Modern Physics – NDA Physics Notes

Pull apart two magnets. The repulsion grows as they get closer. Now imagine trying to push two protons into the same tiny space the size of a nucleus.

The electrostatic repulsion between two protons at nuclear distances would be immense, approximately 230 newtons. Yet the nucleus exists. Protons and neutrons are packed into a space 100,000 times smaller than the atom itself, held together by a force that must be the strongest in nature: the strong nuclear force.

Modern Physics is the story of how scientists in the late 19th and early 20th centuries found that the comfortable picture of atoms as indivisible solid spheres was wrong, and built a far stranger, far more powerful picture. Thomson found the electron. Rutherford found the nucleus. Bohr explained why atoms emit light in discrete wavelengths. Einstein explained the photoelectric effect. Planck quantised energy itself.

1. The Atom: Pre-Modern Picture and Why It Failed

The idea that matter is made of indivisible atoms goes back to Democritus in ancient Greece. By the late 1800s, chemists had strong evidence for atoms, but no one knew what atoms looked like inside.

The pre-modern picture: atoms are like hard billiard balls, indivisible, featureless. This picture collapsed in 1897 when J.J. Thomson discovered that atoms contained smaller, negatively charged particles that could be extracted from them: the electrons. If atoms contain electrons (negative), they must also contain something positive to balance out. The question was: how is that positive charge distributed?

2. Thomson’s Experiment: Discovery of the Electron

J.J. Thomson (1897) passed an electric discharge through gases at low pressure in a sealed tube: the cathode ray tube. He observed rays travelling from the cathode (negative electrode) to the anode (positive electrode). By deflecting these cathode rays with both electric and magnetic fields, he measured the charge-to-mass ratio (e/m) of the particles. He found:

The particles had the same e/m ratio regardless of the cathode material or the gas in the tube. They are universal constituents of matter.

They were negatively charged and had a mass about 1/1836 of the proton mass.

Thomson concluded that these particles: which he called electrons, are fundamental constituents of all atoms. He did NOT discover the nucleus or propose stable orbits.

3. Cathode Rays: Properties and Direction

Cathode rays are streams of electrons emitted from the cathode (negative electrode) of a discharge tube and travelling toward the anode (positive electrode).

Property	Correct Statement	Common Wrong Answer
Composition	Electrons (negatively charged particles)	Neutral particles or protons
Direction of travel	From CATHODE to ANODE (cathode → anode)	From anode to cathode (wrong)
Path without fields	Straight lines	Curved paths
Response to electric field	Deflected toward positive plate	Not deflected
Response to magnetic field	Deflected (as charged particles are)	NOT deflected by magnetic field (wrong)
Application	Television picture tubes (CRT)	Radio transmitters

[NDA 2019-II] The incorrect statement about cathode rays: “Cathode ray particles start from anode and move towards cathode.” Correct: they start from the cathode and move towards the anode.

Electrons are deflected by both electric AND magnetic fields. [NDA 2015-II] The statement “electrons are deflected by electric field but not by magnetic field” is false: Thomson’s own experiment showed magnetic deflection to measure e/m.

4. Rutherford’s Alpha-Particle Scattering Experiment

In 1909–1911, Ernest Rutherford, Hans Geiger, and Ernest Marsden directed a beam of alpha particles (helium nuclei, doubly charged, positive) at a very thin gold foil. They observed where the alpha particles went after hitting the foil.

What Was Observed

Most alpha particles: Passed straight through the foil without deflection, as if the foil were mostly empty space.

Some: Were deflected at small angles.

A very few (about 1 in 8000): Were deflected at very large angles, some even bouncing almost straight back.

What Rutherford Concluded

Conclusion 1: Most of the atom is empty space. The alpha particles passed through unimpeded.

Conclusion 2: Nearly all the mass of the atom resides in a tiny, dense, positively charged nucleus. The concentrated positive charge is what deflects alpha particles at large angles.

Conclusion 3: The radius of the atom is about 10⁵ times the radius of the nucleus. [NDA 2021-I]

What Rutherford Did NOT Conclude

NOT discovered from this experiment: Electrons moving in circular orbits of fixed energy; that conclusion came from Niels Bohr’s model (1913). [NDA 2021-I]

NOT discovered: the electron (discovered by Thomson). NOT discovered: proton (discovered by Rutherford from nitrogen bombardment, separate experiment). NOT discovered: helium (already known). [NDA 2017-I | NDA 2021-II]

★  IMPORTANT Rutherford’s experiment discovered: THE ATOMIC NUCLEUS. [NDA 2017-I | NDA 2021-II]

5. Rutherford’s Nuclear Model

Rutherford proposed that every atom has a tiny, dense, positively charged nucleus at its centre, containing most of the atom’s mass. Electrons are scattered around the nucleus in the mostly empty space outside.

