A Deep Dive into the Fundamentals of Physics: From Lasers to X-rays

Posted on Jun 26, 2024 in Physics

Q. What is the working principle of a Laser? Write some applications of Laser.

A Laser (Light Amplification by Stimulated Emission of Radiation) operates based on three fundamental processes: stimulated emission, population inversion, and optical resonance.

1. Stimulated Emission: When an electron in an excited state of an atom or molecule drops to a lower energy state, it emits a photon (light particle). If this process is stimulated by an incident photon whose energy matches the energy difference between the excited and lower state, it results in the emission of a second photon with the same energy, phase, and direction as the incident photon. This is the principle of stimulated emission.

2. Population Inversion: For stimulated emission to dominate over absorption, there must be more electrons in the excited state than in the lower energy state. This condition is known as population inversion. Achieving population inversion requires an external energy source, such as electrical current or another light source, to excite electrons.

3. Optical Resonance: The laser medium (a solid, liquid, or gas) is placed between two mirrors that form an optical cavity. One of these mirrors is partially transparent. Photons produced by stimulated emission travel back and forth between the mirrors, stimulating more emission with each pass and amplifying the light. When the light intensity builds up sufficiently, it escapes through the partially transparent mirror as a coherent and collimated laser beam.

Lasers are used in medicine (surgery, dermatology), industry (cutting, welding), communication (optical fibers), scientific research (spectroscopy), military (targeting, weapons), entertainment (light shows), and consumer electronics (barcode scanners, optical drives).

Section A

1. What is Poisson’s ratio? Why is the negative sign usually used in the expression of Poisson’s ratio?

   • Poisson’s Ratio (ν): It is the ratio of the transverse strain to the axial strain in a material subjected to axial stress.
   • Negative Sign: The negative sign indicates that the transverse strain is in the opposite direction to the axial strain, meaning that when a material is stretched, it tends to contract in the directions perpendicular to the stretch.

2. Show the relation between dielectric constant (k) and susceptibility (χ).

   • Relation: The dielectric constant k (also denoted as ε_r) is related to the electric susceptibility χ_e by the equation: k = 1 + χ_e

3. Define Reverberation and Reverberation time.

   • Reverberation: The persistence of sound in a particular space after the original sound is removed due to multiple reflections.
   • Reverberation Time (RT₆₀): The time taken for the sound to decay by 60 decibels after the source has stopped producing sound.

4. What is Double refraction? Give examples of materials exhibiting double refraction.

   • Double Refraction (Birefringence): The splitting of a light wave into two separate rays when it passes through certain anisotropic materials.
   • Examples: Calcite, quartz, and mica.

5. What is Bragg’s law?

   • Bragg’s Law: It describes the condition for constructive interference of X-rays scattered by a crystalline lattice: nλ = 2dsinθ where n is the order of reflection, λ is the wavelength of X-rays, d is the distance between crystal planes, and θ is the angle of incidence.

6. What is Heisenberg’s uncertainty principle? What are its applications?

   • Heisenberg’s Uncertainty Principle: It states that it is impossible to simultaneously measure the exact position and momentum of a particle with absolute precision. The more accurately one is known, the less accurately the other can be known: Δx · Δp ≥ h/4π
   • Applications: Quantum mechanics, electron microscopy, and the explanation of atomic structure.

7. Write a short note on Semiconductor Laser.

   • Semiconductor Laser: It is a type of laser diode where the active medium is a semiconductor, typically made of materials like GaAs. It emits coherent light through electron-hole recombination when a p-n junction is forward biased.
   • Applications: Fiber-optic communications, barcode readers, and laser pointers.

8. What is Numerical aperture and Acceptance angle?

   • Numerical Aperture (NA): It is a measure of the light-gathering ability of an optical fiber or lens and is defined as: NA = n sin(θ) where n is the refractive index of the medium and θ is the half-angle of the maximum cone of light that can enter the fiber.
   • Acceptance Angle: The maximum angle at which light can enter the fiber and still be guided through it.

Section B

2. Explain the Stress-Strain curve. What is Hooke’s law?

   • Stress-Strain Curve: It shows the relationship between the stress applied to a material and the strain that results. Key regions include the elastic region, yield point, and plastic region.
   • Hooke’s Law: Within the elastic region, the stress (σ) is proportional to the strain (ε): σ = E · ε where E is the modulus of elasticity (Young’s modulus).

