Dear Students,

Please read Chapters 28 and 30 and review the Power-Points of Chapters 28 and 30 posted in this Module-8 and then review the following conceptual questions and answers as given in this Conceptual-Questions-8 page as listed below. After that take the Quiz-Concepts-8 in the next page, which has been created using the Conceptual Questions of this page. So, please read these conceptual questions with answers very carefully that I gave you here below, so that you can learn well and score high in the Quiz-Concepts-8.

-Dr. Khan

================================

Module-8: Chapters (28, 30):

Ch-28: Special Relativity

Ch-30: Atomic Physics

Ch-28: Special Relativity:

Conceptual Questions

28.1 Einstein’s Postulates

Relativity is the study of how different observers measure the same event. Modern relativity is divided into two parts:

Special relativity deals with observers who are in uniform (un-accelerated) motion, whereas general relativity includes accelerated relative motion and gravity.

Modern relativity is correct in all circumstances and

in the limit of low velocity and weak gravitation, gives the same predictions as classical relativity.

An inertial frame of reference is a reference frame in which

a body at rest remains at rest and a body in motion moves at a constant speed in a straight line unless acted on by an outside force.

Modern relativity is based on Einstein’s two postulates as follows:

The first postulate of special relativity is the idea that the laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. The second postulate of special relativity is the idea that the speed of light c is a constant, independent of the relative motion of the source.

The Michelson-Morley experiment demonstrated that

the speed of light in a vacuum is independent of the motion of the Earth about the Sun.

28.2 Simultaneity And Time Dilation

Two events are defined to be simultaneous if an observer measures them as occurring at the same time.

Those two events are not necessarily simultaneous to all observers—simultaneity is not absolute.

Time dilation is the phenomenon of

time passing slower for an observer who is moving relative to another observer.

Observers moving at a relative velocity v do not measure the same elapsed time for an event. Proper time Δt_o is the time measured by an observer at rest relative to the event being observed. Proper time is related to the time Δt measured by an Earth-bound observer by the equation

Δt = Δt_o / (1-v²/c²)^1/2 = γ Δt_o where γ = 1 / (1 - v² / c²)^1/2

These equations relating proper time and time measured by an Earth-bound observer implies that

relative velocity cannot exceed the speed of light.

The twin paradox asks why a twin traveling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is

faulty because the traveling twin is accelerating. Special relativity does not apply to accelerating frames of reference.

28.3 Length Contraction

All observers agree upon relative speed. Distance depends on an observer’s motion. Proper length L_o is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth. Length contraction L is the shortening of the measured length of an object moving relative to the observer’s frame:

L = L_o (1 - v²/c²)^1/2 = L_o / γ, γ = 1 / (1 - v² / c²)^1/2

What happens to the Length L when v approached the speed of light c?

Length L approached zero (Length Contraction)

28.4 Relativistic Addition of Velocities

With classical velocity addition, velocities add like regular numbers in one-dimensional motion: u = v + u' where v is the velocity between two observers, u is the velocity of an object relative to one observer, and u' is the velocity relative to the other observer. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed:

u = (v + u') / (1 + v u' /c²)

From this relativistic velocity equation, can the velocities be greater than the speed of light, regardless of how fast the object moves?

No

28.5 Relativistic Momentum

The law of conservation of momentum is valid whenever the net external force is zero and for relativistic momentum. Relativistic momentum P is classical momentum multiplied by the relativistic factor γ where γ = 1 / (1 - v² / c²)^1/2 ; So, P = γ mu, where m is the rest mass of the object, u is its velocity relative to an observer, and γ is the relativistic factor. At low velocities, relativistic momentum is equivalent to classical momentum. Relativistic momentum approaches infinity as u approaches c. This implies that

an object with mass cannot reach the speed of light.

28.6 Relativistic Energy

Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy. Total Energy is defined as: E = γ m c² where the relativistic factor γ = 1 / (1 - v² / c²)^1/2. Rest energy is E_o = mc², meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy. No object with mass can attain the speed of light because

an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.

