CSS Physics Paper-I 2024 Solved

Question 2: Stokes Theorem and Motion in xy-plane

Q2(a): Statement of Stokes Theorem

Stokes theorem states that the line integral of a vector field around a closed curve is equal to the surface integral of the curl of that vector field over any open surface bounded by that curve.

Here, C is the closed boundary curve, S is any open surface bounded by C, A is a vector field, dl is the small line element along the boundary, n is the unit normal to the surface, and ∇ × A is the curl of the vector field.

Q2(a): Proof of Stokes Theorem

Consider a smooth surface S bounded by a closed curve C. Divide the surface into a large number of small surface elements. Each small element has its own small boundary. For one such small element, the circulation of the vector field around the small boundary is approximately equal to the normal component of the curl multiplied by the small area:

If we add the circulation around all small surface elements, the line integrals along internal boundaries cancel each other. This happens because every internal boundary is traversed twice in opposite directions by adjacent small elements. Therefore, only the outer boundary curve C remains.

At the same time, the sum of all curl fluxes over small elements becomes the surface integral over the whole surface:

Therefore, after cancellation of internal boundaries, we obtain:

This proves Stokes theorem.

Q2(a): Significance of Stokes Theorem

Stokes theorem is important because it connects a local property of a vector field with a global property. The curl ∇ × A tells the local tendency of rotation at a point, while the line integral ∮ A · dl tells the total circulation around a closed curve.

In physics, Stokes theorem is used in electromagnetism, fluid dynamics and vector calculus. Maxwell’s equations in integral and differential forms are connected through theorems like Stokes theorem. For example, the circulation of an electric or magnetic field around a loop can be related to the curl of that field over a surface.

Q2(b): Position, Velocity and Acceleration

Given:

Position at t = 3 s

Velocity at t = 3 s

Velocity is the time derivative of position.

The speed is:

Acceleration at t = 3 s

Acceleration is the time derivative of velocity.

The magnitude of acceleration is:

Question 3: Relativity, Energy-Momentum Relation and Rotational Work

Q3(a): Relativity of Length According to Einstein

According to Einstein’s special theory of relativity, length is not absolute. The length of an object measured by an observer depends on the relative motion between the object and the observer. The proper length L0 is the length measured in the frame in which the object is at rest. If the object moves with speed v relative to an observer, the length measured parallel to the direction of motion is contracted.

Since γ ≥ 1, the moving length L is always smaller than or equal to the proper length L0. This effect is called length contraction.

Consequences of Lorentz Transformation

The Lorentz transformations are:

These equations replace the Galilean transformation at speeds close to the speed of light. Their major consequences are:

1. Relativity of simultaneity: Two events simultaneous in one frame may not be simultaneous in another frame.
2. Time dilation: A moving clock appears to run slow. The relation is Δt = γΔt0.
3. Length contraction: A moving rod contracts along the direction of motion. The relation is L = L0/γ.
4. No object can exceed light speed: As velocity approaches c, γ becomes very large.
5. Mass-energy relation: Energy and mass are connected through E = mc².

Q3(b): Proof of Energy-Momentum Relation

The relativistic total energy of a particle is:

The relativistic momentum is:

Now calculate E² - p²c²:

But:

Therefore:

Rearranging:

Q3(c): Work and Kinetic Energy in Rotational Motion

In linear motion, work is:

In rotational motion, force is replaced by torque and linear displacement is replaced by angular displacement:

Therefore, total work is:

For a rigid body rotating about a fixed axis:

Since:

We can write:

Now substitute into work:

Integrating from rest to angular speed ω:

This work appears as rotational kinetic energy:

Question 4: Conservative Field, Direction Cosines and Earth Angular Momentum

Q4(a): Conservative Field

A force field is called conservative if the work done by the force in moving a particle from one point to another is independent of path. In a conservative field, work depends only on initial and final positions.

