Download Modern Physics

‫اململكة العربية السعودية‬
‫جامعة اإلمام محمد بن سعود اإلسالمية‬
‫كلية العلوم‬
‫قسم الفيزياء‬
‫‪Modern Physics‬‬
‫‪Level: Three‬‬
‫‪Course Code and Number: PHY 250‬‬
‫‪Modern Physics‬‬
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AIM
• There are two main objectives of the course:
1. First, to provide simple, clear, and
mathematically uncomplicated
explanations of physical concepts and
theories of modern physics
2. And, to clarify and show support for
these theories through a broad range of
current applications and examples
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Modern Physics
PROLOGUE
• Prerequisites: PHY 101, PHY 105, MAT 101 & MAT 102
• Main Resources: Modern Physics, 3rd ed., R
Serway et al., Thomson Learning, 2005. (First 5
chapters, 9, 11, 13)
Concepts of Modern Physics, 5th ed., A
Beiser, McGraw-Hill, 2003. (First 4 chapters, 6, 8,11)
• The course is divided into 6 main chapters:
Relativity, the Quantum Theory of Light, Introduction to
Quantum Physics, Atomic Structure, Molecular Structure,
Nuclear Structure
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Modern Physics
Contents
•
•
•
•
•
•
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Relativity: Einstein’s principle of special relativity, consequences of special
relativity, the Lorentz transformation equations, relativistic momentum and the
relativistic form of Newton’s laws, relativistic energy, equivalence of mass and
energy.
The Quantum Theory of Light: Particle properties of waves, blackbody radiation
and Planck’s hypothesis, the photoelectric effect, explanation of the photoelectric
effect, the x-rays and some applications, the Compton effect, pair production.
Introduction to Quantum Physics: Photons and electromagnetic waves, wave
properties of particles, De Broglie waves, matter waves, the electron microscope,
the uncertainty principle.
Atomic Structure: the particle nature of matter, early models of the atom, Bohr’s
quantum model of the hydrogen atom, atomic spectra and transitions, nuclear
effects on spectral lines, the Franck-Hertz experiment.
Molecular Structure: Molecular bonding, energy states and spectra, molecular
vibration and rotation, electronic transitions in molecules.
Nuclear Structure: Nuclear composition, some properties of nuclei, binding
energy, radioactivity.
Modern Physics
INTRODUCTION
• The end of Physics!
Newton’s laws of motion and his universal
theory of gravitation, Maxwell’s theoretical
work in unifying electricity and magnetism,
and the laws of thermodynamics and kinetic
theory employed mathematical methods to
successfully explain a wide variety of
phenomena
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Modern Physics
Introduction
• Max Planck 1900 & Albert Einstein 1905:
Planck provided the basic ideas led to the
quantum theory & Einstein formulated his
special theory of relativity
• These developments led to understand the
nature, behavior, structure and properties of
many materials
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Modern Physics
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Modern Physics
Introduction
• So, what is Modern Physics. It is a group of
theoretical concepts and principles that
perfectly explains many of experimental
physical phenomenon which classical physics
fails with. In addition to Planck and Einstein,
many other scientists during the 20th century
contributed to modern physics by discovering
the theoretical foundations led to the
development of new physics fields such as
nuclear, molecular, particle and solid state
physics
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Modern Physics
Introduction
• Examples of technologies based on modern
physics: High Temperature Superconductors
(HTS), Positron Emission Tomography (PET),
Magnetic Resonance Imaging (MRI), Particle
Accelerators (PA), Global Positioning Systems
(GPS) and TV Displays (TVD)
• Applications in chemistry, astronomy, biology,
geology, and engineering have also made use
of modern physics
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Modern Physics
Special Theory of Relativity
• Special theory of relativity is a “general” theory!
