CSS Physics Paper-I 2026

Question 2: Divergence, Gradient, Curl and Accelerated Motion

Q2(a): Difference Between Gradient, Divergence and Curl

Gradient, divergence and curl are three important vector operators used in physics. They are built from the del operator ∇, but each one has a different meaning and acts on a different kind of field.

Gradient

The gradient of a scalar function φ(x,y,z) is:

It points in the direction in which φ increases most rapidly. Its magnitude gives the maximum rate of change of φ.

Divergence

The divergence of a vector field A = Axi + Ayj + Azk is:

Divergence tells whether the field behaves like a source or sink at a point. Positive divergence means field lines are spreading outward. Negative divergence means field lines are converging inward.

Curl

The curl of a vector field A is:

Curl measures the local tendency of a vector field to rotate around a point.

Q2(b): Total Velocity Vector at Any Time

The particle moves in the xy-plane. Initial velocity components are:

The acceleration is only in the x-direction:

Velocity in the x-direction after time t is:

Velocity in the y-direction is constant because there is no y-acceleration:

Therefore, the total velocity vector is:

The speed at any time is the magnitude of velocity:

Question 3: Angular Momentum, Kinetic Friction and Normal Force Work

Q3(a): Law of Conservation of Angular Momentum

Angular momentum is the rotational analogue of linear momentum. For a particle, angular momentum about a point is:

where r is position vector and p is linear momentum.

Torque is the rate of change of angular momentum:

If net external torque is zero:

then:

Therefore:

This is the law of conservation of angular momentum.

Examples of Conservation of Angular Momentum

1. Spinning skater: A skater spins faster when pulling arms inward because moment of inertia decreases and angular velocity increases to keep Iω constant.
2. Planetary motion: A planet speeds up near perihelion and slows down near aphelion because angular momentum is conserved under the Sun’s central gravitational force.
3. Diver in air: A diver curls the body to rotate faster and stretches out to rotate slower.

Q3(b): Speed of Block With Kinetic Friction

Given:

Step 1: Normal Force

Since the block is on a horizontal surface and there is no vertical acceleration:

Step 2: Kinetic Friction

Step 3: Net Force

Step 4: Net Work

Step 5: Work-Energy Theorem

According to work-energy theorem:

The block starts from rest, so:

Therefore:

Q3(c): Can a Normal Force Do Work?

Work is defined as:

A normal force does no work when it is perpendicular to displacement. For example, when a block slides along a fixed horizontal table, the normal force is vertical while displacement is horizontal. The angle is 90°, so:

When Normal Force Can Do Work

A normal force can do work if the surface itself moves and the displacement has a component along the normal force.

Question 4: Kepler’s Third Law and Satellite Speed

Q4(a): Kepler’s Third Law

Kepler’s third law states that the square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit.

For a circular orbit, the semimajor axis a becomes the orbital radius r.

Proof of Kepler’s Third Law

For a planet or satellite moving in a circular orbit, gravitational force provides the centripetal force.

Gravitational force is:

Centripetal force is:

Equate both:

Cancel m:

Therefore:

For circular motion:

Substitute into v²=GM/r:

Rearrange:

Since 4π²/(GM) is constant for a given central mass:

For an elliptical orbit, r is replaced by semimajor axis a:

Q4(b): Speed of Satellite at Altitude h

The satellite is at altitude h above Earth’s surface. If Earth’s radius is R, orbital radius is:

For circular orbit:

Cancel m:

Therefore:

Substitute r=R+h:

So:

Question 5: Bernoulli Theorem and Hydraulic Lift

Q5(a): Bernoulli’s Theorem

Bernoulli’s theorem states that for steady, incompressible, non-viscous flow along a streamline, the sum of pressure energy, kinetic energy and potential energy per unit volume remains constant.

Assumptions of Bernoulli’s Theorem

1. The fluid is incompressible.
2. The fluid is non-viscous.
3. The flow is steady.
4. The equation applies along a streamline.
5. No energy is added or removed by pumps or turbines between the points considered.

