Modern Physics PYQ Theory

phy-1109 PHY-1109 Physics

Physics Theory & Explanations (Modern Physics)

Based on provided KUET Physics PYQ (Subtopics A-C)

Subtopic A: Special Theory of Relativity

1. Discuss the Michelson-Morley experiment and its consequences.

The Michelson-Morley experiment was designed to detect the motion of the Earth through the hypothetical “luminiferous ether” (the medium thought to be required for light propagation). They used an interferometer to look for a shift in interference fringes caused by the difference in the speed of light in different directions.

• Result: The experiment yielded a null result. No fringe shift was observed, meaning no motion through the ether was detected.

• Consequences:

  1. Non-existence of Ether: The concept of a stationary ether frame is invalid.

  2. Constancy of the Speed of Light: The speed of light in a vacuum ($c$) is constant and invariant for all observers, regardless of the motion of the source or the observer.

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2. Differentiate between Inertial and Absolute Frames.

• Inertial Frame: A frame of reference in which Newton’s First Law holds true (law of inertia). An object remains at rest or moves with constant velocity unless acted upon by an external force. All inertial frames move at constant velocity relative to each other.

• Absolute Frame: A hypothetical, unique, stationary frame of reference (often associated with the “ether”) against which all other motions could be measured absolutely. Relativity disproved the existence of such a frame; all motion is relative.

3. Explain the physical significance of Einstein’s Mass-Energy relationship ($E=m{c}^2$).

This relation states that mass and energy are interconvertible and are essentially two forms of the same physical entity.

• Significance:

  1. Mass is concentrated Energy: A small amount of mass corresponds to a tremendous amount of energy ($c^2$ is a huge number).

  2. Source of Energy: It explains the energy source of stars (nuclear fusion) and nuclear reactors (fission), where a small loss in mass results in a massive release of energy.

  3. Conservation: The separate laws of conservation of mass and conservation of energy are replaced by a single law of conservation of mass-energy.

4. What is Proper Time?

Proper time ($\tau$ or $t_0$) is the time interval between two events as measured by an observer who is at rest relative to the events (i.e., the events occur at the same location in that observer’s frame). It is always the shortest possible time interval measured between those two events.

5. Show/Explain why a material object cannot travel as fast as light.

According to the relativistic mass variation formula:

$m=\frac{m_0}{\sqrt{1-v^2/c^2}}$

As the velocity $v$ of an object approaches the speed of light $c$, the denominator approaches zero, causing the relativistic mass $m$ to approach infinity.

• Conclusion: Accelerating an infinite mass would require an infinite amount of energy. Since infinite energy is not available, no material object with non-zero rest mass can ever reach or exceed the speed of light.

Subtopic B: Quantum Mechanics and Wave Mechanics

2. Discuss the drawbacks of classical physics in explaining the photoelectric effect.

Classical wave theory failed to explain three key observations:

1. Threshold Frequency: Classical theory predicted emission at any frequency if intensity was high enough. Experimentally, no emission occurs below a specific cutoff frequency (${\nu }_0$), regardless of intensity.

2. Instantaneous Emission: Classical theory predicted a time lag for energy to accumulate. Experimentally, emission is instantaneous ($<10^{-9}$ s).

3. Kinetic Energy vs. Intensity: Classical theory predicted that higher light intensity would increase electron kinetic energy. Experimentally, KE depends only on frequency, not intensity.

3. Write down Einstein’s hypothesis about the photoelectric effect.

Einstein proposed that light is not a continuous wave but consists of discrete packets of energy called photons (or quanta).

• The energy of each photon is proportional to its frequency: $E=h\nu$.

• The interaction is one-to-one: One photon is completely absorbed by one electron. Part of the energy overcomes the work function ($\varphi$), and the rest becomes the kinetic energy of the electron.

4. Define key Photoelectric terms.

• Photoelectric Current: The rate of flow of photoelectrons emitted from the cathode to the anode. It is directly proportional to the intensity of incident light (provided frequency > threshold).

• Work Function ($\varphi$ or $W$): The minimum amount of energy required by an electron to just escape from the metal surface at zero velocity.