Because it describes the motion of ALL objects at ALL
speeds. The Newtonian mechanics is therefore an
approximation of the special relativity
• Measurements of time and space are not absolute,
they are influenced by the dynamical state of an
observer and what is being observed
• No exaggeration in saying that special theory of
relativity had revolutionized science in general so
that our understanding of the physical universe has
been significantly improved
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Modern Physics
Special Theory of Relativity
• Relativity connects between all phenomena of nature:
space and time, matter and energy, electricity and
magnetism
• The beauty of this marvelous theory also originates
from the fact that conclusions can readily be reached
with only the simplest of mathematics
• Einstein once said: “The relativity theory arose from
necessity, from serious and deep contradictions in the
old theory from which there seemed no escape. The
strength of the new theory lies in the consistency and
simplicity with which it solves all these difficulties,
using only a few very convincing assumptions”
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Modern Physics
The General Theory
• What is the general theory of relativity (Einstein 1915)?
It describes the relationships between gravity and the
geometrical structure of space and time. Remarkable
results include: light rays are affected by gravity, and
the big bang theory (the universe is continually
expanding)
• The general theory of relativity concerns with
accelerating frames of reference
• Special theory of relativity, on the contrary, is only
concerned with inertial frames of reference, that is,
frames moving with constant velocities (no
acceleration)
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Modern Physics
Postulates of The Special Theory
• The laws of physics must be the same for all
inertial reference frames: these laws have the
same mathematical form for all observers moving
at constant velocity with respect to one another
• The speed of light is always constant: The
measured value (3x108 m/s) is independent of
the motion of the observer or of the motion of
the source of light
• Some relativistic consequences had immediately
originated from the theory; the most important
will be presented here.
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Modern Physics
Before we advance, we must agree on
the following:
• The special theory of relativity has more to do
with philosophy than with exact science,
therefore, it may most of the time not agree
with human intuition and sensibility.
• Relativity is most successful for objects
moving only with speed close to c, i.e.
relativistic speeds. The effect the theory has
on daily-life objects is barley noticeable, if
any!
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Modern Physics
Time Dilation
• The time interval for a physical event is measured
differently by observers in different inertial frames of
reference
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Modern Physics
Time Dilation
• O concludes that, because of the motion of the vehicle, if the
light is to hit the mirror, it must leave the laser at an angle
with respect to the vertical direction
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Modern Physics
Time Dilation
• Since both observers must measure c for the speed of light, it
follows that the time interval ∆t measured by O is longer than
the time interval ∆t’ measured by O’. The Pythagorean
theorem gives:
Time Dilation
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Modern Physics
Time Dilation
• t’ is usually written as tp and called the proper time (the time
interval between two events as measured by an observer who
sees the events occur at the same point in space)
i.e. ∆t is always > ∆tp
because γ is always > 1
• “A moving clock runs slower than a clock at rest by a factor of
γ ”. In fact, we can generalize these results by stating that all
physical processes, including chemical reactions and biological
processes, slow down when observed from another reference
frame
• The heartbeat rate of an astronaut on earth and through
space!
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Modern Physics
Example
• The period of a pendulum is measured to be 3.0 s in
the rest frame of the pendulum. What is the period
of the pendulum when measured by an observer
moving at a speed of 0.95c with respect to the
pendulum? What would be the period if the speed of
the observer is increased to 1c, 1000 km/hr.?
Hint: ALL matter objects can never have speeds faster
than or even equal to the speed of light!
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Modern Physics
Solution
Proper time ∆tp=3 sec •
Moving pendulum takes longer to complete a •
period than a pendulum at rest does
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Modern Physics
Doppler Effect (Sound)
• It is the change in frequency of sound waves as
the source approaches or recedes from a
stationary observer who hears different pitch
than that occurs in normal situations
• The separation (wavelength) between emitted
waves varies and hence the frequency
• The effect does not depend on the loudness
(amplitude energy) of the waves
• We get the same effect with the observer moving
while the source remains stationary
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Modern Physics
Doppler Effect (Light)
• Spectral lines emitted from distant stars and galaxies (that
are billions of years away from us!) are broadened (spread)
and red shifted (toward the low frequency end of the EM
spectrum)
• The measurements indicate that these objects are receding
from us (with speeds ≈ 104 km.s-1!) and from one another
too and the recession speed is directly proportional to
distance (Hubble’s law)
• Every 106 years, the recession speed increases on average
by 20 km.s-1!
