CSS Physics Paper-I 2021 Solved

Question 2: Special Relativity and Mass-Energy Relation

Q2(a): Einstein’s Postulates of Special Theory of Relativity

Einstein’s special theory of relativity is based on two fundamental postulates.

First Postulate: Principle of Relativity

The laws of physics are the same in all inertial frames of reference. An inertial frame is a frame moving with constant velocity, where Newton’s first law is valid. This means no inertial frame is physically preferred over another inertial frame.

Second Postulate: Constancy of Speed of Light

The speed of light in vacuum is constant for all inertial observers and is independent of the motion of the source or the observer.

This postulate rejected the old idea that light needed a material medium called ether. It also required a new transformation between space and time coordinates, known as Lorentz transformation.

Consequences of Special Relativity

1. Time dilation: Moving clocks appear to run slow.
2. Length contraction: Moving objects contract along the direction of motion.
3. Relativity of simultaneity: Events simultaneous in one inertial frame may not be simultaneous in another.
4. Velocity addition changes: Speeds do not add by simple Galilean rules near light speed.
5. Mass-energy equivalence: Mass and energy are connected by E=mc².

Difference Between Special and General Relativity

Q2(b): Establish Energy-Mass Relationship

The relativistic energy of a particle is:

where:

and m0 is rest mass. Relativistic momentum is:

The energy-momentum relation is:

For a particle at rest:

Therefore:

Taking positive root:

This is Einstein’s mass-energy relationship. It means mass is a form of energy. If a mass m is converted into energy, the energy released is:

Significance of Mass-Energy Relationship

1. Nuclear energy: A small mass defect in nuclear fission or fusion appears as a large amount of energy.
2. Binding energy: Nuclear binding energy is explained through mass difference.
3. Pair production: Energy can convert into matter, such as electron-positron pair formation.
4. Annihilation: Matter and antimatter can annihilate into radiation.
5. Stellar energy: The Sun produces energy through nuclear fusion, where mass is converted into energy.

Question 3: Quantum Statistics and Nuclear Reactor

Q3(a): Difference Between Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann Statistics

Statistical mechanics studies the distribution of particles among available energy states. The correct statistics depend on whether the particles are classical or quantum, distinguishable or indistinguishable, and whether they obey Pauli exclusion principle.

Maxwell-Boltzmann Statistics

Maxwell-Boltzmann statistics applies when particles are classical and distinguishable, and the probability of two particles occupying the same quantum state is very small. It is useful for ordinary gases at high temperature and low pressure.

Bose-Einstein Statistics

Bose-Einstein statistics applies to bosons. Bosons do not obey Pauli exclusion principle, so many bosons can occupy the same quantum state. This explains phenomena such as blackbody radiation and Bose-Einstein condensation.

Fermi-Dirac Statistics

Fermi-Dirac statistics applies to fermions such as electrons, protons and neutrons. Fermions obey Pauli exclusion principle, so no two identical fermions can occupy the same quantum state simultaneously.

Q3(b): Nuclear Reactor with Labelled Diagram

A nuclear reactor is a device in which a controlled nuclear fission chain reaction is maintained. The heat produced by fission is removed by coolant and used to produce steam, which can drive a turbine to generate electricity.

Important Parts of Nuclear Reactor

Working of a Nuclear Reactor

When a uranium-235 nucleus absorbs a slow neutron, it splits into two lighter nuclei and releases energy and additional neutrons. These neutrons can cause further fission events. The moderator slows neutrons, control rods regulate neutron population, and coolant carries away heat. In a power reactor, this heat is used to produce steam for electricity generation.

Question 4: Linear/Angular Momentum and Electronic Filter Circuits

Q4(a): Linear Momentum and Angular Momentum

Linear momentum is the quantity of motion of a body moving along a straight or curved path. It is defined as the product of mass and velocity.