Ch-30: Atomic Physics:

30.1 Discovery of the Atom

Atoms are the smallest unit of elements; atoms combine to form molecules, the smallest unit of compounds. The first direct observation of atoms was in Brownian motion. Analysis of Brownian motion gave accurate sizes for atoms (10-10m on an average) and a precise value for Avogadro’s number N_A, which is:

NA = 6.02 x 10²³ / mol

30.2 Discovery of the Parts of the Atom: Electrons and Nuclei

Atoms are composed of negatively charged electrons, first proved to exist in cathode-ray-tube experiments, and a positively charged nucleus. All electrons are identical and have a charge-to-mass ratio of q_e / m_e = - 1.76 x 10¹¹ C/kg. The positive charge in the nuclei is carried by particles called protons, which have a charge-to-mass ratio of q_p / m_p = 9.57 x 10⁷ C/kg. The electron proton masses and charge are:

The electron mass is m_e = 9.11 x 10^-31 kg and the proton mass is m_p = 1.67 x 10^-27 kg, and q_e = q_p = 1.60 x 10^-19 C

The planetary model of the atom pictures electrons orbiting the nucleus in the

same way that planets orbit the sun.

30.3 Bohr’s Theory of the Hydrogen Atom

The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula 1 / λ = R ( 1/n_f² - 1 / n_i²) where λ is the wavelength of the emitted EM radiation, and R is the

Rydberg constant which has the value R = 1.097 x 10⁷ / m. The constants n_i and n_f are positive integers, and n_i must be greater than n_f.

Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by ΔE = h f = E_i - E_f , where ΔE is the change in energy between the initial and final orbits, and h f is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram. Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by L = m_e v r_n, where L is the angular momentum, r_n is the radius of the nth orbit, and h is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by r_n = n2 / Z a_B (allowed orbits n = 1, 2, 3, 4, ...) where Z is the atomic number of an element (the number of electrons is has when neutral) and a_B is defined to be the

Bohr radius, which is a_B = 0.529 x 10^-10 m

30.4 X Rays: Atomic Origins and Applications

X rays are relatively high-frequency EM radiation. They are produced by transitions between inner-shell electron levels, which produce x rays characteristic of the atomic element, or by decelerating electrons. X rays have many uses, including

the medical diagnostics and x-ray diffraction.

30.5 Applications of Atomic Excitations and De-Excitations

An important atomic process is fluorescence, defined to be any process in which

an atom or molecule is excited by absorbing a photon of a given energy and de-excited by emitting a photon of a lower energy.

Some states live much longer than others and are termed meta-stable. Phosphorescence is the

de-excitation of a metastable state.

Lasers produce coherent single-wavelength EM radiation by stimulated emission, in which a meta-stable state is stimulated to decay. Lasing requires a population inversion, in which

a majority of the atoms or molecules are in their meta-stable state.

30.6 The Wave Nature of Matter Causes Quantization

Quantization of orbital energy is caused by the wave nature of matter. Allowed orbits in atoms occur for constructive interference of electrons in the orbit, requiring an integral number of wavelengths to fit in an orbit’s circumference; that is, n λ_n = 2 π r_n (n = 1, 2, 3, ..., where

λ_n is the electron’s de Broglie wavelength.

Owing to the wave nature of electrons and the Heisenberg uncertainty principle, there are no well-defined orbits; rather,

there are clouds of probability.

30.7 Patterns in Spectra Reveal More Quantization

The Zeeman effect—the splitting of lines when a magnetic field is applied—is caused by other quantized entities in atoms. Both the magnitude and direction of orbital angular momentum are quantized. The same is true for

the magnitude and direction of the intrinsic spin of electrons.

30.8 Quantum Numbers and Rules

The state of a system is completely described by a complete set of quantum numbers. This set is written as (n, l, m_l, m_s). Quantum numbers are used to express the allowed values of quantized entities. Here are the 4 quantum numbers:

Principal quantum number n = 1, 2, 3, ...

Angular momentum quantum numbers: L = {l (l + 1)}^1/2 (h / 2π) ( l = 0, 1, 2, , n-1)

Angular momentum projection (z-component L_z) quantum numbers m_l:

m_l= - l, -l +1, ... 0,1, 2, ...l)

Spin projection (z-component S_z) quantum number m_s = -1/2, 1/2

Spin projections are referred to as Spin-Up and Spin-Down

30.9 The Pauli Exclusion Principle

The state of a system is completely described by a complete set of quantum numbers. This set is written as (n, l, m_l, m_s). The Pauli exclusion principle says that no two electrons can have the same set of quantum numbers; that is, no two electrons can be in the same state. This exclusion limits the number of electrons in atomic shells and sub-shells. Each value of n corresponds to a shell, and each value of l corresponds to a sub-shell. The maximum number of electrons that can be in a shell and in a sub-shell are

2 n² and 2 (2 l + 1) respectively