CSS Physics Paper-I 2025 Solved

Question 2: Stokes Theorem, Conservative Field, Velocity and Acceleration

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 the curve.

Here, C is the closed boundary curve, S is the surface bounded by C, A is the vector field, dl is a line element along the closed curve, and n is the unit normal to the surface.

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. For one small element, the circulation of the vector field around its boundary is approximately equal to the normal component of the curl times the small area:

Now add the circulation over all small elements. The line integrals along internal boundaries cancel because each internal boundary is traversed twice in opposite directions by neighbouring elements. Therefore, only the outer boundary curve C remains.

The sum of all curl fluxes becomes the surface integral over S:

Hence:

Physical Significance of Stokes Theorem

Stokes theorem connects circulation around a closed boundary with rotation of a field over the surface bounded by that boundary. It is widely used in electromagnetism, fluid mechanics and vector calculus. Maxwell’s equations in integral and differential form are connected through Stokes theorem.

Q2(b): Line Integral of a Gradient Field Around a Closed Curve

Let a vector field A be the gradient of a scalar function φ:

The line integral of A around a closed curve is:

But:

Therefore:

When we move around a closed curve, the initial and final points are the same. Therefore, the scalar function returns to its original value:

Hence:

Q2(c): Velocity and Acceleration Components Along Given Direction

The position vector is:

Velocity

Velocity is the first derivative of position:

At t=1:

Acceleration

Acceleration is the derivative of velocity:

At t=1, acceleration is still:

Component Along the Direction 2i – 3j + 2k

The direction vector is:

Magnitude of u is:

Unit vector along the given direction is:

Velocity Component Along u

Acceleration Component Along u

Question 3: Quantum Statistics and Pressure After Paddle-Wheel Work

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

Statistical mechanics explains how particles are distributed among available energy states. The type of statistics depends on whether the particles are classical or quantum, distinguishable or indistinguishable, and whether Pauli exclusion principle applies.

Q3(b): Classical Statistical Mechanics and Quantum Statistical Mechanics

Classical Statistical Mechanics

Classical statistical mechanics studies systems of many particles using classical mechanics. Particles are treated as distinguishable, and their positions and momenta can vary continuously in phase space.

It is suitable for ordinary gases at high temperature and low density, where quantum effects are negligible. Maxwell-Boltzmann distribution is the main example.

Quantum Statistical Mechanics

Quantum statistical mechanics studies many-particle systems using quantum mechanics. Particles are treated as indistinguishable, and energy states are discrete. Quantum statistics also includes spin and the Pauli exclusion principle.

It is necessary for electrons in metals, blackbody radiation, low-temperature gases, semiconductors and nuclear systems.

Q3(c): Pressure in Vessel After Paddle-Wheel Work

Given:

Step 1: Torque on Shaft

Step 2: Angular Speed

Convert 1800 rpm into revolutions per second:

Angular speed is:

Step 3: Work Done by Paddle Wheel

Power delivered is:

Work done in 10 s is:

Step 4: Pressure Rise at Constant Volume

The vessel is closed and volume is constant. Paddle-wheel work increases the internal energy of air. For an ideal gas at constant volume:

For an ideal gas:

For air:

So:

Step 5: Final Pressure

Question 4: Viscosity, Capillarity and Nozzle Continuity

Q4(a): Viscosity

Viscosity is the internal resistance of a fluid to flow. It is caused by friction between adjacent layers of fluid moving with different velocities. A highly viscous fluid flows slowly, while a less viscous fluid flows easily.

Newton’s law of viscosity is:

Here, η is the coefficient of viscosity, dv/dy is velocity gradient and τ is shear stress.

Effect of Temperature on Viscosity of Liquids

In liquids, viscosity generally decreases when temperature increases. This happens because cohesive forces between molecules become weaker at higher temperature, allowing layers of liquid to slide more easily.

Examples:

• Honey flows more easily when heated.
• Engine oil becomes thinner at high temperature.
• Water viscosity decreases as temperature rises.

Effect of Temperature on Viscosity of Gases

In gases, viscosity generally increases when temperature increases. This happens because gas molecules move faster at higher temperature and transfer momentum more effectively between layers.

Q4(b): Mercury Depression and Water Rise in Capillary Tube

Capillarity is the rise or fall of a liquid in a narrow tube due to surface tension and adhesive/cohesive forces.

Water in Glass Capillary

Water rises in a glass capillary because adhesive force between water and glass is greater than cohesive force between water molecules. Therefore, water wets glass and forms a concave meniscus.