• The expansion started 13 billion years ago when a very
small, dense and hot mass of matter explodes violently (the
big bang theory)
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Modern Physics
Length Contraction
• Like time interval, the measurement of length
interval (distance between two points) is not also
absolute but depends on the frame of reference
in which it is measured
• An object whose length at rest is Lp (the proper
length) APPEARS to be contracted to a new length
L (where L < Lp) when it moves relative to a
stationary observer
• Lp is defined similarly as tp as the length of the
object measured by someone who is at rest with
respect to the object
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Modern Physics
Length Contraction
• Consider a spaceship traveling with a speed v
from one star to another and two observers, one
on Earth and the other in the spaceship. The
space traveler claims to be at rest and sees the
destination star as moving toward the spaceship
with speed v. He then measures a smaller time of
travel: ∆tp = ∆t/ γ. On the other hand, the
distance Lp between the stars as measured by the
earth observer is Lp = v∆t.
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Modern Physics
Length Contraction
• Because the space traveler reaches the star in the
shorter time ∆tp , he concludes that the distance, L,
between the stars is shorter than Lp and is given by: L
= v∆tp = v ∆t/ γ = vLp / γ v = Lp / γ
Length contraction
where (1-v2/c 2)1/2 is a factor less than 1. So L is always
< Lp.
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Modern Physics
Length Contraction
• If an object has a proper length
Lp when it is measured by an
observer at rest with respect to
the object, when it moves with
speed v in a direction parallel
to its length, its length L is
measured to be shorter by a
factor of 1/γ
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Modern Physics
Length Contraction: Simulation
When the shutter of the camera is opened, it records the shape of the object at
a given instant of time. Because light from different parts of the object must
arrive at the shutter at the same time, light from more distant parts of the
object must start its journey earlier than light from closer parts as in (a). This is
not the case in (b), and the camera records different parts of the object at
different times. This results in a highly distorted image, which shows horizontal
length contraction, vertical curvature, and image rotation.
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Modern Physics
The Twin Paradox
• It is a famous relativistic effect, which involves an
identical twin one of them (X) remains on earth
while the other (Y) is taken on a trip into a distant
star at speed v and eventually brought back
• Y is 20 years old when he takes off at a speed of
0.8c to the star which is 20 light-years away
• To Y, the distance L he has covered is shortened
to: Lp / γ = 12 light years only!
• Although time goes by the usual rate, Y’s two-way
voyage to the star has taken L/v = 30 years
• But for X, he had to wait (t/tp)x30=50 years!
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Modern Physics
The Twin Paradox
• So, according to each one’s clock, Y is now 50 years
old while X is 70 years old!
• Amazingly, the relativistic paradox effect has been
verified experimentally on earth by sending clocks on
board of airplanes that goes around the world with
non-relativistic speeds. Each single travelling clock
has always shown to be delayed with respect to the
clocks left behind (although brief but noticeable)
• Theoretically, life processes such as heartbeats &
respiration will be less for Y than X for the same
period of time; i.e. the biological clocks of X & Y will
be different
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Modern Physics
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Modern Physics
Lorentz Transformations
• The Lorentz transformation formulas provide a formal and
concise method of solution of relativistic problems
• The Lorentz coordinate transformation is a set of formulas
that relates the space and time coordinates of two inertial
observers moving with a relative speed v. We have already
seen two consequences of the Lorentz transformation in the
time dilation and length contraction formulas
• The Lorentz velocity transformation is the set of formulas that
relate the velocity components ux, uy, uz of an object moving
in frame S to the velocity components u’x, u’y, u’z of the same
object measured in frame S’, which is moving with a speed v
relative to S
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Modern Physics
Lorentz Coordinate Transformations
• the complete coordinate transformations between
an event found to occur at (x, y, z, t) in S and (x’, y’, z’,
t) in S’ are
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Modern Physics
Inverse Lorentz Coordinate Transformations
• If we wish to transform coordinates of an event in
the S’ frame to coordinates in the S frame, we simply
replace v by -v and interchange the primed and
unprimed coordinates in the previous equations. The
resulting inverse transformation is given by
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Modern Physics
Lorentz → Galilean
• When v << c, the Lorentz transformations should
reduce to the Galilean transformation, i.e.