Angular momentum is the moment of linear momentum about a point or axis. It describes rotational motion and is defined as:

For a rigid body rotating about a fixed axis:

Difference Between Linear and Angular Momentum

Newton’s Second Law in Linear Motion

Newton’s second law states that the net force acting on a body is equal to the rate of change of its linear momentum.

If mass is constant:

Newton’s Second Law in Angular Motion

The rotational analogue of force is torque. The rotational form of Newton’s second law is:

For a rigid body rotating about a fixed axis, if moment of inertia is constant:

Q4(b): Acceptor and Rejecter Electronic Circuits

Acceptor and rejecter circuits are frequency-selective resonant circuits. They are usually made from resistance, inductance and capacitance. Their function depends on resonance.

Here, f0 is the resonant frequency, L is inductance and C is capacitance.

Acceptor Circuit

An acceptor circuit is a series RLC circuit that accepts or allows maximum current at resonance. At resonance:

The impedance of a series RLC circuit is:

At resonance, XL=XC, so:

Thus impedance is minimum and current is maximum:

Uses of Acceptor Circuit

1. Radio tuning circuits.
2. Band-pass filters.
3. Selecting a desired frequency from many signals.
4. Communication receivers.

Rejecter Circuit

A rejecter circuit is usually a parallel resonant circuit that rejects or blocks a particular frequency at resonance. At resonance, the impedance of a parallel LC circuit becomes very high, so line current becomes minimum.

At resonance:

The circuit offers maximum impedance to the resonant frequency and therefore rejects that frequency.

Uses of Rejecter Circuit

1. Band-stop filters.
2. Removing unwanted frequency.
3. Noise rejection in communication circuits.
4. Filtering hum or interference.

Difference Between Acceptor and Rejecter Circuit

Question 5: Miller Indices and Close Packed Structures

Q5(a): Miller Indices

Miller indices are a set of three integers used to describe the orientation of a crystal plane. They are written as (hkl). Miller indices are obtained from the intercepts made by a plane on the crystallographic axes.

Procedure to Find Miller Indices

1. Find the intercepts of the plane on the crystallographic axes a, b and c.
2. Express the intercepts in terms of unit cell lengths.
3. Take reciprocals of the intercepts.
4. Clear fractions to get the smallest integers.
5. Write the result as (hkl).

Example

If a plane cuts the axes at a, b and c, the intercepts are:

Taking reciprocals:

So Miller indices are:

Recognition of Symbols

Important Formula for Cubic Crystals

For a cubic crystal with lattice constant a, the spacing between planes (hkl) is:

For the (111) plane:

Q5(b): Closest Packed Crystal Structures

Closest packing is the arrangement of atoms or spheres in a crystal so that empty space is minimized and packing efficiency is maximized. In close-packed structures, each atom touches twelve nearest neighbours.

Hexagonal Close Packing: HCP

In hexagonal close packing, the arrangement of layers follows:

The third layer is placed directly above the first layer. HCP has coordination number 12 and packing fraction about 0.74.

Cubic Close Packing / Face-Centered Cubic: CCP or FCC

In cubic close packing, the stacking sequence is:

The third layer does not lie directly above the first or second. The structure is equivalent to face-centered cubic arrangement. FCC also has coordination number 12 and packing fraction about 0.74.

Comparison of HCP and FCC/CCP

Packing Fraction

The packing fraction of close-packed structures is:

For both HCP and FCC:

This means 74 percent of the crystal volume is occupied by atoms and 26 percent is empty space.

Question 6: Three-Dimensional Diffraction Grating and Dual Nature of Light

Q6(a): Three-Dimensional Diffraction Grating

A three-dimensional diffraction grating is a periodic arrangement of scattering centres in three dimensions. A crystal is the best natural example of a three-dimensional diffraction grating because its atoms are arranged periodically in space.

In an ordinary optical grating, lines are arranged periodically in one dimension. In a crystal, atoms or lattice planes repeat periodically in three dimensions, so X-rays, electrons or neutrons can be diffracted by the crystal lattice.

