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Modern Physics1EE
Dr. Hania Farag
Dr. Amr El-Sherif
Course Contents
β’ 1- Special theory of relativity
β’ 2- The particle properties of waves
β’ References:
β’ 1- Modern Physics by Kenneth Krane (third edition)
β’ 2-Concepts of modern Physics by Arthur Beiser (fifth edition)
What is Relativity?β’ Until the end of the 19th century it was believed that Newtonβs three
Laws of Motion and the associated ideas about the properties of space and time provided a basis on which the motion of matter could be completely understood. (Classical Physics)
β’ However, the formulation by Maxwell was inconsistent with certain aspects of the Newtonian ideas of space and time.
β’ Albert Einstein combined the experimental results and physical arguments of others with his own unique insights, first formulated the new principles in terms of which space, time, matter and energy were to be understood. These principles, and their consequences constitute the Special Theory of Relativity. Later, Einstein was able to further develop this theory, leading to what is known as the General Theory of Relativity
β’ Relativity (both the Special and General) theories, quantum mechanics, and thermodynamics are the three major theories on which modern physics is based
β’ What the principle of relativity essentially states is the following:
β’ The laws of physics take the same form in all frames of reference moving with constant velocity with respect to one another.
β’ the laws of physics are expressed in terms of equations, and the form that these equations take in different reference frames moving with constant velocity with respect to one another can be calculated by use of transformation equations β the so-called Galilean transformation
β’ the applications of those theories to understanding the atom, the atomic nucleus and the particles of which it is composed, collections of atoms in molecules and solids, and, on a cosmic scale, the origin and evolution of the universe.
What is GPS?
Global Positioning System (GPS).
GPS was developed by the United States Department of Defense to provide a satellite-based navigation system for the U.S. military.
β’ Commercial airliner, luxury cars now come with built-in navigation systems that include GPS receivers with digital maps
β’ Hand-held GPS navigation units
β’ The current GPS configuration consists of a network of 24 satellites in high orbits around the Earth. Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/h
How does it work?
β’ Each satellite carries with it an atomic clock that "ticks" with an accuracy of 1 nanosecond . A GPS receiver in an airplane determines its current position and heading by comparing the time signals it receives from a number of the GPS satellites (usually 6 to 12). The precision achieved is remarkable: even a simple hand-held GPS receiver can determine your absolute position on the surface of the Earth to within 5 to 10 meters in only a few seconds. A GPS receiver in a car can give accurate readings of position, speed, and heading in real-time!
β’ The clock ticks from the GPS satellites must be known to an accuracy of 20-30 nanoseconds. However, because the satellites are constantly moving relative to observers on the Earth, effects predicted by the Special and General theories of Relativity must be taken into account to achieve the desired 20-30 nanosecond accuracy.
Review of Classical Mechanics
Mechanics:
β’ A particle of mass βmβ moving with velocity βvβ has a kinetic energy defined by
πΎ =1
2ππ£2 (Joules) (1)
β’ and linear momentum β π"
π = π π£ (kgΒ·m/s) or (NΒ·s) (2)
β’ In terms of the linear momentum, the kinetic energy can be written
πΎ =π2
2π(3)
Fundamental Conservation Laws
β’ When one particle collides with another, we analyze the collision by applying two fundamental conservation laws:
β’ 1. Conservation of Energy: The total energy of an isolated system (on which no net external force acts) remains constant. In the case of a collision between particles, this means that the total energy of the particles before the collision is equal to the total energy of the particles after the collision.
β’ 2. Conservation of Linear Momentum: The total linear momentum of an isolated system remains constant. For the collision, the total linear momentum of the particles before the collision is equal to the total linear momentum of the particles after the collision. Because linear momentum is a vector, application of this law usually gives us two equations, one for the x components and another for the y components.
β’ The importance of these conservation laws is both so great and so fundamental that, even though the special theory of relativity modifies Eqs. (1), (2), and (3), the laws of conservation of energy and linear momentum remain valid.
Example
β’ A helium atom (m = 6.6465 Γ 10β27 kg) moving at a speed of π£π»π= 1.518 Γ 106 m/s collides with an atom of nitrogen (m = 2.3253 Γ 10β26
kg) at rest. After the collision, the helium atom is found to be moving with a velocity of π£π»π= 1.199 Γ 106 m/s at an angle of Οπ»π= 78.75β¦relative to the direction of the original motion of the helium atom.
(a) Find the velocity (magnitude and direction) of the nitrogen atom after the collision.
(b) Compare the kinetic energy before the collision with the total kinetic energy of the atoms after the collision.
Solutionβ’ (a) The law of conservation of momentum for this collision can
β’ be written in vector form as
πππππ‘πππ = ππππππ
which is equivalent to
ππ₯,ππππ‘πππ = ππ₯,πππππ and ππ¦,ππππ‘πππ = ππ¦,πππππ
β’ Letβs choose the x axis to be the direction of the initial motion of the
helium atom. Then, from figure (1), the initial values of the momentum are,
ππ₯,ππππ‘πππ = ππ»ππ£π»π
ππ¦,ππππ‘πππ = 0
β’ And the final momentum can be written as
ππ₯,πππππ= ππ»ππ£π»πβ² cos Οπ»π+πππ£π
β² cos Οπ= ππ»ππ£π»π
ππ¦,πππππ= ππ»ππ£π»πβ² sin Οπ»π+πππ£π
β² sin Οπ= 0
β’ Solving for the unknown terms, we find
π£πβ² cos Οπ=
ππ»π(π£π»πβπ£π»πβ² cos Οπ»π)
ππ
π£πβ² sin Οπ= -
ππ»ππ£π»πβ² sin Οπ»π)
ππ
β’ We now solve for π£πβ² and Οπ
π£πβ² = (π£π
β² sin Οπ)2+(π£πβ² cos Οπ)2
=4.977 *105 m/s
Οπ= tanβ1 π£πβ² sin Οπ
π£πβ² cos Οπ
= -42.48Λ
β’ (b) The initial kinetic energy is
πΎππππ‘πππ =1
2ππ»ππ£
β²π»π
2= 7.658 Γ 10-15 J
and the total final kinetic energy is
πΎπππππ =1
2ππ»ππ£β²π»π
2+1
2πππ£β²π
2= 7.658 Γ 10-15 J
β’ Note that the initial and final kinetic energies are equal.