Problem with Rutherford’s model: classical physics says an electron moving in a circle would continuously radiate energy (electromagnetic radiation), lose speed, and spiral into the nucleus within nanoseconds. Atoms would be unstable, but they clearly are not. This problem required Bohr’s revolutionary solution.

6. Bohr’s Atomic Model

In 1913, Niels Bohr proposed a model that fixed Rutherford’s stability problem by combining classical mechanics with quantum ideas.

Postulate 1: Stable Orbits: Electrons revolve in certain stable orbits (stationary states) around the nucleus without emitting radiant energy. [NDA 2024-I] Energy is not lost while the electron stays in its orbit.

Postulate 2: Quantised Angular Momentum: The angular momentum of an electron in an allowed orbit is an integer multiple of h/2π: L = nh/2π (n = 1, 2, 3, …). This explains which orbits are allowed.

Postulate 3: Emission and Absorption: Radiation is emitted or absorbed only when an electron jumps between orbits. The frequency of radiation: hf = E_higher − E_lower.

Who Proposed What: Critical Attribution Table

Discovery/Proposal	Scientist	What They Did NOT Do
Discovered the electron (cathode ray experiments, 1897)	J.J. Thomson	Did not discover nucleus; did not propose stable orbits
Discovered the atomic nucleus (alpha scattering, 1909–11)	Ernest Rutherford	Did not discover electron; did not propose stable orbits
Proposed stable electron orbits (atomic model, 1913)	Niels Bohr	Did not discover nucleus or electron
Explained photoelectric effect (1905)	Albert Einstein	Did not discover nucleus or propose atomic orbits
Proposed E = mc² (mass-energy equivalence)	Albert Einstein	—
Quantised energy (Planck’s constant h, 1900)	Max Planck	Did not explain photoelectric effect (that was Einstein)

NDA 2024-I: who proposed that electrons revolve in stable orbits without emitting radiant energy? Answer: Niels Bohr. Not Rutherford, not Thomson, not Einstein.   [NDA 2024-I]

7. Atomic Number, Mass Number, and Neutrons

Atomic number (Z): Number of protons in the nucleus. Equals number of electrons in a neutral atom.

Mass number (A): Number of protons + number of neutrons in the nucleus.

Number of neutrons (N): N = A − Z.

The nucleus contains protons and neutrons only, NOT electrons. [NDA 2010-I] Electrons orbit outside the nucleus in shells.

Worked Example: Aluminium-27: ₁₃Al²⁷: Atomic number Z = 13 (protons = 13). Mass number A = 27. Neutrons = 27 − 13 = 14.   [NDA 2010-I]

Neutron-to-Proton Ratio for Light Nuclei: For stable light nuclei (A < 10): the number of protons and neutrons is approximately equal: neutron-to-proton ratio ≈ 1. [NDA 2011-II] Heavy stable nuclei need more neutrons than protons (ratio > 1) to overcome proton-proton electrostatic repulsion.

★ IMPORTANT Ionisation energy of hydrogen in its ground state = 13.6 eV (electron volts).   [NDA 2017-II] NOT 13.6 MeV (that would be 10⁶ × 13.6 eV, the scale of nuclear binding energies, not atomic ionisation). NOT 13.6 Joules. NOT zero (the electron is bound).

8. Nuclear Forces: What Holds the Nucleus Together

Two forces act between protons in a nucleus:

1. Electrostatic repulsion: Pushes protons apart. Acts over long range.

2. Strong nuclear force: Pulls protons and neutrons together. Overwhelmingly stronger at short range (~10⁻¹⁵ m), but virtually zero beyond that distance.

Protons and neutrons are bound in the nucleus by the short-range strong interaction. [NDA 2011-II] NOT the weak interaction (handles beta decay). NOT the electromagnetic force (that would repel protons). NOT gravity (far too weak: gravity between two protons is ~10³⁶ times weaker than their electromagnetic repulsion).

9. The Electromagnetic Spectrum: Complete Order

All electromagnetic waves travel at the same speed in vacuum, the speed of light c ≈ 3 × 10⁸ m/s. [NDA 2010-I] X-ray and gamma ray speed ratio = exactly 1. Speed does not depend on frequency or wavelength in vacuum.

