CSS Physics Paper-II 2020 Solved

Question 2: Coulomb Law, Poisson Equation and Electric Field

Q2(a): Electric Field of Point Charges

Electric field is the force experienced by a unit positive test charge placed at a point in space. If a test charge q0 experiences force F, then electric field is:

According to Coulomb’s law, the force between a source charge q and a test charge q0 separated by distance r is:

Therefore, electric field due to point charge q is:

In vector form:

The direction of electric field depends on the sign of the source charge. For a positive charge, electric field is radially outward. For a negative charge, electric field is radially inward.

Electric Field Due to Multiple Point Charges

Electric field obeys the principle of superposition. If several point charges are present, the net electric field at a point is the vector sum of electric fields due to all charges:

Q2(b): Derivation of Poisson’s Equation from Gauss’s Law

Gauss’s law in differential form is:

where ρ is volume charge density.

Electric field is related to electric potential V by:

Substitute this into Gauss’s law:

But:

Therefore:

So:

Laplace’s Equation

In a charge-free region:

Therefore, Poisson’s equation becomes:

Q2(c): Electric Field Due to Charge 2e

Given:

The electric field due to a point charge is:

Using:

Substitute:

Calculate denominator:

Calculate numerator:

Now:

Question 3: Poynting Vector, Maxwell Equations and Vector Potential

Q3(a): Energy Transport and Poynting Vector

Electromagnetic fields carry energy and momentum. When an electromagnetic wave travels through space, energy is transported by the electric and magnetic fields. The rate of energy flow per unit area is represented by the Poynting vector.

The Poynting vector is defined as:

In vacuum, since:

we can write:

The direction of S is the direction of electromagnetic energy propagation. Its SI unit is:

Energy Density of Electromagnetic Field

The energy density stored in electric and magnetic fields is:

For an electromagnetic wave in vacuum, electric and magnetic energy densities are equal on average.

Poynting Theorem

Poynting theorem is the energy conservation law of electromagnetism:

This means that the decrease in electromagnetic field energy inside a volume equals the energy flowing out through the surface plus the work done on charges inside the volume.

Q3(b): Maxwell’s Equations in Integral and Differential Forms

Importance of Maxwell’s Equations

1. They unify electricity and magnetism.
2. They predict electromagnetic waves.
3. They show that light is an electromagnetic wave.
4. They provide the foundation of radio, optics, antennas and communication technology.

Q3(c): Vector Potential

The magnetic vector potential is a vector field A defined such that magnetic field is the curl of A:

This definition automatically satisfies Gauss’s law for magnetism:

because divergence of curl is always zero:

Scalar and Vector Potentials

In electromagnetism, electric and magnetic fields can be expressed in terms of scalar potential V and vector potential A:

Importance of Vector Potential

1. It simplifies electromagnetic field calculations.
2. It is used in radiation and antenna theory.
3. It is important in quantum mechanics, especially the Aharonov-Bohm effect.
4. It helps express Maxwell’s equations in potential form.

Question 4: Uncertainty Principle, Tunneling and Electron Momentum

Q4(a): Heisenberg’s Uncertainty Principle

Heisenberg’s uncertainty principle states that it is impossible to know simultaneously the exact position and exact momentum of a microscopic particle. The more accurately position is known, the less accurately momentum can be known, and vice versa.

The mathematical form is:

Since:

we can also write:

Energy-Time Uncertainty

Another form of uncertainty principle is:

Physical Meaning

The uncertainty principle is not due to defective instruments. It is a fundamental property of quantum systems. A particle localized sharply in space must be represented by a wave packet containing many wavelengths, which creates uncertainty in momentum.

Significance

1. It explains why electrons cannot exist inside the nucleus as ordinary confined particles.
2. It supports the wave nature of matter.
3. It explains zero-point energy.
4. It limits classical determinism at microscopic scales.

Q4(b): Barrier Tunneling

Barrier tunneling is a quantum phenomenon in which a particle passes through a potential barrier even when its energy is less than the barrier height. In classical physics, this is impossible. In quantum mechanics, the particle has a wave function that can penetrate the barrier.

Classical View

If a particle has energy E and faces a barrier of height V0, then classically:

means the particle cannot cross the barrier.

Quantum View

In quantum mechanics, the wave function inside the classically forbidden barrier region does not become zero immediately. Instead, it decays exponentially:

where:

If the barrier is thin enough, the wave function may still have non-zero value on the other side. Therefore, there is a finite probability that the particle will appear beyond the barrier.