For water in glass:

Therefore:

So capillary height is positive:

Mercury in Glass Capillary

Mercury is depressed in a glass capillary because cohesive force between mercury atoms is greater than adhesive force between mercury and glass. Therefore, mercury does not wet glass and forms a convex meniscus.

For mercury in glass:

Therefore:

So h becomes negative, meaning the level is depressed.

Q4(c): Nozzle Diameter and Flow Rate

Given:

Part (i): Nozzle Diameter

For incompressible steady flow, continuity equation is:

Since area is proportional to diameter squared:

Therefore:

Substitute values:

Part (ii): Rate of Water Leaving the Nozzle

Flow rate is:

Area of hose is:

Therefore:

In litres per second:

Question 5: Double Slit, Grating Resolving Power and SHM

Q5(a): Analytical Treatment of Double Slit Interference

In Young’s double slit experiment, two coherent waves from two slits superpose on a screen and produce interference fringes. Let the intensities due to the two slits be I1 and I2, and the phase difference between them be δ.

The resultant intensity is:

If both slits have equal intensity:

Then:

Using identity:

Therefore:

Phase Difference and Path Difference

If the path difference is Δ, phase difference is:

For double slit:

Condition for Maximum Intensity

Maximum intensity occurs when:

This gives:

or:

At maximum:

Condition for Minimum Intensity

Minimum intensity occurs when:

This gives:

or:

At minimum:

Intensity Pattern

Q5(b): Resolving Power of a Grating

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

For a diffraction grating:

where:

• m is order of spectrum.
• N is number of illuminated grating lines.

Derivation

The grating equation is:

For a grating with N slits, the sharpness of principal maxima increases with the number of illuminated slits. Rayleigh’s criterion gives:

This shows directly that resolving power is proportional to order m. If the order is increased, the resolving power increases.

Q5(c): SHM Acceleration and Velocity

Given:

Angular frequency is:

Velocity

Velocity in SHM is:

Here, x=A, so:

Acceleration

Acceleration in SHM is:

Substitute values:

The negative sign means acceleration is directed toward the mean position.

Question 6: Carnot Cycle, Gibbs Function and Heat Engine Efficiency

Q6(a): Carnot Cycle

The Carnot cycle is an ideal reversible heat-engine cycle operating between two heat reservoirs. It consists of two reversible isothermal processes and two reversible adiabatic processes. It gives the maximum possible efficiency for any heat engine working between the same two temperatures.

Four Steps of Carnot Cycle

1. Isothermal expansion at high temperature TH: Gas absorbs heat QH from the hot reservoir and does work.
2. Adiabatic expansion: Gas expands without heat exchange, and its temperature falls from TH to TC.
3. Isothermal compression at low temperature TC: Gas rejects heat QC to the cold reservoir.
4. Adiabatic compression: Gas is compressed without heat exchange, and its temperature rises from TC back to TH.

Carnot Cycle Diagram

Carnot Efficiency

The efficiency of a Carnot engine is:

where temperatures must be in kelvin.

Q6(b): Gibbs Function

Gibbs free energy is a thermodynamic potential defined as:

Since:

we can write:

Gibbs function is especially useful for processes at constant temperature and constant pressure. It tells whether a process is spontaneous under these conditions.

Proof That Gibbs Function Is Constant for Reversible Isothermal and Isobaric Process

Start from:

Differentiate:

For a reversible process:

Substitute into dG:

Cancel terms:

So:

For an isothermal process:

For an isobaric process:

Therefore:

Hence:

Q6(c): Heat Engine Efficiency and Work Done

Given:

Work done per cycle is:

Efficiency is:

In percentage:

Question 7: Clausius-Clapeyron, Adiabatic Relation and Isothermal Work

Q7(a): Clausius-Clapeyron Equation

The Clausius-Clapeyron equation gives the slope of a phase boundary between two phases in a pressure-temperature diagram. It relates pressure, temperature, latent heat and change in volume during phase transition.

For two phases in equilibrium, Gibbs free energies are equal. Along the phase boundary:

For one mole:

Thus:

Rearrange:

Therefore:

During phase transition, entropy change is:

where L is latent heat per mole. Substitute:

Clausius-Clapeyron Approximation for Vaporization

For vaporization, vapour volume is much greater than liquid volume:

So:

For ideal gas vapour:

Substitute into Clapeyron equation:

Rearrange:

Integrate:

Two Conclusions

1. Vapour pressure increases rapidly with temperature: The equation lnP=-L/(RT)+constant shows that vapour pressure rises strongly as temperature increases.
2. Boiling point depends on external pressure: Lower external pressure lowers boiling point, while higher pressure raises boiling point. This explains boiling at lower temperature on mountains and higher temperature in pressure cookers.