Type	Wavelength Range	Frequency Range	Energy per Photon	NDA Context
Gamma rays	< 0.01 nm	> 3×10¹⁹ Hz	Highest (MeV range)	Nuclear reactions; cancer treatment
X-rays	0.01 nm – 10 nm (≈ 1 Å – 10 nm)	3×10¹⁶ to 3×10¹⁹ Hz	Very high (keV range)	Medical imaging, crystallography: NOT radar
Ultraviolet (UV)	10 nm – 380 nm	7.9×10¹⁴ to 3×10¹⁶ Hz	High	Sunburn, sterilisation
Visible light	380 nm – 780 nm	3.8×10¹⁴ to 7.9×10¹⁴ Hz	Medium	Human vision
Infrared (IR)	780 nm – 1 mm	3×10¹¹ to 3.8×10¹⁴ Hz	Low	Heat, remote controls
Microwaves	1 mm – 30 cm	1×10⁹ to 3×10¹¹ Hz	Very low	RADAR, microwave ovens, mobile phones
Radio waves	> 30 cm	< 1×10⁹ Hz	Lowest	Broadcasting, RADAR (long range)

15 nm radiation → Ultraviolet (UV range: 10–380 nm). [NDA 2010-II]

Smallest wavelength among microwaves, infrared, visible, X-rays: X-rays.   [NDA 2020-I & II]

Maximum energy per photon among radio waves, light waves, microwaves, X-rays: X-rays.   [NDA 2017-II]

10. X-Rays: The Highest-Priority Topic in This Chapter

★ IMPORTANT X-rays account for 9 PYQs, more than any other single topic in this chapter. Know every fact below.

X-Ray Wavelength

X-rays have wavelengths of the order of 1 Ångström (1 Å = 10⁻¹⁰ m), also expressed as 1 nm (the X-ray range spans ~0.01–10 nm, with the most commonly tested value being 1 nm or 1 Å). [NDA 2015-II | NDA 2018-II | NDA 2022-II | NDA 2025-II]

Wavelength Option	Belongs To	NDA Status
1 nm (= 10 Å = 10⁻⁹ m)	X-ray range (0.01–10 nm)	CORRECT: X-ray wavelength
1 Å (= 0.1 nm = 10⁻¹⁰ m)	X-ray range	CORRECT: X-ray wavelength
100 nm	Ultraviolet (UV)	WRONG for X-rays
500 nm	Visible light (green)	WRONG for X-rays
5000 nm	Infrared	WRONG for X-rays
1 μm (= 1000 nm)	Infrared	WRONG for X-rays
1 mm	Microwave	WRONG for X-rays

X-Ray Nature

X-rays are electromagnetic radiation: part of the EM spectrum, travelling at the speed of light with no mass or charge. [NDA 2010-I] X-rays AND gamma rays: both electromagnetic. Alpha rays (helium nuclei: particles, charged, mass) and beta rays (electrons: particles, charged, mass) are NOT electromagnetic. Cathode rays (electron streams) are also not electromagnetic. [NDA 2010-I]

11. X-Ray Production

X-rays are produced when high-energy electrons strike a metal target (anode) inside an evacuated tube. The electrons are accelerated from the cathode (filament) to the anode by a high potential difference.

~99% of kinetic energy: Converted to heat at the target. (The target must be water-cooled to prevent melting.) [NDA 2010-I | NDA 2023-I]

~1% of kinetic energy: Emitted as X-rays (bremsstrahlung radiation and characteristic X-rays).

A brilliant visible light is NOT produced at the target. [NDA 2010-I]

12. Penetrating Power vs Intensity: Voltage vs Filament Current

Two separate controls on an X-ray machine affect different properties:

Control	What It Affects	How	NDA Answer
Accelerating voltage (potential difference between cathode and anode)	PENETRATING POWER (energy) of X-rays	Higher voltage → electrons hit target with more KE → higher energy X-rays → shorter wavelength → more penetrating	Increase voltage to increase penetrating power
Filament current (heating current in cathode filament)	INTENSITY (number) of X-rays	More current → hotter filament → more electrons emitted → more X-ray photons produced per second → higher intensity	Increase filament current to increase intensity

★ IMPORTANT NDA Confusion: what INCREASES PENETRATING POWER of X-rays? Answer: INCREASING THE POTENTIAL DIFFERENCE (VOLTAGE).   [NDA 2012-I] NOT increasing filament current: that increases intensity (number), not energy (penetrating power).

13. Cutoff Wavelength of X-Rays

When an electron is accelerated through potential difference V, it gains kinetic energy:

KE = eV

If all this kinetic energy is converted to a single X-ray photon (maximum energy, minimum wavelength):

eV = hf_max = hc/λ_min

λ_min = hc/(eV)   ← cutoff (minimum) wavelength

λ_min is inversely proportional to V. If voltage is doubled: λ_min_new = hc/(e × 2V) = λ_min/2. The cutoff wavelength is halved.   [NDA 2017-I] Higher voltage → electrons arrive with more energy → can produce photons with more energy → shorter minimum wavelength.

14. Uses of X-Rays: and What They Cannot Do

X-rays are used for: Medical imaging (bone fractures, chest X-rays, dental radiography); Cancer treatment (high-energy X-rays kill tumour cells) [NDA 2023-I]; Quality control in industry (detecting internal defects in materials without cutting); Crystallography (X-ray diffraction reveals atomic structure of crystals).