3. State the difference between Polarization (P) and Polarizability (α). Discuss about the different types of polarization.

   • Polarization (P): The measure of the dipole moment per unit volume of a material in response to an electric field.
   • Polarizability (α): A measure of how easily the electron cloud of a molecule can be distorted by an external electric field, resulting in induced dipoles.
   • Types of Polarization:
     • Electronic Polarization: Displacement of electron clouds relative to nuclei.
     • Ionic Polarization: Displacement of positive and negative ions in opposite directions.
     • Orientation Polarization: Alignment of permanent dipole moments in an external field.

4. What are the different classifications of magnetic materials and its characteristics? Distinguish between soft and hard magnets.

   • Classifications:
     • Diamagnetic: Weakly repelled by a magnetic field (e.g., bismuth).
     • Paramagnetic: Weakly attracted to a magnetic field (e.g., aluminum).
     • Ferromagnetic: Strongly attracted and can retain magnetization (e.g., iron).
     • Antiferromagnetic: Opposing magnetic moments cancel out (e.g., manganese oxide).
     • Ferrimagnetic: Opposing magnetic moments do not fully cancel (e.g., magnetite).
   • Soft Magnets: Easily magnetized and demagnetized (e.g., soft iron).
   • Hard Magnets: Retain magnetization and are difficult to demagnetize (e.g., neodymium magnets).

Section C

3. Discuss the Weber’s law. How did Weber’s law influence Fechner’s law? Justify your answer.

   • Weber’s Law: The principle stating that the smallest detectable change in a stimulus is a constant proportion of the stimulus level. ΔI / I = k where ΔI is the change in intensity, I is the original intensity, and k is a constant.
   • Fechner’s Law: Builds on Weber’s Law by proposing that the perceived intensity of a stimulus is proportional to the logarithm of the actual stimulus intensity: S = k log(I / I₀) where S is the perceived intensity, I is the stimulus intensity, and I₀ is the threshold intensity.

4. What is the phenomenon of Interference of light? What are the conditions of interference of light and condition of constructive and destructive interference? Write about the Young’s double slit experiment.

   • Interference of Light: When two light waves superimpose, they create a resultant wave with varying intensity based on their phase relationship.
   • Conditions for Interference: Coherent light sources and constant phase difference.
   • Constructive Interference: Occurs when the path difference between waves is an integer multiple of the wavelength (nλ).
   • Destructive Interference: Occurs when the path difference is a half-integer multiple of the wavelength ((n+1/2)λ).
   • Young’s Double Slit Experiment: Demonstrated interference by passing light through two closely spaced slits, creating a pattern of bright and dark fringes on a screen due to constructive and destructive interference.

5. What do you mean by Polarization of light? Write about the different types of polarization. What is Brewster’s law?

   • Polarization of Light: Orientation of the oscillations of the electric field vector of a light wave in a particular direction.
   • Types of Polarization:
     • Linear Polarization: Electric field oscillates in a single plane.
     • Circular Polarization: Electric field rotates in a circular motion.
     • Elliptical Polarization: Combination of linear and circular polarization.
   • Brewster’s Law: The angle of incidence at which light with a particular polarization is perfectly transmitted through a surface with no reflection is given by: tan(θ_B) = n₂ / n₁ where θ_B is the Brewster angle, and n₁ and n₂ are the refractive indices of the two media.

Section D

4. What is the origin of X-rays? How are X-rays produced in Coolidge tube? Distinguish between Continuous and Characteristic X-rays.

   • Origin of X-rays: Emitted when high-energy electrons strike a metal target and undergo rapid deceleration.
   • Production in Coolidge Tube: Electrons are emitted from a heated cathode and accelerated towards a metal anode. When they collide with the anode, X-rays are produced.
   • Continuous X-rays (Bremsstrahlung): Produced by the deceleration of electrons, resulting in a continuous spectrum.
   • Characteristic X-rays: Produced when an electron from a higher energy level fills a vacancy in a lower energy level of the target atom, resulting in discrete wavelengths.

5. What is De-Broglie’s hypothesis? Explain the Davisson-Germer experiment. Distinguish between group velocity and phase velocity. Express De-Broglie’s wavelength in terms of group velocity and phase velocity.

   • De-Broglie’s Hypothesis: Particles such as electrons have wave-like properties, with a wavelength λ given by: λ = h/p where h is Planck’s constant and p is the momentum.
   • Davisson-Germer Experiment: Confirmed De-Broglie’s hypothesis by showing that electrons scattered off a crystal produce an interference pattern, indicating wave-like behavior.
   • Group Velocity: The speed at which the envelope of a wave packet moves: v_g = dω/dk
   • Phase Velocity: The speed at which a single wave phase propagates: v_p = ω/k
   • De-Broglie’s Wavelength: Relates to the group and phase velocities as: λ = h/mv where m is the mass and v is the velocity of the particle.

Q. What are Einstein’s coefficients A and B? Derive Einstein’s relation between them.

Einstein’s coefficients describe the probabilities of absorption, spontaneous emission, and stimulated emission of radiation by atoms. These are crucial in understanding the interaction of light with matter.

1. Coefficient A₂₁ (Spontaneous Emission):

   • It represents the probability per unit time that an excited atom in state 2 will spontaneously decay to a lower energy state 1, emitting a photon.

2. Coefficient B₁₂ (Absorption):

   • It represents the probability per unit time that an atom in state 1 will absorb a photon and transition to a higher energy state 2.

3. Coefficient B₂₁ (Stimulated Emission):

   • It represents the probability per unit time that an atom in state 2 will, due to the presence of a photon, transition to state 1 and emit a photon identical to the stimulating photon.

Derivation of Einstein’s Relation

Einstein’s relation between these coefficients can be derived by considering a system in thermal equilibrium where the rate of absorption equals the rate of emission.

In thermal equilibrium:

9. Define stress and strain produced on a body.

   • Stress: Stress is the internal resistance offered by a material to the external force applied per unit area. It is measured in pascals (Pa). Stress = Force/Area
   • Strain: Strain is the deformation or displacement of a material relative to its original length due to applied stress. It is a dimensionless quantity. Strain = Change in length/Original length

10. Describe the nature of hard magnetic and soft magnetic material, with reference to the hysteresis curve.

    • Hard Magnetic Materials: These materials have a wide hysteresis loop, high coercivity, and retain significant magnetization even after the external magnetic field is removed. They are used in permanent magnets.
    • Soft Magnetic Materials: These materials have a narrow hysteresis loop, low coercivity, and can be easily magnetized and demagnetized. They are used in transformer cores and electromagnets.

11. What is phon?

    • A phon is a unit of loudness level for pure tones, equal to the sound pressure level (in decibels) of a 1 kHz tone perceived as equally loud by the average listener.

12. What is reverberation?

    • Reverberation is the persistence of sound in a particular space after the original sound is produced. It is caused by multiple reflections of sound waves from the surfaces in an environment.

13. State the Bragg’s law of X-ray diffraction.

    • Bragg’s law describes the condition for constructive interference of X-rays scattered by the crystal lattice planes: nλ = 2d sin θ where n is an integer, λ is the wavelength of X-rays, d is the distance between crystal planes, and θ is the angle of incidence.

14. State and explain Heisenberg’s Uncertainty Principle.

    • Heisenberg’s Uncertainty Principle states that it is impossible to simultaneously measure the exact position and momentum of a particle. The more precisely one quantity is known, the less precisely the other is known: Δx · Δp ≥ h/4π where Δx is the uncertainty in position and Δp is the uncertainty in momentum.

15. What do you understand by pumping in LASER?

    • Pumping in LASER refers to the process of supplying energy to the laser medium to achieve population inversion, where more atoms are in an excited state than in the ground state, necessary for stimulated emission.

16. What is meant by numerical aperture and acceptance angle in optical fiber?

    • Numerical Aperture (NA): It measures the light-gathering ability of an optical fiber and is defined as: NA = √(n₁² – n₂²) where n₁ is the refractive index of the core and n₂ is the refractive index of the cladding.
    • Acceptance Angle: The maximum angle at which light can enter the fiber and still be guided through it, given by: θ₀ = sin^-1(NA)

Q2. Attempt any two of the following:

1. State and explain Hooke’s law in elasticity. Explain the three different types of elasticity.

   • Hooke’s Law: It states that the strain in a solid is proportional to the applied stress within the elastic limit of that material. Stress ∝ Strain ⇒ Stress = E × Strain where E is the modulus of elasticity.
   • Types of Elasticity:
     1. Young’s Modulus (E): Measures the stiffness of a solid material in tension or compression.
     2. Shear Modulus (G): Measures the material’s response to shear stress.
     3. Bulk Modulus (K): Measures the material’s response to uniform pressure.