Transmission Probability

For a rectangular barrier of width a, the tunneling probability roughly behaves like:

This shows that tunneling probability decreases rapidly as barrier height or barrier width increases.

Applications of Tunneling

1. Alpha decay: Alpha particles escape nuclei by tunneling through nuclear potential barriers.
2. Tunnel diode: Semiconductor device based on quantum tunneling.
3. Scanning tunneling microscope: Uses tunneling current to image surfaces at atomic scale.
4. Nuclear fusion in stars: Tunneling helps particles overcome Coulomb repulsion.

Q4(c): Momentum of Electron

Given:

The speed is much less than the speed of light, so non-relativistic momentum is sufficient:

Substitute values:

Question 5: Doping, NPN/PNP Transistors and MOSFET

Q5(a): Doping in Semiconductors

Doping is the deliberate addition of a small controlled amount of impurity atoms to a pure semiconductor to increase its electrical conductivity. Pure silicon or germanium is called an intrinsic semiconductor. After doping, it becomes an extrinsic semiconductor.

n-Type Semiconductor

An n-type semiconductor is made by adding pentavalent impurity atoms to silicon or germanium. Pentavalent atoms have five valence electrons. Four electrons form covalent bonds with neighbouring semiconductor atoms, while the fifth electron becomes free for conduction.

Examples of pentavalent donor impurities are:

• Phosphorus
• Arsenic
• Antimony

p-Type Semiconductor

A p-type semiconductor is made by adding trivalent impurity atoms. Trivalent atoms have three valence electrons, so one bond remains incomplete. This creates a hole, which behaves as a positive charge carrier.

Examples of trivalent acceptor impurities are:

• Boron
• Aluminium
• Gallium
• Indium

Q5(b): NPN and PNP Transistors

A bipolar junction transistor is a three-layer semiconductor device used for amplification and switching. It has three terminals:

• Emitter
• Base
• Collector

NPN Transistor

An NPN transistor consists of a thin p-type base between two n-type regions. Electrons are the majority carriers. In active mode, the emitter-base junction is forward biased and the collector-base junction is reverse biased.

In an NPN transistor, a small base current controls a much larger collector current. Conventional current enters the collector and leaves through the emitter.

PNP Transistor

A PNP transistor consists of a thin n-type base between two p-type regions. Holes are the majority carriers. In active mode, the emitter-base junction is forward biased and the collector-base junction is reverse biased.

Comparison Between NPN and PNP Transistors

Transistor Action

The emitter is heavily doped to inject carriers. The base is very thin and lightly doped, so most carriers pass through it. The collector is moderately doped and collects the carriers. This allows a small base current to control a large collector current.

Q5(c): MOSFET

MOSFET stands for Metal-Oxide-Semiconductor Field-Effect Transistor. It is a voltage-controlled device in which gate voltage controls current between source and drain.

Main Terminals of MOSFET

• Gate: Controls the channel.
• Source: Supplies charge carriers.
• Drain: Collects charge carriers.
• Body/Substrate: Semiconductor base region.

Structure and Working

The gate is separated from the semiconductor channel by a thin insulating oxide layer, usually silicon dioxide. Because the gate is insulated, gate current is extremely small. When a suitable gate voltage is applied, a conducting channel is formed between source and drain.

Types of MOSFET

Applications of MOSFET

1. Digital logic circuits.
2. Microprocessors and memory chips.
3. Power switching circuits.
4. Amplifiers.
5. Voltage regulators and motor control circuits.

Question 6: Natural Radioactivity, Radioactive Decay and Alpha-Decay Energy

Q6(a): Natural Radioactivity

Natural radioactivity is the spontaneous disintegration of unstable atomic nuclei found in nature. During this process, nuclei emit radiation in order to become more stable. The phenomenon was discovered by Henri Becquerel and later studied by Marie Curie and Pierre Curie.

Types of Natural Radioactive Emissions

Alpha Decay

In alpha decay, a heavy nucleus emits an alpha particle:

Alpha decay occurs mainly in heavy nuclei because emission of an alpha particle increases nuclear stability.

Beta Decay

In beta-minus decay, a neutron changes into a proton:

In beta-plus decay, a proton changes into a neutron:

Gamma Decay

Gamma decay occurs when an excited nucleus releases energy as a gamma photon:

Q6(b): Radioactive Decay

Radioactive decay is a random nuclear process. It is impossible to predict exactly when one particular nucleus will decay, but the decay of a large number of nuclei follows a definite statistical law.

