CSS Physics Paper-II 2022 Solved

Question 2: Electric Dipole Field, Point-Charge Field and Definitions

Q2(a): Electric Field of a Dipole on the y-axis

Let the dipole lie on the x-axis with charges +q and -q separated by distance 2a. The point P lies on the y-axis at distance y from the origin.

Assume:

The distance of each charge from point P is:

The electric field due to +q at P is directed away from +q. Its vector is:

The electric field due to -q at P is directed toward -q. Its vector is:

Now add both fields:

Therefore:

The dipole moment magnitude is:

So:

For y >> a:

Hence:

Q2(b): Electric Field of a Point Charge in Vector Form

The electric field at a distance r from a point charge Q is:

In full vector form:

where r is the position vector from the source charge to the field point.

Graph of Electric Field with Distance

The magnitude is:

Therefore, electric field decreases rapidly as distance increases.

Q2(c): Definitions

Electric Field

Electric field at a point is the force experienced by a unit positive test charge placed at that point.

Its SI unit is:

Electric Dipole

An electric dipole is a system of two equal and opposite charges separated by a small distance.

If charges +q and -q are separated by distance 2a, dipole moment is:

The direction of dipole moment is from negative charge to positive charge.

Question 3: Photoelectric Effect and Stopping Potential

Q3(a): Photoelectric Effect

The photoelectric effect is the emission of electrons from a metal surface when light of sufficiently high frequency falls on it. The emitted electrons are called photoelectrons.

Important observations are:

1. Emission occurs only if incident light has frequency greater than a threshold frequency.
2. Photoemission is almost instantaneous.
3. Maximum kinetic energy of photoelectrons depends on frequency, not intensity.
4. Photoelectric current depends on intensity if frequency is above threshold.

Einstein’s Photoelectric Equation

Einstein explained the photoelectric effect by assuming that light consists of photons. Each photon has energy:

When one photon strikes one electron, its energy is used in two parts:

1. To overcome the work function φ of the metal.
2. To give maximum kinetic energy Kmax to the emitted electron.

Therefore:

So:

If stopping potential is Vs, then:

Thus:

Q3(b): Inadequacy of Wave Theory

The classical wave theory of light could not explain several facts of the photoelectric effect.

Q3(c): Stopping Potential Numerical

Given:

Einstein’s equation gives:

Substitute values:

Stopping potential is related to kinetic energy by:

When kinetic energy is in electron-volts, stopping potential in volts has the same numerical value:

Question 4: Semiconductors, Magnetic Materials and Landé g Factor

Q4(a): Intrinsic and Extrinsic Semiconductors

A semiconductor is a material whose conductivity lies between that of a conductor and an insulator. Silicon and germanium are common examples.

Intrinsic Semiconductor

An intrinsic semiconductor is a pure semiconductor with no intentional impurity added. In an intrinsic semiconductor, electron-hole pairs are generated thermally.

Important properties are:

1. It is chemically pure.
2. Number of electrons equals number of holes.
3. Conductivity is low at low temperature.
4. Conductivity increases with temperature.

Extrinsic Semiconductor

An extrinsic semiconductor is formed by adding a small controlled amount of impurity to an intrinsic semiconductor. This process is called doping.

n-Type Semiconductor

n-type semiconductor is made by doping silicon or germanium with pentavalent impurity atoms such as phosphorus, arsenic or antimony. These atoms donate extra electrons.

p-Type Semiconductor

p-type semiconductor is made by doping with trivalent impurity atoms such as boron, aluminium, gallium or indium. These atoms create holes.

Q4(b): Diamagnetic, Paramagnetic and Ferromagnetic Materials

Diamagnetic Materials

Diamagnetic materials are weakly repelled by a magnetic field. They develop induced magnetic moments opposite to the applied field.

• Magnetic susceptibility is small and negative.
• Relative permeability is slightly less than 1.
• They do not retain magnetism after field removal.
• Examples: bismuth, copper, silver, water.

Paramagnetic Materials

Paramagnetic materials are weakly attracted by a magnetic field because they have unpaired electrons.