• Threshold Frequency (${\nu }_0$): The minimum frequency of incident light required to just eject electrons from a specific metal surface.

• Stopping Potential ($V_s$): The minimum negative (retarding) potential applied to the anode that stops even the most energetic photoelectrons, reducing the photoelectric current to zero.

5. What is Phase Velocity and Group Velocity?

• Phase Velocity ($v_p$): The velocity at which a single phase (like a crest) of a monochromatic wave travels. $v_p=\omega /k$.

• Group Velocity ($v_g$): The velocity at which a wave packet (formed by the superposition of waves) travels. This represents the velocity of energy transport and particle motion. $v_g=d\omega /dk$.

6. What is Compton Shift? Briefly describe the theory.

Compton shift is the increase in wavelength ($\Delta \lambda$) of X-rays or gamma rays when they are scattered by loosely bound (free) electrons.

• Theory: It is treated as an elastic collision between a photon and an electron. The incident photon transfers some of its energy and momentum to the electron (which recoils). Since the scattered photon has less energy ($E^{\prime }<E$), it must have a lower frequency and thus a longer wavelength (${\lambda }^{\prime }>\lambda$).

7. What is Quanta?

A quantum (plural: quanta) is the minimum amount of any physical entity (like energy) involved in an interaction. In light, a quantum of energy is called a photon.

8. Give the significance of the Schrodinger Wave Equation.

The Schrodinger equation is the fundamental equation of quantum mechanics (analogous to Newton’s laws in classical mechanics).

• Significance: It describes how the quantum state of a physical system changes with time. Its solution gives the Wave Function ($\Psi$).

• $\Psi$: Contains all possible information about the system.

• $|\Psi {|}^2$: Represents the probability density of finding the particle at a specific position and time.

9. State the Correspondence Principle.

Proposed by Niels Bohr, the Correspondence Principle states that the predictions of quantum mechanics must agree with the predictions of classical physics in the limit of very large quantum numbers (macroscopic scale).

• Example: As the orbit number $n\to \infty$, the frequency of radiation emitted by an atom approaches the frequency of the electron’s orbital revolution, as predicted by classical electrodynamics.

Subtopic C: Atomic Structure

1. What are the basic postulates of the Bohr Atom Model?

1. Stationary Orbits: Electrons revolve around the nucleus in specific stable orbits without radiating energy.

2. Quantization of Angular Momentum: The angular momentum ($L$) of an electron in a stationary orbit is an integral multiple of $h/2\pi$.

   $mvr=\frac{nh}{2\pi }$

3. Frequency Condition (Energy Transition): Radiation is emitted or absorbed only when an electron jumps from one orbit to another. The energy difference determines the frequency ($f$) of the photon.

   $h\nu =E_2-E_1$

2. Mention the names of quantum numbers associated with the vector atom model.

1. Principal Quantum Number ($n$): Determines the size and energy of the orbit (shell).

2. Orbital (Azimuthal) Quantum Number ($l$): Determines the shape of the orbital and orbital angular momentum.

3. Magnetic Quantum Number ($m_l$): Determines the orientation of the orbit in a magnetic field.

4. Spin Quantum Number ($s$ or $m_s$): Determines the intrinsic spin direction of the electron.

3. Discuss Magnetic Quantum Number and Total Angular Quantum Number.

• Magnetic Quantum Number ($m_l$): It explains the splitting of spectral lines in a magnetic field (Zeeman Effect). It specifies the allowed orientations of the orbital angular momentum vector in space. Values range from $-l$ to $+l$.

• Total Angular Momentum Quantum Number ($j$): In the vector atom model, the orbital angular momentum ($l$) and spin angular momentum ($s$) couple together. $j$ represents the resultant total angular momentum of the electron.

  $j=l+s\text{or}j=|l-s|$

4. Define Space Quantization.

Space quantization is the concept that the angular momentum vector of an atom cannot orient itself in any arbitrary direction in space. It can only take specific, discrete orientations with respect to an external magnetic field (or a quantization axis). These allowed orientations are determined by the magnetic quantum number ($m_l$).