Crystal as a Three-Dimensional Grating

The wavelength of visible light is much larger than atomic spacing, so visible light cannot resolve crystal planes effectively. X-rays have wavelengths comparable to interatomic spacings, so crystals act as diffraction gratings for X-rays.

Bragg’s Law

Constructive interference from crystal planes occurs when the path difference between rays reflected from adjacent planes is an integral multiple of wavelength.

Here:

• d is spacing between crystal planes.
• θ is glancing angle.
• n is order of diffraction.
• λ is wavelength.

Derivation of Bragg’s Law

Consider two X-ray beams reflected from two adjacent crystal planes. The lower beam travels an extra distance compared with the upper beam. The total path difference is:

For constructive interference:

Therefore:

Uses of Three-Dimensional Diffraction Grating

1. Determination of crystal structure.
2. Measurement of interplanar spacing.
3. Study of metals, semiconductors and minerals.
4. Protein crystallography.
5. Material identification using X-ray diffraction.

Q6(b): Dual Nature of Light

The dual nature of light means that light shows both wave-like and particle-like behaviour. Some experiments can be explained only by treating light as a wave, while others require light to be treated as particles called photons.

Wave Nature of Light

Wave nature of light is shown by interference, diffraction and polarization.

Interference

Young’s double slit experiment shows interference fringes. Bright and dark fringes are formed because light waves superpose constructively and destructively.

Diffraction

Diffraction is the bending or spreading of light around obstacles or through narrow apertures. It proves that light behaves as a wave.

Polarization

Polarization shows that light is a transverse wave. Only transverse waves can be polarized.

Particle Nature of Light

Particle nature of light is shown by the photoelectric effect and Compton scattering.

Photoelectric Effect

In the photoelectric effect, light falling on a metal surface ejects electrons if its frequency is greater than a threshold frequency. The energy of each photon is:

Einstein’s photoelectric equation is:

This shows that light transfers energy in packets called photons.

Compton Effect

In Compton scattering, X-rays collide with electrons and undergo a change in wavelength. This can be explained by treating photons as particles with momentum:

Examples of Dual Nature

Question 7: Laws of Thermodynamics and Heat Engine Efficiency

Q7(a): Three Laws of Thermodynamics

Thermodynamics studies heat, work, internal energy and the direction of natural processes. The three laws of thermodynamics describe energy conservation, entropy and behaviour near absolute zero.

First Law of Thermodynamics

The first law is the law of conservation of energy applied to thermodynamic systems. It states that heat supplied to a system is used to increase internal energy and to do work.

Here:

• ΔU is change in internal energy.
• Q is heat supplied to the system.
• W is work done by the system.

If heat is supplied and the system does work, part of heat increases internal energy and part appears as work.

Second Law of Thermodynamics

The second law gives the direction of natural processes and introduces entropy. It has several equivalent statements.

Kelvin-Planck Statement

No heat engine operating in a cycle can convert all the heat taken from a hot reservoir completely into work without rejecting some heat to a cold reservoir.

Clausius Statement

Heat cannot flow spontaneously from a colder body to a hotter body without external work.

Entropy Statement

The entropy of an isolated system never decreases.

Third Law of Thermodynamics

The third law states that the entropy of a perfect crystal approaches zero as its temperature approaches absolute zero.

This law implies that absolute zero cannot be reached by a finite number of physical processes.

Q7(b): Heat Engine and Efficiency

A heat engine is a cyclic device that takes heat from a hot reservoir, converts part of it into work and rejects the remaining heat to a cold reservoir.

For a heat engine:

Efficiency is:

Given:

Therefore:

Convert into percentage:

The rejected heat is:

Question 8: Michelson-Morley, Unification and Large Hadron Collider Notes

Q8(a): Michelson-Morley Experiment and Modern Nobel-Award Interferometry

The Michelson-Morley experiment was designed to detect the motion of the Earth through the hypothetical luminiferous ether. In the nineteenth century, many physicists believed that light required a medium for propagation, just as sound requires air. This assumed medium was called ether.