β’ This is the characteristic of an elastic collision, in which no energy is lost to, for example, internal excitation of the particles.
Exampleβ’ An atom of uranium (m = 3.9529 Γ 10β25 kg) at rest decays
spontaneously into an atom of helium (m = 6.6465 Γ 10β27 kg) and an atom of thorium (m = 3.8864 Γ10β25 kg). The helium atom is observed to move in the positive x direction with a velocity of 1.423 Γ 107 m/s (see Figure)
β’ (a) Find the velocity (magnitude and direction) of the thorium atom.
β’ (b) Find the total kinetic energy of the two atoms after the decay.
Velocity AdditionSuppose a jet plane is moving at a velocity of vPG = 650m/s, as measured by an observer on the ground. The subscripts on the velocity mean βvelocity of the plane relative to the groundβ.
The plane fires a missile in the forward direction; the velocity of the missile relative to the plane is vMP = 250m/s. According to the observer on the ground, the velocity of the missile is:
.
We can generalize this rule as follows:
Let π£π΄π΅ represent the velocity of A relative to B, and let π£π΅πΆ represent the velocity of B relative to C. Then the velocity of A relative to C is
β’ π£π΄πΆ = π£π΄π΅ +π£π΅πΆ (4)
β’ This equation is written in vector form to allow for the possibility that the velocities might be in different directions; for example, the missile might be fired not in the direction of the planes velocity but in some other direction. This seems to be a very βcommon-senseβ way of combining velocities, but this common-sense rule can lead to contradictions with observations when we apply it to speeds close to the speed of light.
The failure of Classical Concept of Timeβ’ In high-energy collisions between two protons, many new particles can be
produced, one of which is a pi meson (also known as a pion). When the pions are produced at rest in the laboratory, they are observed to have an average lifetime (the time between the production of the pion and its decay into other particles) of 26.0 ns . On the other hand, pions in motion are observed to have a very different lifetime.
β’ In one particular experiment, pions moving at a speed of 2.737Γ108 m/s (91.3% of the speed of light) showed a lifetime of 63.7ns.
β’ Let us imagine this experiment as viewed by two different observers . Observer O1, at rest in the laboratory, sees the pion moving relative to the laboratory at a speed of 91.3% of the speed of light and measures its lifetime to be 63.7ns. Observer O2 is moving relative to the laboratory at exactly the same velocity as the pion, so according to O2 the pion is at rest and has a lifetime of 26.0ns. The two observers measure different values for the time interval between the same two events (the formation of the pion and its decay).
(a) The pion experiment according to O1 markers A and B respectively show the locations of the pions creation and decay.(b) The same experiment as viewed by O2,relative to whom the pion is at rest and the laboratory moves with velocity βv.
According to Newton, time is the same for all observers. Newtons laws are based on this assumption. The pion experiment clearly shows that time is not the same for all observers, which indicates the need for a new theory that relates time intervals measured by different observers who are in motion with respect to each other.
The failure of Classical Concept of Spaceβ’ The pion experiment also leads to a failure of the classical ideas about space.
Suppose O1 erects two markers in the laboratory, one where the pion is created and another where it decays. The distance D1 between the two markers is equal to the speed of the pion multiplied by the time interval from its creation to its decay:
D1 = (2.737 Γ 108m/s)(63.7 Γ 10β9s) = 17.4m.
β’ To O2, traveling at the same velocity as the pion, the laboratory appears to be rushing by at a speed of 2.737Γ 108m/s and the time between passing the first and second markers, showing the creation and decay of the pion in the laboratory, is 26.0ns. According to O2, the distance between the markers is
β’ D2 = (2.737 Γ 108m/s)(26.0 Γ 10β9s) = 7.11m.
β’ Once again, we have two observers in relative motion measuring different values for the same interval, in this case the distance between the two markers in the laboratory. The physical theories of Galileo and Newton are based on the assumption that space is the same for all observers, and so length measurements should not depend on relative motion. The pion experiment again shows that this cornerstone of classical physics is not consistent with modern experimental data.
The failure of Classical Concept of Velocity
β’ Classical physics places no limit on the maximum velocity that a particle can reach. One of the basic equations of kinematics,
β’ π£ = π£π + ππ‘, (5)
β’ shows that if a particle experiences an acceleration a for a long enough time t, velocities as large as desired can be achieved, perhaps even exceeding the speed of light.
β’ For another example, when an aircraft flying at a speed of 200 m/s relative to an observer on the ground launches a missile at a speed of 250 m/s relative to the aircraft, a ground-based observer would measure the missile to travel at a speed of 200 m/s + 250 m/s = 450 m/s, according to the classical velocity addition rule (Eq. 4).
β’ We can apply that same reasoning to a spaceship moving at a speed of 2.0 Γ 108m/s (relative to an observer on a space station), which fires a missile at a speed of 2.5 Γ 108m/s relative to the spacecraft. We would expect that the observer on the space station would measure a speed of 4.5 Γ 108m/s for the missile. This speed exceeds the speed of light (3.0 Γ 108m/s).
β’ Allowing speeds greater than the speed of light leads to a number of conceptual and logical difficulties, such as the reversal of the normal order of cause and effect for some observers.