Q7(b): Proof of TV^(γ-1)=constant

For a reversible adiabatic process:

Using ideal gas equation:

So:

Substitute into adiabatic equation:

Simplify:

Since n and R are constants for a fixed amount of gas:

Q7(c): Work Done on Gas in Two-Step Process

Given:

Step 1: Find Volume After Isothermal Expansion

For isothermal expansion:

Therefore:

Step 2: Work Done by Gas During Isothermal Expansion

For isothermal expansion:

Since V2/V1 = P1/P2:

Substitute values:

Step 3: Work During Constant-Pressure Cooling Back to Initial Volume

During constant-pressure cooling at P2, the gas volume decreases from V2 to V1.

Step 4: Net Work by Gas

Step 5: Work Done on Gas

Work done on gas is the negative of work done by gas:

Question 8: Newton’s Rings, Polarization by Reflection and Doppler Effect

Q8(a): Newton’s Rings

Newton’s rings are circular interference fringes formed due to interference of light reflected from the upper and lower surfaces of a thin air film between a plano-convex lens and a plane glass plate.

Formation

When a plano-convex lens is placed on a glass plate, a thin air film of varying thickness is formed between the two surfaces. The thickness is zero at the point of contact and increases outward. When monochromatic light falls normally, reflected rays from the upper and lower surfaces of the air film interfere and form circular bright and dark rings.

Why Rings Are Circular

The air film has the same thickness at all points located at the same distance from the point of contact. Therefore, the fringes are circular.

Condition for Dark Rings in Reflected Light

For reflected light, one ray undergoes a phase reversal of π. Therefore, the condition for dark rings is:

For a plano-convex lens of radius of curvature R:

Substitute:

Diameter D=2r, so:

Condition for Bright Rings in Reflected Light

Uses of Newton’s Rings

1. Determination of wavelength of monochromatic light.
2. Measurement of radius of curvature of a lens.
3. Testing flatness of optical surfaces.
4. Determination of refractive index of a liquid.

Q8(b): Polarization by Reflection

Polarization is the phenomenon in which vibrations of light are restricted to one plane. Since light is a transverse electromagnetic wave, it can be polarized.

When unpolarized light is reflected from a transparent dielectric surface such as glass or water, the reflected light becomes partially polarized. At a particular angle of incidence, called Brewster angle, the reflected light becomes completely plane-polarized.

Brewster’s Law

Brewster’s law states that the tangent of the polarizing angle is equal to the refractive index of the second medium relative to the first medium.

If light travels from air into glass, then:

where n is the refractive index of glass.

Relation Between Reflected and Refracted Rays

At Brewster angle, the reflected and refracted rays are perpendicular to each other.

Explanation

At Brewster angle, the component of the electric field parallel to the plane of incidence is not reflected. The reflected beam therefore contains vibrations mainly perpendicular to the plane of incidence and becomes plane-polarized.

Applications

1. Polarizing sunglasses reduce glare from roads and water surfaces.
2. Photography filters remove unwanted reflections.
3. Optical instruments use polarization control.
4. Polarization helps study stress patterns and material properties.

Q8(c): Doppler Effect

The Doppler effect is the apparent change in frequency or wavelength of a wave due to relative motion between the source and the observer.

If the source and observer move closer, the observed frequency increases. If they move away from each other, the observed frequency decreases.

Doppler Effect for Sound

For sound waves, the general formula is:

Here:

• f' is observed frequency.
• f is source frequency.
• v is speed of sound.
• vo is speed of observer.
• vs is speed of source.

Sign Convention

1. If observer moves toward source, use v+vo.
2. If observer moves away from source, use v-vo.
3. If source moves toward observer, use v-vs.
4. If source moves away from observer, use v+vs.

Examples

1. The pitch of an ambulance siren increases as it approaches and decreases as it moves away.
2. Police radar uses Doppler shift to measure vehicle speed.
3. Astronomy uses red shift and blue shift to study motion of stars and galaxies.
4. Medical Doppler ultrasound measures blood flow.

Doppler Effect for Light

For light, the Doppler effect appears as red shift or blue shift. If a source moves away, wavelength increases and light shifts toward red. If a source moves closer, wavelength decreases and light shifts toward blue.