chapter 24 electromagnetic waves. electromagnetic waves, introduction electromagnetic (em) waves...
TRANSCRIPT
Chapter 24
Electromagnetic Waves
Electromagnetic Waves, Introduction Electromagnetic (em) waves permeate
our environment EM waves can propagate through a
vacuum Much of the behavior of mechanical
wave models is similar for em waves Maxwell’s equations form the basis of
all electromagnetic phenomena
Conduction Current A conduction current is carried by charged
particles in a wire The magnetic field associated with this
current can be calculated using Ampère’s Law:
The line integral is over any closed path through which the conduction current passes
od I B s
Conduction Current, cont. Ampère’s Law in this
form is valid only if the conduction current is continuous in space
In the example, the conduction current passes through only S1
This leads to a contradiction in Ampère’s Law which needs to be resolved
Displacement Current Maxwell proposed the resolution to the
previous problem by introducing an additional term This term is the displacement current
The displacement current is defined as
Ed o
dI
dt
Displacement Current, cont. The changing electric field may be considered
as equivalent to a current For example, between the plates of a capacitor
This current can be considered as the continuation of the conduction current in a wire
This term is added to the current term in Ampère’s Law
Ampère’s Law, General The general form of Ampère’s Law is
also called the Ampère-Maxwell Law and states:
Magnetic fields are produced by both conduction currents and changing electric fields
( ) Eo d o o o
dd I I I
dt
B s
Ampère’s Law, General – Example The electric flux through
S2 is EA S2 is the gray circle A is the area of the
capacitor plates E is the electric field
between the plates If q is the charge on the
plates, then Id = dq/dt This is equal to the
conduction current through S1
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 Saturn’s rings
Died of cancer
Maxwell’s Equations, Introduction In 1865, James Clerk Maxwell provided a
mathematical theory that showed a close relationship between all electric and magnetic phenomena
Maxwell’s equations also predicted the existence of electromagnetic waves that propagate through space
Einstein showed these equations are in agreement with the special theory of relativity
Maxwell’s Equations In his unified theory of
electromagnetism, Maxwell showed that electromagnetic waves are a natural consequence of the fundamental laws expressed in these four equations:
0o
B Eo o o
qd d
d dd d I
dt dt
E A B A
E s B s
Maxwell’s Equations, Details The equations, as shown, are for free space
No dielectric or magnetic material is present The equations have been seen in detail in
earlier chapters: Gauss’ Law (electric flux) Gauss’ Law for magnetism Faraday’s Law of induction Ampère’s Law, General form
Lorentz Force Once the electric and magnetic fields
are known at some point in space, the force of those fields on a particle of charge q can be calculated:
The force is called the Lorentz force
q q F E v B
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
Plane em Waves We will assume that the
vectors for the electric and magnetic fields in an em wave have a specific space-time behavior that is consistent with Maxwell’s equations
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
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
We also assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only
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
Properties of EM Waves The solutions of Maxwell’s are wave-like, with
both E and B satisfying a wave equation Electromagnetic waves travel at the speed of
light
This comes from the solution of Maxwell’s equations
oo
1c
Properties of em Waves, 2 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
Properties of em Waves, 3 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 Maxwell’s Equations
Electromagnetic waves obey the superposition principle
BEc
Derivation of Speed – Some Details From Maxwell’s equations applied to empty
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 o
E E B Band
x t x t
1 o ov c
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 ƒ ƒ is the wave frequency
E to B Ratio – Details, cont The speed of the electromagnetic
wave is
Taking partial derivations also gives
2 ƒƒ
2c
k
max
max
E Ec
B k B
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
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
Doppler Effect, cont. 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 ƒ’ is the apparent frequency of the light seen by
the observer ƒ is the frequency emitted by the source
ƒ ' ƒc v
c v
Doppler Effect, final For galaxies receding from the Earth v
is entered as a negative number Therefore, ƒ’<ƒ and the apparent
wavelength, ’, is greater than the actual wavelength
The light is shifted toward the red end of the spectrum
This is what is observed in the red shift
Heinrich Rudolf Hertz 1857 – 1894 Greatest discovery
was radio waves 1887
Showed the radio waves obeyed wave phenomena
Died of blood poisoning