• Magnetic susceptibility is small and positive.
• Relative permeability is slightly greater than 1.
• Magnetization disappears when field is removed.
• Examples: aluminium, platinum, oxygen.

Ferromagnetic Materials

Ferromagnetic materials are strongly attracted by a magnetic field and can retain magnetization after the external field is removed.

• Magnetic susceptibility is large and positive.
• They contain magnetic domains.
• They show hysteresis.
• They can be permanently magnetized.
• Examples: iron, cobalt, nickel.

Q4(c): Landé g Factor

The Landé g factor is a dimensionless factor that relates the total angular momentum of an atom to its magnetic moment. It is important in atomic spectroscopy, especially in the Zeeman effect.

The Landé g factor for an atom in LS coupling is:

where:

• L is total orbital angular momentum quantum number.
• S is total spin angular momentum quantum number.
• J is total angular momentum quantum number.

Use in Zeeman Effect

The energy shift of an atomic level in a magnetic field is:

where μB is Bohr magneton, mJ is magnetic quantum number and B is magnetic field.

Question 5: Square Charges, Capacitor Area and Definitions

Q5(a): Electric Field at the Location of Charge q

Let the square have side a. Place charge q at the origin:

We calculate electric field at the position of q due to the other three charges only.

Let:

Field Due to 2q

Charge 2q is at distance a along +x direction from the origin. At the origin, its field points in the negative x-direction.

Field Due to 4q

Charge 4q is at distance a along +y direction from the origin. At the origin, its field points in the negative y-direction.

Field Due to 3q

Charge 3q is at the opposite corner (a,a). Its distance from the origin is:

Magnitude of field is:

Direction from 3q to the origin is along:

Therefore:

Total Electric Field at q

Add all components:

Total Force on q

Force on charge q is:

Therefore:

Q5(b): Surface Area for 1 F Parallel Plate Capacitor

For a parallel plate capacitor:

Given:

Rearrange for area:

Substitute values:

Is it Possible in the Lab?

No, it is not practical for an ordinary lab capacitor. The required area is extremely large. This shows why a simple air-filled parallel plate capacitor cannot easily provide capacitance as large as 1 F. Practical capacitors use very thin dielectrics, rolled plates, porous electrodes, electrolytic structures or supercapacitor designs.

Q5(c): Definitions

Capacitance

Capacitance is the ability of a conductor or capacitor to store charge per unit potential difference.

Unit of Capacitance

The SI unit of capacitance is farad.

Surface Charge Density

Surface charge density is charge per unit area on a surface.

Its SI unit is:

Question 6: Particle in a One-Dimensional Box

Q6(a): Schrödinger Equation for Particle in a Box

A particle in a one-dimensional infinite potential well is confined between x=0 and x=L. The potential energy is:

Inside the box, the time-independent Schrödinger equation is:

Rearrange:

Let:

Then:

The general solution is:

Boundary Conditions

Since the walls are infinitely rigid, wave function must be zero at the walls:

Apply x=0:

So:

Thus:

Now apply ψ(L)=0:

For a non-zero wave function:

Therefore:

Energy Eigenvalues

Since:

we get:

Substitute k=nπ/L:

Using ℏ=h/2π:

Normalized Eigenfunctions

The normalized wave functions are:

Q6(b): First Three Eigen Wave Functions

The first wave function has one half-wave, the second has two half-waves, and the third has three half-waves.

Q6(c): Probability Density Graphs

Probability density is:

Therefore:

Question 7: Charged Particle in Magnetic Field and Dielectric Slab Capacitor

Q7(a): Motion of a Charged Particle in Magnetic Field

The magnetic force on a moving charged particle is:

If velocity v is perpendicular to magnetic field B, the magnitude of force is:

This magnetic force is always perpendicular to velocity. Therefore, it does not change the speed of the particle. It only changes the direction of motion and acts as centripetal force.

For circular motion:

Cancel one v:

So radius is:

Angular Frequency and Period

Angular speed is:

Substitute r=mv/(|q|B):

Time period is:

Direction of Motion

If the magnetic field is directed into the plane of the paper, the direction of force is found by the right-hand rule for a positive charge. A negative charge moves in the opposite sense.