2. What is Hysteresis Loss? Derive the expression for Hysteresis loss in B-H loop.

   • Hysteresis Loss: Energy loss due to the lag between the magnetic flux density (B) and the magnetic field strength (H) in a ferromagnetic material when it is subjected to cyclic magnetization.
   • Expression for Hysteresis Loss: The area of the B-H loop represents the energy loss per cycle per unit volume. It can be derived using the integral of the B-H curve over one cycle: W = ∮ H dB

3. Explain briefly the various types of polarization.

   • Polarization of Light:
     1. Linear Polarization: Electric field oscillates in one plane.
     2. Circular Polarization: Electric field rotates in a circle while the magnitude remains constant.
     3. Elliptical Polarization: Electric field describes an ellipse, combining linear and circular polarization.

Q3. Attempt any two of the following:

1. Analyze the mechanism of Nicol’s prism.

   • Nicol’s prism is used to produce polarized light by exploiting the birefringence of calcite crystals. It splits an incoming light beam into two orthogonally polarized beams (ordinary and extraordinary rays). The ordinary ray is absorbed or reflected, allowing only the extraordinary ray (polarized light) to pass through.

2. Distinguish between interference due to amplitude division and coherent division.

   • Interference by Amplitude Division: Light is split into two or more parts by partial reflection or transmission, which then interfere with each other (e.g., thin-film interference).
   • Interference by Wavefront Division: The wavefront of light is divided by diffraction or reflection into two parts, which then overlap and interfere (e.g., Young’s double-slit experiment).

3. Evaluate the fringe width for monochromatic light of wavelength 525 nm, slit separation 3 mm, and source to screen distance 1.5 m.

   • Fringe Width (β): β = λD/d where λ = 525 nm = 525 × 10^-9 m, D = 1.5 m, and d = 3 mm = 3 × 10^-3 m. β = (525 × 10^-9 × 1.5) / (3 × 10^-3) = 2.625 × 10^-4 m = 0.2625 mm

Q4. Attempt any two of the following:

1. Write in detail about how X-rays are generated from an atom. What are continuous and characteristic X-rays?

   • Generation of X-rays: X-rays are produced when high-energy electrons strike a metal target. The sudden deceleration of electrons upon collision causes the emission of X-rays.
   • Continuous X-rays: Produced by the deceleration of electrons in the target material, resulting in a broad spectrum of X-ray wavelengths (Bremsstrahlung radiation).
   • Characteristic X-rays: Produced when electrons from higher energy levels fill vacancies in inner shells, emitting X-rays with specific wavelengths characteristic of the target material.

2. What are Miller Indices? How to determine Miller indices of a plane? Draw the (111), (101) and (100) Miller plane in a cubic crystal unit cell.

   • Miller Indices: A set of three integers that denote the orientation of a crystal plane in a lattice.
   • Determination:
     1. Find the intercepts of the plane with the crystallographic axes.
     2. Take the reciprocals of the intercepts.
     3. Clear the fractions to get integer values.
   • Drawing the planes: Refer to crystallography resources or textbooks for detailed sketches of these planes.

3. State the de Broglie Hypothesis. Explain the Davisson-Germer experiment to prove the de Broglie hypothesis.

   • De Broglie Hypothesis: Proposes that particles such as electrons have wave-like properties, with wavelength λ given by: λ = h/p where h is Planck’s constant and p is the momentum.
   • Davisson-Germer Experiment: Demonstrated the wave nature of electrons by showing that electrons scattered off a nickel crystal produced an interference pattern, confirming the de Broglie hypothesis.

Q5. Attempt any two of the following:

1. What are Einstein’s coefficients A and B? Derive Einstein’s relation between them.

   • Already answered in the previous response.

2. How many types of loss are there in optical fibers? Explain briefly.

   • Types of Losses:
     1. Absorption Loss: Due to the absorption of light by the fiber material.
     2. Scattering Loss: Due to microscopic variations in the fiber material.
     3. Bending Loss: Due to macroscopic bends in the fiber.
     4. Dispersion Loss: Due to the spreading of light pulses over time.

3. A ruby laser of wavelength 6940 Å emits 1.00 Joule pulse of light. What is the minimum number of Cr³⁺ ions in the ruby?

   •