is called the amplitude formula. Its value depend on the distance, x
k xA c o s
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20.4 Standing /Stationary waves Antinodes The point with maximum displacement = A
Ak xA c o s1c o s k x
1c o s 1k x, . . .3,2,,0k x
, . . .3,2,1,0mwhere mk x
kmx
2k
,...2
3,,2
,0x2mx Antinodes are occur when Therefore
and
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Nodes The point with minimum displacement = 0
0c o s k xA0c o s k x
0c o s 1k x,...
25,
23,
2kx
, . . .7,5,3,1nwhere 2nkx
knx2
2kand
,...4
5,4
3,4
x4nx Nodes are
occur when Therefore
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determine the time for antinodes and nodes will occur in the stationary wave. Antinodes
The point with maximum displacement = A
ts i n
AtA s i n1s i n t
1s i n 1t,...
25,
23,
2t
, . . .7,5,3,1nwhere 2nt
2nt T
2
Tnt4 ,...
45,
43,
4TTTt
Antinodes occur when the time are
and
Therefore
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Nodes The point with minimum displacement = 0
All the point in the stationary wave at the equilibrium position (y = 0)
0s i n tA0s i n t
0s i n 1t, . . .3,2,,0t
, . . .4,3,2,1,0mwhere mtmt T
2
Tmt2
,...2
3,,2
,0 TTTtNodes occur when the time are
and
Therefore
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20.4 Standing /Stationary waves Displacement-distance graph for stationary wave
A
4Tt
0
A
A
y
x
2 23 2
4 43
45
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N A N A N A N A
T2T0t ,,
4T3t
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Characteristics of Standing Waves Nodes and antinodes remain stationary
Nodes points of least amplitude Antinodes points of maximum amplitude
Resulting from interference Waves of
Equal amplitude Equal wavelength
Pass thru each other In opposite directions Out of phase at nodes (regions of stable destructive interference)
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20.5 Electromagnetic Waves
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Introduction: EM Waves Waves that can transmit energy without a medium or material to travel through (light waves, heat waves, any waves in space, etc.)
http://www.thebrain.mcgill.ca/flash/capsules/images/outil_b leu16_img01.jpg
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Introduction: EM Waves Electromagnetic waves are transverse waves. They consist of both a changing electric field and a changing magnetic field. The fields are at right angles to each other and to the direction of the wave.
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Introduction: EM Waves
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James Clerk Maxwell 1831 1879 Developed the electromagnetic theory of light Developed the kinetic theory of gases Explained the nature of color vision Explained the nature of
Died of cancer
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In 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena
existence of electromagnetic waves that propagate through space Einstein showed these equations are in agreement with the special theory of relativity
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Electromagnetic Waves In empty space, q = 0 and I = 0 Maxwell predicted the existence of electromagnetic waves
The electromagnetic waves consist of oscillating electric and magnetic fields The changing fields induce each other which maintains the propagation of the wave
A changing electric field induces a magnetic field A changing magnetic field induces an electric field
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Electromagnetic Waves Electromagnetic waves are formed when an electric field couples with a magnetic field. The magnetic and electric fields of an electromagnetic wave are perpendicular to each other and to the direction of the wave. James Clerk Maxwell and Heinrich Hertz studied how electromagnetic waves are formed and how fast they travel.
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Electromagnetic Waves When you listen to the radio, watch TV, or cook dinner in a microwave oven, you are using electromagnetic waves. Radio waves, television waves, and microwaves are all types of electromagnetic waves. They differ from each other in wavelength. Wavelength is the distance between wave crests.
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Electromagnetic Waves Electromagnetic Radiation: Electromagnetic waves are produced by the motion of electrically charged particles. These waves are also called "electromagnetic radiation" because they radiate from the electrically charged particles. They travel through empty space as well as through air and other substances.
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Electromagnetic Waves Scientists have observed that electromagnetic radiation has a dual "personality." Besides acting like waves, it acts like a stream of particles (called "photons") that have no mass. The photons with the highest energy correspond to the shortest wavelengths.
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Electromagnetic Waves When matter is heated, it gives off light. For example, turning on an ordinary light bulb causes an electric current to flow through a metal filament that heats the filament and produces light. The electrical energy absorbed by the filament excites the atoms' electrons, causing them to "wiggle". This absorbed energy released from the atoms in the form of light.
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Electromagnetic Waves
wrong, it still provided insight as to why electromagnetic radiation is given off. It is due to accelerating electrons,
contribution is that the radius of the orbit of the electrons must be constrained to certain values, and not just any value. At such a place the electron could orbit the nucleus and not give off any radiation.