Q7(b): Atomic Description of Dielectrics

A dielectric is an insulating material that becomes polarized when placed in an electric field. It does not allow free flow of charge like a conductor, but its internal charges shift slightly under an applied field.

Atomic Polarization

In an atom, the positively charged nucleus and negatively charged electron cloud have the same centre in the absence of an external electric field. When an electric field is applied, the electron cloud shifts slightly opposite to the field and the nucleus shifts slightly along the field. This creates an induced dipole.

Polar Molecules

Some molecules already have permanent dipole moments. In the absence of field, these dipoles are randomly oriented. In an electric field, they tend to align with the field, producing orientational polarization.

Types of Polarization

Effect on Capacitor

When a dielectric is placed inside a capacitor, it reduces the effective electric field for the same free charge and increases capacitance.

This expression applies when the space between plates is completely filled with dielectric.

Q7(c): Capacitance with Dielectric Slab of Thickness t

Let plate separation be d, plate area be A, and original capacitance without dielectric be:

A dielectric slab of thickness t and dielectric constant K is inserted between the plates. The remaining air gap is:

The system behaves like two capacitors in series:

• Air region of thickness d-t
• Dielectric region of thickness t

The effective separation becomes:

Therefore, capacitance is:

Using C0=ε0A/d, we can write:

So:

Special Case

If dielectric completely fills the space, then t=d:

This matches the standard result for a fully filled dielectric capacitor.

Question 8: Antiparticles, Beta Radiation Detection and Electron Outside Nucleus

Q8(a): Subatomic Particles and Their Antiparticles

An antiparticle has the same mass as its corresponding particle but opposite charge and opposite relevant quantum numbers. When a particle meets its antiparticle, they may annihilate and convert mass into energy.

Annihilation Example

Electron and positron can annihilate:

This process converts rest mass into electromagnetic radiation while conserving energy, momentum and charge.

Q8(b): Detection of Beta-Minus Radiation

Beta-minus radiation consists of high-speed electrons emitted from unstable nuclei. In beta-minus decay:

Beta particles can be detected because they ionize matter and are deflected by electric and magnetic fields.

1. Geiger-Muller Counter

A Geiger-Muller tube contains gas at low pressure. When beta particles enter the tube, they ionize gas atoms. The resulting ions and electrons create an electrical pulse, which is counted by the circuit.

Detection Steps

1. Beta particle enters the GM tube through a thin window.
2. It ionizes gas molecules.
3. Electrons accelerate toward the anode wire.
4. An avalanche of ionization occurs.
5. A voltage pulse is produced and counted.

2. Cloud Chamber

In a cloud chamber, beta particles ionize supersaturated vapour. Droplets form along the ionized path, making the track visible. Beta particles produce thin, irregular tracks because they are light and easily scattered.

3. Scintillation Counter

In a scintillation detector, beta radiation strikes a scintillating material and produces tiny flashes of light. These flashes are converted into electrical pulses by a photomultiplier tube.

4. Magnetic Deflection

Beta-minus particles are electrons, so they are deflected in magnetic and electric fields. Their deflection direction confirms their negative charge.

5. Absorption Method

Beta particles have moderate penetrating power. They can pass through paper but are absorbed by a few millimetres of aluminium. Absorption tests help distinguish beta radiation from alpha and gamma radiation.

Q8(c): Proof That Electron Does Not Exist Inside Nucleus

One strong argument comes from Heisenberg’s uncertainty principle. If an electron existed inside the nucleus, it would be confined to a very small region of size about:

Uncertainty principle states:

So:

Using ℏ = 1.055×10⁻³⁴ J s:

The corresponding relativistic energy would be approximately:

Convert to electron-volts:

This energy is far greater than typical beta-decay electron energies. Therefore, an electron cannot be sitting inside the nucleus before decay.

Modern Explanation of Beta Decay

In beta-minus decay, the electron is created during the decay process through weak interaction:

The electron is not a pre-existing nuclear constituent. It is produced at the moment of beta decay.