Hertz’s Experiment An induction coil is
connected to a transmitter
The transmitter consists of two spherical electrodes separated by a narrow gap
Hertz’s Experiment, cont The coil provides short voltage surges to the
electrodes As the air in the gap is ionized, it becomes a
better conductor The discharge between the electrodes
exhibits an oscillatory behavior at a very high frequency
From a circuit viewpoint, this is equivalent to an LC circuit
Hertz’s Experiment, final Sparks were induced across the gap of the
receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter
In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties Interference, diffraction, reflection, refraction and
polarization He also measured the speed of the radiation
Poynting Vector Electromagnetic waves carry energy As they propagate through space, they
can transfer that energy to objects in their path
The rate of flow of energy in an em wave is described by a vector called the Poynting vector
Poynting Vector, cont The Poynting Vector is
defined as
Its direction is the direction of propagation
This is time dependent Its magnitude varies in
time Its magnitude reaches a
maximum at the same instant as the fields
1
o S E B
Poynting Vector, final The magnitude of the vector represents
the rate at which energy flows through a unit surface area perpendicular to the direction of the wave propagation This is the power per unit area
The SI units of the Poynting vector are J/s.m2 = W/m2
Intensity The wave intensity, I, is the time
average of S (the Poynting vector) over one or more cycles
When the average is taken, the time average of cos2(kx-t) = ½ is involved
2 2max max max max
2 2 2avgo o o
E B E c BI S
c
Energy Density The energy density, u, is the energy per
unit volume For the electric field, uE= ½ oE2
For the magnetic field, uB = ½ oB2
Since B = E/c and oo1c 2
21
2 2B E oo
Bu u E
Energy Density, cont The instantaneous energy density
associated with the magnetic field of an em wave equals the instantaneous energy density associated with the electric field In a given volume, the energy is shared
equally by the two fields
Energy Density, final The total instantaneous energy density is the
sum of the energy densities associated with each field u =uE + uB = oE2 = B2 / o
When this is averaged over one or more cycles, the total average becomes uav = o (Eavg)2 = ½ oE2
max = B2max / 2o
In terms of I, I = Savg = c uavg The intensity of an em wave equals the average
energy density multiplied by the speed of light
Momentum Electromagnetic waves transport momentum
as well as energy As this momentum is absorbed by some
surface, pressure is exerted on the surface Assuming the wave transports a total energy
U to the surface in a time interval t, the total momentum is p = U / c for complete absorption
Pressure and Momentum Pressure, P, is defined as the force per unit
area
But the magnitude of the Poynting vector is (dU/dt)/A and so P = S / c For complete absorption An absorbing surface for which all the incident
energy is absorbed is called a black body
1 1F dp dU dtP
A A dt c A
Pressure and Momentum, cont For a perfectly reflecting surface,
p = 2 U / c and P = 2 S / c For a surface with a reflectivity somewhere
between a perfect reflector and a perfect absorber, the momentum delivered to the surface will be somewhere in between U/c and 2U/c
For direct sunlight, the radiation pressure is about 5 x 10-6 N/m2
Determining Radiation Pressure This is an apparatus for
measuring radiation pressure
In practice, the system is contained in a high vacuum
The pressure is determined by the angle through which the horizontal connecting rod rotates
Space Sailing A space-sailing craft includes a very large sail
that reflects light The motion of the spacecraft depends on the
pressure from light From the force exerted on the sail by the reflection
of light from the sun Studies concluded that sailing craft could
travel between planets in times similar to those for traditional rockets
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
The EMSpectrum
Note the overlap between types of waves
Visible light is a small portion of the spectrum
Types are distinguished by frequency or wavelength
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 10-4 m Well suited for radar systems Microwave ovens are an application
Notes on the EM Spectrum, 2 Infrared waves
Wavelengths of about 10-3 m to 7 x 10-7 m Incorrectly called “heat waves” 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)
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)
Visible Light – Specific Wavelengths and Colors
Notes on the EM Spectrum, 3 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
Notes on the EM Spectrum, final 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
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
Polarization of Light Waves The electric and magnetic vectors associated
with an electromagnetic wave are perpendicular to each other and to the direction of wave propagation
Polarization is a property that specifies the directions of the electric and magnetic fields associated with an em wave
