Modern phyiscs lecture 1
- 1. Modern Physics 1EE Dr. Hania Farag Dr. Amr El-Sherif
- 2. Course Contents • 1- Special theory of relativity • 2- The particle properties of waves • References: • 1- Modern Physics by Kenneth Krane (third edition) • 2-Concepts of modern Physics by Arthur Beiser (fifth edition)
- 3. What is Relativity? • Until the end of the 19th century it was believed that Newton’s 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) • However, the formulation by Maxwell was inconsistent with certain aspects of the Newtonian ideas of space and time. • 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
- 4. • Relativity (both the Special and General) theories, quantum mechanics, and thermodynamics are the three major theories on which modern physics is based • What the principle of relativity essentially states is the following: • The laws of physics take the same form in all frames of reference moving with constant velocity with respect to one another. • 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 – the so-called Galilean transformation
- 5. • 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.
- 7. 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. • Commercial airliner, luxury cars now come with built-in navigation systems that include GPS receivers with digital maps • Hand-held GPS navigation units • 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
- 9. How does it work? • 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! • 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.
- 11. Review of Classical Mechanics Mechanics: • A particle of mass “m” moving with velocity “v” has a kinetic energy defined by 퐾 = 1 2 푚푣2 (Joules) (1) • and linear momentum “ 푝 " 푝 = 푚푣 (kg·m/s) or (N·s) (2) • In terms of the linear momentum, the kinetic energy can be written 퐾 = 푝2 2푚 (3)
- 12. Fundamental Conservation Laws • When one particle collides with another, we analyze the collision by applying two fundamental conservation laws: • 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. • 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. • 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.
- 13. Example • A helium atom (m = 6.6465 × 10−27 kg) moving at a speed of 푣퐻푒= 1.518 × 106 m/s collides with an atom of nitrogen (m = 2.3253 × 10−26 kg) at rest. After the collision, the helium atom is found to be moving with a velocity of 푣퐻푒= 1.199 × 106 m/s at an angle of ϑ퐻푒= 78.75◦ 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.
- 15. Solution • (a) The law of conservation of momentum for this collision can • be written in vector form as 푝 푖푛푖푡푖푎푙 = 푝 푓푖푛푎푙 which is equivalent to 푝 푥,푖푛푖푡푖푎푙 = 푝 푥,푓푖푛푎푙 and 푝 푦,푖푛푖푡푖푎푙 = 푝 푦,푓푖푛푎푙 • Let’s 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, 푝 푥,푖푛푖푡푖푎푙 = 푚퐻푒푣퐻푒 푝 푦,푖푛푖푡푖푎푙 = 0
- 16. • And the final momentum can be written as ′ cos ϑ퐻푒+푚푁푣푁′ 푝 푥,푓푖푛푎푙= 푚퐻푒푣퐻푒 cos ϑ푁= 푚퐻푒 푣퐻푒 ′ sin ϑ퐻푒+푚푁푣푁′ 푝 푦,푓푖푛푎푙= 푚퐻푒푣퐻푒 sin ϑ푁= 0 • Solving for the unknown terms, we find 푚퐻푒(푣퐻푒−푣퐻푒 푣푁′ cos ϑ푁= ′ cos ϑ퐻푒) 푚푁 푣푁′ sin ϑ푁= - ′ sin ϑ퐻푒) 푚푁 푚퐻푒푣퐻푒
- 17. • We now solve for 푣푁′ and ϑ푁 푣푁′ = (푣푁′ sin ϑ푁)2+(푣푁′ cos ϑ푁)2 =4.977 *105 m/s ϑ푁= tan−1 푣푁′ sin ϑ푁 푣푁′ cos ϑ푁 = -42.48˚ • (b) The initial kinetic energy is 퐾푖푛푖푡푖푎푙 = 1 2 푚퐻푒푣′ 퐻푒 2= 7.658 × 10-15 J and the total final kinetic energy is 퐾푓푖푛푎푙 = 1 2 푚퐻푒 푣′퐻푒 2+ 1 2 푚푁푣′푁 2= 7.658 × 10-15 J • Note that the initial and final kinetic energies are equal. • This is the characteristic of an elastic collision, in which no energy is lost to, for example, internal excitation of the particles.
- 18. Example • An atom of uranium (m = 3.9529 × 10−25 kg) at rest decays spontaneously into an atom of helium (m = 6.6465 × 10−27 kg) and an atom of thorium (m = 3.8864 ×10−25 kg). The helium atom is observed to move in the positive x direction with a velocity of 1.423 × 107 m/s (see Figure) • (a) Find the velocity (magnitude and direction) of the thorium atom. • (b) Find the total kinetic energy of the two atoms after the decay.
- 19. Velocity Addition Suppose 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 ”velocity of the plane relative to the ground”. 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 푣퐴퐵 represent the velocity of A relative to B, and let 푣퐵퐶 represent the velocity of B relative to C. Then the velocity of A relative to C is • 푣퐴퐶 = 푣퐴퐵 +푣퐵퐶 (4) • 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 ”common-sense” 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.
- 20. The failure of Classical Concept of Time • 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. • In one particular experiment, pions moving at a speed of 2.737×108 m/s (91.3% of the speed of light) showed a lifetime of 63.7ns. • 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).
- 21. (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 −v. 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.
- 22. The failure of Classical Concept of Space • 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 × 108m/s)(63.7 × 10−9s) = 17.4m. • To O2, traveling at the same velocity as the pion, the laboratory appears to be rushing by at a speed of 2.737× 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 • D2 = (2.737 × 108m/s)(26.0 × 10−9s) = 7.11m. • 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.
- 23. The failure of Classical Concept of Velocity • Classical physics places no limit on the maximum velocity that a particle can reach. One of the basic equations of kinematics, • 푣 = 푣표 + 푎푡, (5) • 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. • 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). • We can apply that same reasoning to a spaceship moving at a speed of 2.0 × 108m/s (relative to an observer on a space station), which fires a missile at a speed of 2.5 × 108m/s relative to the spacecraft. We would expect that the observer on the space station would measure a speed of 4.5 × 108m/s for the missile. This speed exceeds the speed of light (3.0 × 108m/s). • 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.