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Modern Physics
Lorentz Velocity Transformations
• The relativistic form of the velocity transformation (S frame) is
• If the object has velocity components uy and uz along y and z
respectively (also in S frame), the components in S’ are
• For obtaining the inverse transformation (S’ frame), we apply
the previous rules to get
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Modern Physics
Example
• (a) Imagine a motorcycle rider moving with a speed
of 0.80c past a stationary observer. If the rider
throws a ball in the forward direction with a speed of
0.70c with respect to himself, what is the speed of
the ball as seen by the stationary observer?
(b) Suppose that the motorcyclist turns on a beam of
light in the same direction as he moves. What would
the stationary observer measure for the speed of the
beam of light?
Hint: v (S’ frame) = 0.80c; u’x = 0.70c
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Modern Physics
Example
• An observer on Earth observes two spacecrafts
moving in the same direction toward the Earth.
Spacecraft A appears to have a speed of 0.50c, and
spacecraft B appears to have a speed of 0.80c. What
is the speed of spacecraft A measured by an observer
in spacecraft B?
Hint: ux = 0.50c and v (S’ frame) = 0.80c
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Modern Physics
Applications of Relativity: (1) Relativistic
Momentum
• The conservation of linear momentum states that when two
bodies collide, the total momentum remains constant
assuming the bodies are isolated (that is, they interact only
with each other)
• Now suppose the collision is described in a reference frame S
in which momentum is conserved. If the velocities of the
colliding bodies are calculated in a second moving inertial
frame S’ using the Lorentz transformation, and the classical
definition of momentum p=mu applied, one finds that
momentum is not conserved in the second reference frame S’
• However, because the laws of physics are the same in all
inertial frames, momentum must be conserved in all frames if
it is conserved in any one!
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Modern Physics
Relativistic Momentum
• It is found that momentum is conserved in both S and S’, (and
indeed in all inertial frames), if we redefine momentum as
• where u is the velocity of the particle and m is the proper
(rest) mass, that is, the mass measured by an observer at rest
with respect to the mass (relativistic mass=mγ)
• When u is much less than c, the above equation reduces to
the classical form of momentum
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Modern Physics
Classical and Relativistic Momentum
• The figure depicts how p
varies with u/c for both γmu and
Mu. When u/c is small, mu and
γmu are very much the same.
As u approaches c, the curve for
γmu rises more steeply. If u=c,
Then p=∞, which is impossible
This is another reason why we
can not accelerate an object to the
speed of light.
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Modern Physics
u
u
u
Applications of Relativity: (2) Relativistic Form of
Newton’s Second Law
• The relativistic form of Newton’s second law is given by the
expression
• This expression is logical because it protects classical
mechanics in the limit of low velocities and requires the
momentum of an isolated system (Fext= 0) to be conserved
relativistically as well as classically
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Modern Physics
Example
• An electron, which has a mass of 9.11x10-31 kg,
moves with a speed of 0.750c. Find its relativistic
momentum and compare this with the momentum
calculated from the classical expression. Take
c=3x10m8 m/s
• A particle is moving at a speed of less than c/2. If the
speed of the particle is doubled, what happens to its
momentum?