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Plane em Waves
Assume that the vectors for the electric and magnetic fields in an em wave have a specific space-time behavior that is consistent with
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Plane em Waves
Assume an em wave that travels in the x direction with the electric field in the y direction and the magnetic field in the z direction
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Plane em Waves, cont The x-direction is the direction of propagation Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves Assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only
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Rays A ray is a line along which the wave travels All the rays for the type of linearly polarized waves that have been discussed are parallel The collection of waves is called a plane wave A surface connecting points of equal phase on all waves, called the wave front, is a geometric plane
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Comparing EM vs Mechanical Waves
EM Waves Mechanical Waves Can propagate through vacuum
Need a medium
Transverse waves Transverse or longitudinal
Originate from changing electric / magnetic fields
Originate from the oscillation of the particles of a medium 78
Electromagnetic Vibrations
Electric vibration, E = E0sin (wt kx) Magnetic vibration, B = B0 sin (wt kx) E is perpendicular to B
http://physicsclub.net/physletIndex/waves.html
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No charges, no currents Changing magnetic field creates electric field Changing electric field creates magnetic field
dtdldB
dtdldE
AdB
AdE
E
pathclosed
B
pathclosed
surfaceclosed
surfaceclosed
00
0
0
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EM wave
vBEvv
tkxBB
tkxEE
x
z
y
/
)sin(
)sin(
0
0
vk
f
f
k
2
2
smv /100.31041085.8
11 8
71200
vB
vEBE
The speed of light!!
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EM spectrum
f
c speed of light (m/s) f frequency (Hz=1/s)
wavelength (m)
fc
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Energy in EM wave EM waves transport energy Energy density: Poynting vector (energy transported by EM wave per unit time per unit area) Average energy per unit time per unit area
2
0
20 2
121 BEu
BES0
1
rmsrms BES0
1
cEB
BE
/2
121
00
20
0
200
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Average intensity Displacement D follows harmonic oscillation: Intensity (brightness for light) I is proportional to electric field squared Average over time (one period of oscillation) I:
)sin(0 tDD
)(sin20
2 tIIDI
2)2cos1(
21
21sin
21
)(sin1)(sin1
02
00
2
0
20
0
20
0
20
IdxxIxdxI
tdtT
IdttT
IITT
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Energy transported by waves Intensity of oscillation I (energy per unit area/ per sec) is proportional to amplitude squared D2
3D wave (from energy conservation):
D12 4pr1
2= D22 4pr2
2
D1/D2=r2/r1
Amplitude of the wave is inversely proportional to the distance to the source: r
D 1
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Properties of EM Waves -like,
with both E and B satisfying a wave equation Electromagnetic waves travel at the speed of light
This comes from the solution of
oo
1c
86
Properties of em Waves The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation
This can be summarized by saying that electromagnetic waves are transverse waves
87
Properties of em Waves The magnitudes of the fields in empty space are related by the expression
This also comes from the solution of the partial differentials obtained from
Electromagnetic waves obey the superposition principle
BEc
88
Derivation of Speed: Some Details
space, the following partial derivatives can be found: These are in the form of a general wave equation, with Substituting the values for o and o gives c = 2.99792 x 108 m/s
2 2 2 2
2 2 2 2o o o oE E B Band
x t x t
1 o ov c
89
E to B Ratio Some Details The simplest solution to the partial differential equations is a sinusoidal wave:
E = Emax cos (kx t) B = Bmax cos (kx t)
The angular wave number is k = 2 is the wavelength
The angular frequency is = 2
)sin()sin(
00
00tkxBBtkxEE
90
E to B Ratio Details, cont The speed of the electromagnetic wave is Taking partial derivations also gives
2 ?2
ck
max
max
E E cB k B
91
em Wave Representation This is a pictorial representation, at one instant, of a sinusoidal, linearly polarized plane wave moving in the x direction E and B vary sinusoidally with x 92
Doppler Effect for Light Light exhibits a Doppler effect
Remember, the Doppler effect is an apparent change in frequency due to the motion of an observer or the source
Since there is no medium required for light waves, only the relative speed, v, between the source and the observer can be identified
93
Doppler Effect The equation also depends on the laws of relativity v is the relative speed between the source and the observer
c is the speed of light
light seen by the observer
source
? ? c vc v
94
Doppler Effect For galaxies receding from the Earth v is entered as a negative number
wavelength, actual wavelength The light is shifted toward the red end of the spectrum
This is what is observed in the red shift
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Electromagnetic Waves When normal white light, such as that from the sun, is passed through a prism, the light separates into a continuous spectrum of colors: Continuous (white light) spectra
Bohr knew that when pure elements were excited by heat or electricity, they gave off distinct colors rather than white light. This phenomenon is most commonly seen in modern-day neon lights, tubes filled with gaseous elements (most commonly neon).