Decay Law

The rate of decay is proportional to the number of undecayed nuclei:

where λ is decay constant.

Separate variables:

Integrate:

At t=0, N=N0. Therefore:

Activity

Activity is the number of decays per second:

Since dN/dt=-λN:

Therefore:

Half-Life

Half-life is the time in which half of the radioactive nuclei decay:

Mean Life

Mean life is:

Q6(c): Energy Released in Alpha Decay of Uranium-238

The decay is:

Given atomic masses:

Step 1: Mass Defect

Step 2: Convert Mass Defect to Energy

Use:

Therefore:

Step 3: Energy in Joules

Since:

Therefore:

Question 7: Nuclear Fission, Reactors and Radiation Detection

Q7(a): Nuclear Fission

Nuclear fission is the process in which a heavy nucleus splits into two medium-mass nuclei with the release of energy and neutrons. It usually occurs when a heavy nucleus such as uranium-235 absorbs a slow neutron.

A typical fission reaction is:

Features of Fission

1. A heavy nucleus splits into two lighter fragments.
2. Two or three neutrons are usually released.
3. A large amount of energy is released due to mass defect.
4. The released neutrons can cause further fission events.
5. This can lead to a chain reaction.

Chain Reaction

A chain reaction occurs when neutrons released in one fission event trigger further fission events. If controlled, it is used in nuclear reactors. If uncontrolled, it can produce a nuclear explosion.

Energy Released in Fission

The energy appears as kinetic energy of fragments, kinetic energy of neutrons, gamma rays and later heat. The average energy released per fission of uranium-235 is about 200 MeV.

Q7(b): Basic Principles of Nuclear Reactors

A nuclear reactor is a device in which a controlled nuclear chain reaction is maintained. The heat produced by fission is used to generate steam, which can drive turbines for electricity production.

Main Parts of a Nuclear Reactor

Principle of Operation

Fission releases neutrons and energy. The moderator slows neutrons. Control rods adjust neutron population. Coolant removes heat. The heat produces steam, and the steam drives a turbine connected to a generator.

Q7(c): Methods of Detection of Nuclear Radiation

Nuclear radiation is detected through ionization, excitation, photographic effect or semiconductor charge production. Common detectors are described below.

Question 8: Dielectric Polarization, Ampere Law and Accelerators

Q8(a): Dielectric Medium and Electric Polarization

A dielectric medium is an insulating material that does not conduct electricity easily but becomes polarized when placed in an electric field. Examples include glass, mica, rubber, plastic, paper, oil and ceramics.

Electric Polarization

Electric polarization is the dipole moment per unit volume of a dielectric material:

In vector form:

where p is electric dipole moment.

How Polarization Occurs

When a dielectric is placed in an electric field, positive and negative charges inside atoms or molecules shift slightly in opposite directions. This creates induced dipoles. In polar molecules, permanent dipoles tend to align with the applied field.

Types of Polarization

Effect on Capacitor

When a dielectric is inserted between capacitor plates, capacitance increases:

where κ is dielectric constant.

Q8(b): Ampere’s Law

Ampere’s circuital law states that the line integral of magnetic field around a closed path is equal to μ0 times the current enclosed by that path.

Proof for a Long Straight Wire

Consider a long straight wire carrying current I. Magnetic field at distance r is circular and constant in magnitude on a circular Amperian loop.

For a circular path:

The circumference of the circular path is:

Therefore:

Ampere’s law gives:

So:

Ampere-Maxwell Law

Maxwell corrected Ampere’s law by adding displacement current:

In differential form:

Applications

1. Magnetic field of a long straight wire.
2. Magnetic field inside a solenoid.
3. Magnetic field of a toroid.
4. Foundation of electromagnetic wave theory.

Q8(c): Accelerators

Particle accelerators are machines that increase the kinetic energy of charged particles using electric fields and guide them using magnetic fields. They are used in nuclear physics, particle physics, medicine, industry and materials research.

Basic Principle

A charged particle in an electric field experiences force:

The electric field does work on the particle and increases its kinetic energy:

Magnetic fields are used to bend and focus charged particles:

Types of Accelerators

Applications of Accelerators

1. Study of atomic nuclei and elementary particles.
2. Cancer treatment using particle beams.
3. Production of medical isotopes.
4. Material analysis and ion implantation.
5. Synchrotron radiation for biology, chemistry and materials science.