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Modern Physics
Applications of Relativity: (3) Relativistic
Energy
• The definition of momentum and the laws of motion
required generalization to make them compatible
with the principle of relativity. This implies that the
relativistic form of the kinetic energy must also be
modified
• We begin with the fact that the work done (W) on an
object by a constant force (F) through a distance (s) is
W=F.s. If no other forces act on the object and it
starts motion from rest, then W=K.E.=F.s
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Modern Physics
Applications of Relativity: (3) Relativistic
Energy
• Now, if (F) is not constant then we can write the general form
for classical K.E. as
s
s
0
0
K .E.   Fds  m  ads
s
u
ds
1
 K .E.  m  du  m  udu  mu 2
dt
2
0
0
• The relativistic form of K.E. is
K .E. 
mc2
 mc2  mc2  mc2
u2
1 2
c
 mc2  Etotal  K .E.  mc2
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Modern Physics
Applications of Relativity: (3) Relativistic
Energy
• The last equation E=γmc2 is Einstein’s famous mass–energy
equivalence equation, which shows that mass is a measure of
the total energy in all forms. It not only applies to particles
but also to macroscopic objects
• It has the remarkable implication that any kind of energy
added to a “brick” of matter—electric, magnetic, elastic,
thermal, gravitational, chemical—actually increases the mass!
• Another implication of Equation’s equation is that a small
mass corresponds to an enormous amount of energy because
c2 is a very large number. This concept has revolutionized the
field of nuclear physics
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Modern Physics
Kinetic Energy at Low Speeds
• The classical formula for KE [KE=(1/2)mv2] for speeds much
smaller than c has experimentally been already verified. Let
us check if this is true by considering the relativistic formula
for KE:
• Since v2/c2 << 1, we can use the binomial approximation
(1 + x)n ≈ 1 + nx, and this is only valid for |x| << 1:
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Modern Physics
Energy-Momentum Relation
• If the object at rest (u=0 & K.E.=0), i.e. γ=1, then its
total energy is called the rest energy and termed E0 =
mc2
• In many situations, the momentum or energy of a
particle is measured rather than its speed. It is
therefore useful to have an expression relating the
total energy E to the relativistic momentum p. This is
accomplished using E= γmc2 and p=γmu. This will be
done on the next slide
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Modern Physics
Total Energy-Momentum Relation
• We show that the total energy–momentum relationship is
given by E2= p2c2+(mc2)2 . Take E= γmc2 and p=γmu. By
squaring the two equations and then subtracting, we get:
2 4
2 2
2
m
c
m
u
c
2
E2 
&
p

 2
2
2
u
u
c
(1  2 )
(1  2 )
c
c
2
u
m 2 c 4 (1  2 )
2 2 2
mu c
2 2
2
2 2
c
p c 

E

p
c

u2
u2
(1  2 )
(1  2 )
c
c
2
 E 2  p 2 c 2  (mc 2 ) 2  Etotal
 E02  p 2 c 2
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Modern Physics
Total Energy-Momentum Relation
• Because the rest energy (E0) is invariant quantity, the quantity
will also be so, i.e. has the same value in all inertial frames
of reference
• The energy-momentum relationship holds true also for a system of
many particles provided that (m) represents the entire system.
• However, E0 of an isolated system may be greater than or less than
the sum of the rest energies of its constituents. Examples include
neutrons and protons within an atomic nucleus
• Except for the hydrogen atom, this difference in energy is called the
“binding energy” of the nucleus (energy needed to break up)
• For comparison purposes, a typical binding energy is 1012 kJ/kg of
nuclear matter, while the binding energy of water molecules is only
in the order of 103 kJ/kg of liquid water
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Modern Physics
Massless Particles: The Photon
• In classical physics, any particle that does not have mass is
considered absent. The reason is that both its total energy
(Etotal=KE+PE) and momentum (p=mu) are functions in mass.
• Considered to be the general theory, relativistic mechanics provides
the same result when we substitute m=0 and u<<c in the equations
Etotal=γmc2 and p=γmu.
• However, when m=0, but u=c: Etotal=0/0 and p=0/0, which are
indeterminate, i.e. Etotal and p can have any values
• In this case, the total energy of such particles is given by
Etotal2= p2c2+(mc2)2= p2c2+0
Etotal = pc
For Photons
• So, massless particles do exist and they exhibit particle like
properties as energy and momentum.
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Modern Physics
The Quantum Theory of Light
• In classical physics, the mechanics of particles and the optics of
waves are traditionally independent disciplines, each with its own
chain of experiments and principles.
• On the other hand, in the underworld (world of molecules, atoms,
electrons and nuclei), there is no such differentiation. Here, a
moving object, like an electron, can be described as a particle (i.e.
has a mass and charge) or as a wave
• We regard electromagnetic waves (em) as waves because under
suitable circumstances they exhibit diffraction, interference and
polarization phenomena
• Under other circumstances, em waves behave as they consist of
streams of particles. This wave-particle duality is central to a
thorough understanding of modern physics
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Modern Physics
Foundation of Electromagnetic Waves
• In 1831, the British physicist Faraday and American physicist Henry
found out that a changing magnetic field can induce an electrical
current in a wire loop. The British physicist James Clerk Maxwell in
1864, knowing this discovery, noted that a changing magnetic field
is equivalent in its effect to an electric field, which also indicates a
current in the same wire
• This result led Maxwell to suggest the converse: a changing electric
field is associated with a changing magnetic field, confirming what
had the Danish physicist Orested discovered in 1819. Maxwell was
able to show that accelerated electric charges (like moving
electrons under the effect of electric field) generate “linked”
electric and magnetic disturbances that can travel indefinitely
through space
• Because these electrons oscillate periodically (i.e. their intensity
and direction change with time), the resulting disturbances are
waves whose electric and magnetic components are perpendicular
to each other and to the direction of propagation
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Modern Physics
Electromagnetic Waves
• Maxwell was able to show that the speed c of such electromagnetic
waves in free space is given by
where the constant quantities ε0 and μ0 are the electric permittivity
and magnetic permeability of free space, respectively constant
quantities
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Modern Physics
Electromagnetic Waves
• Maxwell finally correctly concluded that light rays is em waves. He
died of cancer in 1879 (the same year that Einstein was born! It is
also amazing to know that Newton was born in the same year that
Galileo died!) before he could see the experimental confirmation of
his magnificent theory by the German physicist Hertz in 1888
• Hertz showed that em waves exist and behave exactly as Maxwell
had described. He used alternating currents (ac) in generating his
waves and determined their wavelength (λ), frequency (ν) and
speed (v), and demonstrated that they could be reflected, refracted
and deflected (exactly as light rays)
• Although all types of em waves have the same fundamental nature,
their interaction with matter depends upon their frequencies
(wavelengths)
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Basic Properties of EM Waves
• EM waves have the same (c) speed in free space
• EM waves obey the principle of superposition (interference): “If two
or more waves of the same nature pass at a point at the same time,
they form a resultant wave whose amplitude is the sum of
amplitudes of the individual waves”.