96
Electromagnetic Waves
97
Electromagnetic Waves White light spectra:
98
Electromagnetic Spectrum The EM spectrum is the entire range of wavelengths (or frequencies) of EM waves, including the visible spectrum
99
Electromagnetic radiation with wavelengths between 400 nm and 700 nm is visible. In order of decreasing wavelength (increasing frequency), the colors are red (700 nm), orange, yellow, green, blue, violet (400 nm).
X-ray UV Infrared wave
Visible
ray radio
7000Å 4000Å
The Electromagnetic Spectrum
100
The Spectrum of EM Waves Various types of electromagnetic waves make up the em spectrum There is no sharp division between one kind of em wave and the next All forms of the various types of radiation are produced by the same phenomenon accelerating charges
101
The spectrum of electromagnetic waves Various types of electromagnetic waves, distinguished by frequency or wavelength, make up the EM spectrum. Radio waves (104 m to ~ 0.1 m ): Radio and television communication Microwaves (0.3 m to 10-4 m): Radar systems, microwave ovens Infrared waves (10-3 m to 7 ×10-7 m): Produced by hot objects and molecules
102
The spectrum of electromagnetic waves
Visible (700 nm to 400 nm): Different wavelengths = different colors Ultraviolet (4×10-7 m to 6 ×10-10 m) X-rays (10-8 m to 10-12 m) Gamma rays (10-10 m to 10-
14 m) Emitted by radioactive nuclei
103
The EM Spectrum Note the overlap between types of waves Visible light is a small portion of the spectrum Types are distinguished by frequency or wavelength
104
Notes on The EM Spectrum Radio Waves
Wavelengths of more than 104 m to about 0.1 m Used in radio and television communication systems
Microwaves Wavelengths from about 0.3 m to 10-4m Well suited for radar systems Microwave ovens are an application
105
Notes on the EM Spectrum, 2 Infrared waves
Wavelengths of about 10-3 m to 7 x 10-7 m
Produced by hot objects and molecules Readily absorbed by most materials
Visible light Part of the spectrum detected by the human eye Most sensitive at about 5.5 x 10-7 m (yellow-green)
106
More About Visible Light
Different wavelengths correspond to different colors The range is from red ( ~7 x 10-7 m) to violet ( ~4 x 10-
7 m)
107
Visible Light Specific Wavelengths and Colors
108
Notes on the EM Spectrum Ultraviolet light
Covers about 4 x 10-7 m to 6 x10-10 m Sun is an important source of uv light Most uv light from the sun is absorbed in the stratosphere by ozone
X-rays Wavelengths of about 10-8 m to 10-12 m Most common source is acceleration of high-energy electrons striking a metal target Used as a diagnostic tool in medicine
109
Notes on the EM Spectrum Gamma rays
Wavelengths of about 10-10m to 10-14 m Emitted by radioactive nuclei Highly penetrating and cause serious damage when absorbed by living tissue
Looking at objects in different portions of the spectrum can produce different information
110
Wavelengths and Information These are images of the Crab Nebula They are (clockwise from upper left) taken with
x-rays visible light radio waves infrared waves
Summary
112
Summary Vibrating objects are sources of waves, which may be either a pulse or continuous. Wavelength: distance between successive crests. Frequency: number of crests that pass a given point per unit time. Amplitude: maximum height of crest. Wave velocity, v = f
113
Electromagnetic Vibrations
Electric vibration, E = E0sin (wt kx) Magnetic vibration, B = B0 sin (wt kx) E is perpendicular to B
http://physicsclub.net/physletIndex/waves.html
114
c
oo
1c C = 2.998 X 108 ms-1
0 = 8.85 X 10-12 Fm-1 (Free space permittivity)
0 = 4 x 10-7 Hm-1 (Free space permittivity)
115
Electromagnetic Waves Spectrum Electromagnetic waves come in many wavelengths and frequencies. Each one is useful in different ways.
http://science.hq.nasa.gov/kids/imagers/ems/index.html
116
Spectrum of Electromagnetic Radiation Region Wavelength
(Angstroms) Wavelength
(centimeters) Frequency
(Hz) Energ
y (eV)
Radio > 109 > 10 < 3 x 109 < 10-5
Microwave 109 - 106 10 - 0.01 3 x 109 - 3 x 1012 10-5 - 0.01
Infrared 106 - 7000 0.01 - 7 x 10-5 3 x 1012 - 4.3 x 1014 0.01 - 2
Visible 7000 - 4000 7 x 10-5 - 4 x 10-
5 4.3 x 1014 - 7.5 x
1014 2 - 3
Ultraviolet 4000 - 10 4 x 10-5 - 10-7 7.5 x 1014 - 3 x 1017 3 - 103
X-Rays 10 - 0.1 10-7 - 10-9 3 x 1017 - 3 x 1019 103 - 105
Gamma Rays < 0.1 < 10-9 > 3 x 1019 > 105
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