For a conservative force:

and the force can be written in terms of potential energy:

Gravitational Force as Negative Derivative of Potential Energy

For two masses M and m separated by distance r, gravitational potential energy is:

For radial motion, the force is:

Differentiate U(r):

Therefore:

The negative sign shows that gravitational force is attractive and acts toward the center of the attracting mass.

Q4(b): Direction Cosines of (2, -1, 2)

For a vector:

Magnitude is:

Direction cosines are:

Q4(c): Earth’s Rotational and Orbital Angular Momentum

Angular momentum associated with Earth’s rotation about its own axis is:

Approximating Earth as a solid sphere:

So:

Angular momentum of Earth in its orbital motion around the Sun is:

Now compare by taking the ratio:

Given:

Now:

Therefore:

Exam Explanation: Even though Earth spins once per day, the radius of its orbit around the Sun is enormously larger than Earth’s own radius. Since angular momentum depends on r², the orbital angular momentum dominates.

Question 5: Fluid Pressure and Doppler Effect

Q5(a): Variation of Pressure in a Fluid at Rest

In a fluid at rest, pressure increases with depth. Consider a small vertical fluid element of height dz, area A and density ρ. The pressure at the bottom is slightly greater than the pressure at the top because the lower layer supports the weight of the fluid above it.

For equilibrium:

If vertical height z is measured upward:

Simplifying:

Therefore:

This is the hydrostatic pressure equation. It means pressure decreases as height increases and increases as depth increases.

If density is constant, integration gives:

Here, P0 is atmospheric pressure at the surface, h is depth below the surface and ρgh is gauge pressure.

Pressure Variation in the Atmosphere

In the atmosphere, pressure also follows:

However, air density is not constant. Using the ideal gas relation:

So:

Substitute into hydrostatic equation:

For an isothermal atmosphere, temperature T is constant:

Integrating from sea level to height z:

Therefore:

This is the barometric formula. It shows that atmospheric pressure decreases exponentially with height.

Q5(b): Doppler Effect Numerical

Given:

The police car is behind and chasing the observer. The observer is moving away from the source, while the source is moving toward the observer. Therefore:

Substitute values:

Exam Note: If the speed of sound is taken as 343 m/s, the answer remains approximately 1.23 kHz. The result is higher than the emitted frequency because the source is approaching the observer faster than the observer is moving away.

Question 6: Standing Waves and Grating Resolving Power

Q6(a): Formation of Standing Waves

Yes. If two waves of the same amplitude, same frequency and same wavelength travel in opposite directions through a string, they superpose to produce a standing wave.

Let the two waves be:

The resultant displacement is:

Using the identity:

We get:

This is the equation of a standing wave. It is not of the form f(x-vt) or f(x+vt), so it does not represent a travelling wave. Instead, each point of the string oscillates with amplitude 2A sin(kx).

Nodes and Antinodes

Nodes occur where displacement is always zero:

Since k = 2π/λ:

Antinodes occur where amplitude is maximum:

Is Energy Transported?

In a standing wave, there is no net transport of energy along the string. The two travelling waves carry equal energy in opposite directions, so the average energy flow cancels. Energy remains localized and changes between kinetic and potential forms.

Q6(b): Grating Dispersion

For a diffraction grating, the grating equation is:

Here, d is grating spacing, θ is diffraction angle, m is order of spectrum and λ is wavelength.

Differentiate with respect to wavelength:

Therefore, angular dispersion is:

If a lens of focal length f is used, linear dispersion is:

Resolving Power of a Grating

Resolving power is the ability of a grating to separate two closely spaced wavelengths. It is defined as:

For a diffraction grating:

where m is order of spectrum and N is the total number of illuminated slits.

Question 7: Real Gas, Heat Pump and Fermi-Dirac Statistics

Q7(a): Ideal Gas and Real Gas

A real gas is often described by the van der Waals equation:

Here, a corrects for intermolecular attraction and b corrects for finite molecular volume.