Derivation of Bernoulli’s Equation

Consider a fluid element moving along a streamline from point 1 to point 2. According to work-energy principle, work done by pressure forces equals change in kinetic energy plus change in gravitational potential energy.

Pressure work done on volume ΔV at point 1 is:

Pressure work done by the fluid at point 2 is:

Net pressure work is:

Mass of the fluid element is:

Change in kinetic energy is:

Change in potential energy is:

Using work-energy principle:

Divide by ΔV:

Rearrange:

Therefore:

Q5(b): Hydraulic Lift Numerical

A hydraulic lift works on Pascal’s law. Pressure applied to an enclosed liquid is transmitted equally in all directions.

Therefore:

Given:

Since area is proportional to radius squared:

So:

Substitute values:

Question 6: Young Double Slit and Spring-Block Period

Q6(a): Young’s 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 a small distance d. The two slits behave as coherent sources. Their waves overlap on a distant screen and produce alternating bright and dark fringes.

Path Difference

At a point on the screen making angle θ with the central axis, the path difference between waves from the two slits is:

For small angles:

where D is the distance between slits and screen, and y is distance of the fringe from the central maximum.

So:

Constructive Interference

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

Therefore:

For small angles:

So position of bright fringe is:

Destructive Interference

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

Therefore:

For small angles:

Fringe Width

The distance between two consecutive bright or dark fringes is:

Fringe Pattern

Q6(b): Period of Spring-Block Oscillation

Given:

For a mass-spring oscillator, the period is:

Substitute values:

Question 7: X-Ray Diffraction and Grating Angle

Q7(a): Diffraction of Light

Diffraction is the bending or spreading of waves when they pass through an aperture or around an obstacle. Diffraction becomes prominent when the size of the aperture or obstacle is comparable to the wavelength of the wave.

Light shows diffraction because it has wave nature. Diffraction patterns consist of bright and dark regions due to constructive and destructive interference of secondary wavelets.

Types of Diffraction

Diffraction of X-Rays by Crystals

X-rays have wavelengths comparable to the spacing between atoms in crystals. Therefore, a crystal can act as a three-dimensional diffraction grating for X-rays.

Crystal planes reflect X-rays. Constructive interference occurs when the path difference between rays reflected from adjacent crystal planes equals an integral multiple of wavelength.

Bragg’s Law

The condition for constructive interference is:

Here:

• d is spacing between crystal planes.
• θ is glancing angle.
• n is order of reflection.
• λ is wavelength of X-rays.

Derivation of Bragg’s Law

Consider X-rays reflected from two adjacent parallel crystal planes separated by distance d. The ray reflected from the lower plane travels an extra path:

For constructive interference:

Uses of X-Ray Diffraction

1. Determination of crystal structure.
2. Measurement of interplanar spacing.
3. Identification of unknown crystalline substances.
4. Study of metals, minerals and semiconductors.
5. Protein crystallography and molecular structure analysis.

Q7(b): Diffraction Grating Angle for Second-Order Maximum

Given:

Step 1: Convert Grooves per Centimeter to Grooves per Meter

Step 2: Grating Spacing

Grating spacing is reciprocal of number of grooves per meter:

Step 3: Grating Equation

For normal incidence:

Therefore:

Substitute values:

Therefore:

Since maxima appear symmetrically on both sides of the central maximum:

Question 8: First Law and Isothermal Expansion Work

Q8(a): First Law of Thermodynamics

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

Using the sign convention where Wby is work done by the gas:

or:

Isothermal Expansion of an Ideal Gas

In an isothermal process, temperature remains constant:

For an ideal gas, internal energy depends only on temperature. Therefore, in an isothermal process:

From the first law:

Derivation of Work Done by Gas

Small work done by gas is:

For an ideal gas:

So:

Substitute into work expression:

Integrate from initial volume Vi to final volume Vf:

Since nRT is constant in isothermal process:

Work Done on Gas

Work done on the gas is opposite in sign to work done by the gas:

Therefore:

Q8(b): Work Done During Expansion from 3 L to 10 L

Given:

Work done by the gas is:

Substitute values:

Now:

Therefore:

Work done on the gas is: