Modern Physics

Level: ThreeCourse Code and Number: PHY 250

1 Modern Physics

المملكة العربية السعوديةجامعة اإلمام محمد بن

سعود اإلسالميةكلية العلوم

قسم الفيزياء

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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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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Contents• 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.

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INTRODUCTION

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• 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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• 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

Introduction

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• 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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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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Introduction

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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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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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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Postulates of The Special Theory

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• 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.

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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Time Dilation

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• The time interval for a physical event is measured differently by observers in different inertial frames of reference

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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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:

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Time Dilation

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)

• “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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i.e. ∆t is always > ∆tp because γ is always > 1

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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Solution

•Proper time ∆tp=3 sec

•Moving pendulum takes longer to complete a period than a pendulum at rest does

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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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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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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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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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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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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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Length Contraction: Simulation

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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.

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!Modern Physics28

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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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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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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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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Lorentz → Galilean

• When v << c, the Lorentz transformations should reduce to the Galilean transformation, i.e.

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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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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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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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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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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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Classical and Relativistic Momentum

• The figure depicts how pvaries with u/c for both γmu andMu. When u/c is small, mu andγmu are very much the same.As u approaches c, the curve forγmu rises more steeply. If u=c,Then p=∞, which is impossibleThis is another reason why wecan not accelerate an object to thespeed of light.

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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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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 thespeed of the particle is doubled, what happens to its momentum?

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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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Applications of Relativity: (3) Relativistic Energy

• Now, if (F) is not constant then we can write the general form for classical K.E. as

• The relativistic form of K.E. is

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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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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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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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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:

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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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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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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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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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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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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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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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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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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

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4TeAetotal

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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Blackbody Radiation: UV Catastrophic

• According to the previous formula, the energy densityu(ν)dν (J/m3) should increaseas ν2. The experimentalresults however showed thatu(ν)dν falls to zero as ν→∞.This discrepancy becameKnown as the UV catastrophic.

Dr. Mohamed KhaterModern Physics61

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.

Dr. Mohamed KhaterModern Physics62

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

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