Apparatus

The experiment used a Michelson interferometer. A beam of light was split into two perpendicular beams. The beams travelled along two arms, were reflected by mirrors and then recombined to produce interference fringes.

Expected Result

If Earth moved through ether, the speed of light should have been different along different directions. This would produce a shift in interference fringes when the apparatus was rotated.

Observed Result

No significant fringe shift was observed. This was called a null result. It showed that ether drift could not be detected.

Importance

1. It weakened the ether theory.
2. It supported the idea that speed of light is constant.
3. It became one of the experimental backgrounds of special relativity.
4. It showed the power of interferometry for precision measurement.

Latest Usage in Nobel-Award Physics

The modern use of interferometry became central in gravitational wave detection. LIGO uses laser interferometry to detect extremely tiny changes in arm length caused by passing gravitational waves. The 2017 Nobel Prize in Physics was awarded for decisive contributions to the LIGO detector and observation of gravitational waves.

Q8(b): Unification of Forces and Abdus Salam’s Contribution

Unification of forces means describing apparently different fundamental forces under a single theoretical framework. Physics seeks simple and deeper laws behind different interactions in nature.

Four Fundamental Forces

Electroweak Unification

At ordinary energies, electromagnetic and weak nuclear forces appear different. However, at high energies they are understood as two aspects of a single electroweak interaction. This theory was developed by Sheldon Glashow, Abdus Salam and Steven Weinberg.

Abdus Salam’s Contribution

Abdus Salam, the Pakistani theoretical physicist, made a major contribution to electroweak unification. He helped develop the theory showing that electromagnetic and weak interactions can be unified within one gauge theory. For this work, Salam shared the 1979 Nobel Prize in Physics with Sheldon Glashow and Steven Weinberg.

Significance

1. It showed that different forces may have a common origin.
2. It predicted the role of W and Z bosons in weak interaction.
3. It strengthened the Standard Model of particle physics.
4. It made Abdus Salam one of the most important Muslim scientists of the modern era.
5. It encouraged the search for grand unification and deeper theories beyond the Standard Model.

Q8(c): Essay on Large Hadron Particle Accelerator

The Large Hadron Collider, commonly called the LHC, is the world’s largest and most powerful particle accelerator. It is located at CERN near the border of Switzerland and France. It accelerates protons or heavy ions to extremely high energies and collides them so physicists can study fundamental particles and forces.

Purpose of the Large Hadron Collider

The purpose of the LHC is to recreate conditions similar to those that existed shortly after the Big Bang. By colliding particles at very high energies, scientists can study the structure of matter, test the Standard Model, search for new particles and investigate questions about mass, symmetry and the early universe.

Working Principle

Charged particles are accelerated by electric fields and guided by magnetic fields. In the LHC, two proton beams travel in opposite directions inside a circular tunnel. Superconducting magnets bend and focus the beams. When the beams collide at detector points, new particles are produced.

Main Parts of the LHC

Major Experiments

The major LHC experiments include ATLAS, CMS, ALICE and LHCb. ATLAS and CMS are general-purpose detectors. They played a key role in the discovery of the Higgs boson in 2012. ALICE studies heavy-ion collisions and quark-gluon plasma. LHCb studies matter-antimatter differences through particles containing bottom quarks.

Higgs Boson

The discovery of a Higgs-boson-like particle in 2012 was one of the greatest achievements of the LHC. The Higgs field explains why many elementary particles have mass. This discovery strongly supported the Standard Model of particle physics.

Importance of LHC

1. It tests the Standard Model at high energies.
2. It helped discover the Higgs boson.
3. It searches for physics beyond the Standard Model.
4. It studies matter-antimatter asymmetry.
5. It helps understand conditions of the early universe.
6. It advances superconducting magnet technology, computing and detector science.

Limitations and Criticism

The LHC is extremely expensive and technically complex. It has confirmed many Standard Model predictions, but new physics beyond the Standard Model has been difficult to find. Still, it remains one of the most important scientific instruments ever built.