Work Done on Ideal Gas in Thermal Isolation

Thermal isolation means no heat is exchanged with the surroundings:

Such a process is called an adiabatic process. From the first law of thermodynamics:

Since Q=0:

So work done by the gas is:

Work done on the gas is:

For an ideal gas:

Therefore:

For a reversible adiabatic process:

Work done by gas is:

Therefore, work done on gas is:

Q7(b): Maximum Coefficient of Performance of Heat Pump

A heat pump used to heat a house has maximum coefficient of performance:

Convert Celsius temperatures to Kelvin:

Now substitute:

Important Note: If the device were treated purely as a refrigerator for cooling, the refrigerator COP would be COPR = TC/(TH-TC)=264/30=8.8. But because the question says it is used to heat a house, the heat pump COP is 9.8.

Q7(c): Fermi-Dirac Statistics

Fermi-Dirac statistics describes the distribution of identical particles called fermions among available energy states. Fermions have half-integral spin, such as 1/2, 3/2, and so on. Electrons, protons and neutrons are common examples.

The most important condition is the Pauli exclusion principle. It states that no two identical fermions can occupy the same quantum state simultaneously.

The Fermi-Dirac distribution function is:

Here, f(E) is the probability that a state of energy E is occupied, EF is the Fermi energy, k is Boltzmann’s constant and T is absolute temperature.

Important Features of Fermi-Dirac Statistics

1. It applies to fermions, such as electrons.
2. It obeys Pauli exclusion principle.
3. At absolute zero, all states below Fermi energy are filled and all states above it are empty.
4. It explains electron behavior in metals, semiconductors and white dwarfs.
5. It is different from Bose-Einstein statistics, which applies to bosons.

Question 8: Comprehensive Notes

Q8(a): Damped Harmonic Motion

Damped harmonic motion is oscillatory motion in which the amplitude gradually decreases with time due to a resistive force. The resistive force is usually proportional to velocity and acts opposite to the motion.

The damping force is:

The equation of motion is:

Dividing by m:

Let:

For underdamped motion, the solution is:

where:

Types of Damping

1. Underdamping: Oscillations continue but amplitude decreases exponentially.
2. Critical damping: The system returns to equilibrium in the shortest time without oscillating.
3. Overdamping: The system returns slowly to equilibrium without oscillating.

The energy of a damped oscillator decreases with time because the resistive force converts mechanical energy into heat.

Q8(b): Young Double Slit Experiment

Young’s double slit experiment demonstrates the wave nature of light through interference. A monochromatic light source illuminates two narrow slits separated by distance d. The two slits act as coherent sources. Their waves overlap on a screen placed at distance D, producing bright and dark fringes.

The path difference between waves reaching a point on the screen is:

For small angles:

Therefore:

Condition for Bright Fringes

Constructive interference occurs when path difference is an integral multiple of wavelength:

So:

Condition for Dark Fringes

Destructive interference occurs when path difference is an odd half multiple of wavelength:

So:

Fringe Width

The distance between two consecutive bright or dark fringes is:

Q8(c): Longitudinal and Transverse Coherence

Coherence describes the fixed phase relationship between waves. Interference fringes are clear only when waves maintain a stable phase relation. There are two important types of coherence: longitudinal coherence and transverse coherence.

Longitudinal Coherence

Longitudinal coherence, also called temporal coherence, refers to phase correlation along the direction of propagation. It tells how far a wave remains phase-correlated with itself along its direction of travel.

The coherence time is approximately:

The coherence length is:

In terms of wavelength spread:

A source with narrow spectral width has large coherence length. Laser light has high longitudinal coherence, while white light has small coherence length.

Transverse Coherence

Transverse coherence refers to phase correlation across a wavefront, perpendicular to the direction of propagation. It tells whether two points on the same wavefront can act as coherent sources.

For a source of angular size θ, transverse coherence length is approximately:

A smaller source gives greater transverse coherence. This is why narrow slits and pinholes are used in interference experiments.

Difference Between Longitudinal and Transverse Coherence