The direction of polarization is defined to be the direction in which the electric field is vibrating
Unpolarized Light, Example All directions of
vibration from a wave source are possible
The resultant em wave is a superposition of waves vibrating in many different directions
This is an unpolarized wave
The arrows show a few possible directions of the waves in the beam
Polarization of Light, cont A wave is said to be linearly
polarized if the resultant electric field vibrates in the same direction at all times at a particular point
The plane formed by the electric field and the direction of propagation is called the plane of polarization of the wave
Methods of Polarization It is possible to obtain a linearly
polarized beam from an unpolarized beam by removing all waves from the beam expect those whose electric field vectors oscillate in a single plane
The most common processes for accomplishing polarization of the beam is called selective absorption
Polarization by Selective Absorption
Uses a material that transmits waves whose electric field vectors in the plane parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction
Selective Absorption, cont E. H. Land discovered a material that
polarizes light through selective absorption He called the material Polaroid The molecules readily absorb light whose
electric field vector is parallel to their lengths and allow light through whose electric field vector is perpendicular to their lengths
Selective Absorption, final It is common to refer to the direction
perpendicular to the molecular chains as the transmission axis
In an ideal polarizer, All light with the electric field parallel to the
transmission axis is transmitted All light with the electric field perpendicular
to the transmission axis is absorbed
Intensity of a Polarized Beam The intensity of the polarized beam
transmitted through the second polarizing sheet (the analyzer) varies as I = Io cos2 θ
Io is the intensity of the polarized wave incident on the analyzer
This is known as Malus’ Law and applies to any two polarizing materials whose transmission axes are at an angle of θ to each other
Intensity of a Polarized Beam, cont The intensity of the transmitted beam is
a maximum when the transmission axes are parallel = 0 or 180o
The intensity is zero when the transmission axes are perpendicular to each other This would cause complete absorption
Intensity of Polarized Light, Examples
On the left, the transmission axes are aligned and maximum intensity occurs
In the middle, the axes are at 45o to each other and less intensity occurs
On the right, the transmission axes are perpendicular and the light intensity is a minimum
Properties of Laser Light The light is coherent
The rays maintain a fixed phase relationship with one another
There is no destructive interference The light is monochromatic
It has a very small range of wavelengths The light has a small angle of divergence
The beam spreads out very little, even over long distances
Stimulated Emission Stimulated emission is required for laser
action to occur When an atom is in an excited state, an
incident photon can stimulate the electron to fall to the ground state and emit a photon
The first photon is not absorbed, so now there are two photons with the same energy traveling in the same direction
Stimulated Emission, Example
Stimulated Emission, Final The two photons (incident and emitted)
are in phase They can both stimulate other atoms to
emit photons in a chain of similar processes
The many photons produced are the source of the coherent light in the laser
Necessary Conditions for Stimulated Emission For the stimulated emission to occur,
there must be a buildup of photons in the system
The system must be in a state of population inversion More atoms must be in excited states than
in the ground state This insures there is more emission of
photons by excited atoms than absorption by ground state atoms
More Conditions The excited state of the system must be
a metastable state Its lifetime must be long compared to the
usually short lifetimes of excited states The energy of the metastable state is
indicated by E* In this case, the stimulated emission is
likely to occur before the spontaneous emission
Final Condition The emitted photons must be confined
They must stay in the system long enough to stimulate further emissions
In a laser, this is achieved by using mirrors at the ends of the system
One end is generally reflecting and the other end is slightly transparent to allow the beam to escape
Laser Schematic
The tube contains atoms The active medium
An external energy source is needed to “pump” the atoms to excited states
The mirrors confine the photons to the tube Mirror 2 is slightly transparent
Energy Levels, He-Ne Laser This is the energy level
diagram for the neon The neon atoms are
excited to state E3* Stimulated emission
occurs when the neon atoms make the transition to the E2 state
The result is the production of coherent light at 632.8 nm
Laser Applications Laser trapping Optical tweezers Laser cooling
Allows the formation of Bose-Einstein condensates