• The amplitude of em waves is usually considered as the electric
field component (E) since it is the parameter that interacts with
matter and gives rise to nearly all common optical effects
• EM waves diffract. A strong example of diffraction of em waves is
taken from Young’s double slit experiment when secondary waves
spread out from each slit: the original wave is split into many. These
diffracted waves are then superimposed and fall onto a screen
located behind the slits forming an interference pattern on the form
of bright and dark lines
• Young’s experiment is another proof that light consists of waves;
Maxwell’s theory tells us that these waves are electromagnetic in
nature
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Modern Physics
Particle Properties of Waves: Blackbody
Radiation
• Although certain that light is em waves, scientists wanted to
understand the origin of the em radiation emitted by objects of
matter. They eventually failed to do so using existed theories at that
time, which triggered the “physics crisis” during the 19th century
• Thomas Wedgwood, a maker of porcelain, seems to have been the
first to note the universal character of the radiation emitted by
heated objects. In 1792, he observed that all the objects in his
ovens, regardless of their chemical nature, size, or shape, became
red at the same temperature. This crude observation was
sharpened considerably by the advancing state of spectroscopy, so
that by the mid-1800s it was known that glowing solids emit
continuous spectra rather than the bands or lines emitted by
heated gases
• Since a body at a constant temperature is in thermal equilibrium
with surroundings, it must absorb energy from them at the same
rate as it emits energy
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Modern Physics
Blackbody Radiation
• Based upon this simple thermodynamical fact, In 1859 Kirchhoff
(German) formulated his theory that for any body in thermal
equilibrium the emitted power is proportional to the power
absorbed: ef = J(f, T).Af , where ef is the power emitted per unit area
per unit frequency, Af is the absorption power absorbed per unit
area per unit frequency, and J( f, T) is a universal function (the same
for all bodies) that depends only on f, the light frequency, and T, the
absolute temperature of the body.
• A blackbody (perfect absorber) is defined as an object that absorbs
all the em radiation falling on it and consequently appears black. It
has Af = 1 for all frequencies and so Kirchhoff’s theory for a
blackbody becomes: ef = J(f, T). This equation shows that the
power emitted per unit area per unit frequency by a blackbody
depends only on temperature and light frequency and not on the
physical and chemical makeup of the blackbody, in agreement with
Wedgwood’s early observation.
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Modern Physics
Blackbody Radiation: Stefan’s Law
• Emission from a glowing solid. Note that the amount of
radiation Emitted (the area under the curve)
Increases rapidly with Increasing temperature
• The next important development
came from the Austrian physicist Stefan
in 1879. He found experimentally that
the total power per unit area (A) emitted at
all frequencies by a hot solid, etotal, was
proportional to the fourth power of its
absolute temperature. Stefan’s law may
be written as
etotal  eAT 4
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Modern Physics
Blackbody Radiation
where σ is called the Stefan–Boltzmann constant and e is equal
to 1. A body that is not an ideal radiator will obey the same
general law but with a coefficient, e, less than 1.
• In 1893, Wilhelm Wien proposed a general formula that
explained the experimental behavior of λmax with temperature
shown on the previous slide. This law is called Wien’s
displacement law and may be written
• Rayleigh and Jeans came up with a theoretical formula
claimed to explain the experimental radiation curves in the
blackbody spectrum:
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Modern Physics
Blackbody Radiation: UV Catastrophic
• According to the previous
formula, the energy density
u(ν)dν (J/m3) should increase
as ν2. The experimental
results however showed that
u(ν)dν falls to zero as ν→∞.
This discrepancy became
Known as the UV catastrophic.
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Modern Physics
Dr. Mohamed Khater
Blackbody Radiation: Planck’s Formula
• Max Planck, a German physicist, realized the fundamental
effect of the problem and its independence on matter
structure and was able to modify the Rayleigh-Jeans formula
in 1900 to be:
• Planck derived his formula by assuming, correctly, that energy
is emitted and absorbed in discrete steps of (hν) not in a
continuous stream; each hν is called a quantum (plural
quanta) from the Latin word “how much”. He was awarded
the Nobel prize in physics in 1918 for this specific discovery.
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Modern Physics
Dr. Mohamed Khater
Blackbody Radiation: Planck’s Formula
• At high ν, i.e. hν >> kT
ehν/kT → ∞ and u(ν)dν→0 as
experimentally observed. On the other hand, at low ν, i.e. hν
>> kT
ehν/kT ≈ 1+(hν/kT) [If ex=1+x+(x2/2!)+(x3/3!)+…,
and x is very small then ex ≈1+x because the other terms are
very small], and Planck formula becomes:
which is Rayleigh-Jeans formula
63
Modern Physics