★ IMPORTANT X-rays are NOT used for RADAR.   [NDA 2023-I] Radar uses radio waves and microwaves. Radar requires waves that REFLECT off objects. X-rays would PENETRATE objects rather than reflect from them. The incorrect statement: “Due to their shorter wavelengths, they can be used for radar systems.”: FALSE.

15. Photon Energy: E = hf

Light and all EM radiation travel in packets of energy called photons. The energy of each photon:

E = hf = hc/λ

E = photon energy (Joules or eV). h = Planck’s constant = 6.626 × 10⁻³⁴ J s. f = frequency (Hz). c = speed of light (3 × 10⁸ m s⁻¹). λ = wavelength (m). Higher frequency (shorter wavelength) → higher energy per photon.

Energy order (increasing): radio waves < microwaves < infrared < visible < UV < X-rays < gamma rays.

Statement	Correct or Wrong?	Reason
Visible light wavelength > UV wavelength > X-ray wavelength	CORRECT	λ_visible > λ_UV > λ_X-ray
X-ray photons have higher energy than UV photons	CORRECT	Shorter wavelength → higher energy
UV photon energy is LESS than visible light photon energy	WRONG	UV photons have MORE energy (shorter wavelength)

Statements 1 and 2 are correct; Statement 3 is false.   [NDA 2018-II]

16. Planck’s Constant: Dimensions

From E = hf: h = E/f. The dimensions of h:

[h] = [E/f] = J/Hz = J·s = kg·m²·s⁻¹ = [M L² T⁻¹]

This is the same as the dimensions of angular momentum (L = mvr = kg × m/s × m = kg·m²·s⁻¹ = M L² T⁻¹). [NDA 2025-II] NOT linear momentum ([M L T⁻¹]: that is p = mv). NOT displacement ([L]). NOT torque ([M L² T⁻²]).

17. Photoelectric Effect

When light of sufficient frequency falls on a metal surface, electrons are emitted from the surface. This is the photoelectric effect (also called photoelectric emission).

Light must exceed a threshold frequency (ν₀). Below this, no electrons are emitted regardless of intensity.

Above the threshold: electrons are emitted instantaneously when light falls on the surface.

Increasing light intensity above threshold: more electrons emitted per second (not faster electrons).

Increasing frequency above threshold: electrons emitted with greater kinetic energy.

Who Explained It: The photoelectric effect was explained by Albert Einstein in 1905, using Planck’s quantum theory. Einstein proposed that light consists of photons, each with energy E = hf. A photon transfers all its energy to one electron; if E > work function φ, the electron is emitted with KE = hf − φ. [NDA 2017-II | NDA 2019-I] Einstein received the Nobel Prize in Physics specifically for this explanation (not for relativity).

Naming: Electron emission from a metallic surface by application of light is called photoelectric emission (or photoelectric effect). [NDA 2017-II] NOT thermionic emission (due to heat). NOT high field emission (due to strong electric field).

18. Solar Cells, LEDs, and Light-to-Electricity Conversion

Device	Conversion Direction	Principle	NDA Status
Solar cell (photovoltaic cell)	Light → Electricity	Photovoltaic effect (semiconductor photoelectric effect)	CORRECT: light to electricity
LED (Light Emitting Diode)	Electricity → Light	Electroluminescence: electrons recombine with holes in p-n junction	—
Laser diode	Electricity → Light	Stimulated emission of photons	—
Transistor	—	Amplifies or switches electrical signals: no light-electricity conversion	NOT a light converter

NDA 2022-II: which device converts light energy to electrical energy? Answer: Solar cell. Not LED (electricity → light). Not laser diode (electricity → light). Not transistor (no light conversion). [NDA 2022-II]

19. Nuclear Energy: Fission and Fusion

Nuclear Fission

Nuclear fission is the splitting of a heavy nucleus (uranium-235 or plutonium-239) into two or more lighter nuclei, releasing enormous energy. The chain reaction: each fission releases 2–3 neutrons which can trigger further fissions.

Controlled fission: Used in nuclear reactors to generate electricity.

Uncontrolled fission: Basis of nuclear (atomic) bombs.

Nuclear Fusion

Nuclear fusion is the combining of two light nuclei (typically hydrogen isotopes) into a heavier nucleus, releasing even more energy per unit mass than fission.

Uncontrolled fusion: Basis of thermonuclear (hydrogen) bombs, with fusion triggered by a fission bomb as a “spark plug.”

Controlled fusion: Not yet achieved in commercial power plants. Ongoing research (ITER project).

E = mc² (Einstein’s Mass-Energy Equivalence)

In any nuclear reaction (fission or fusion), the products have slightly less mass than the reactants. This mass defect (Δm) is converted to energy according to:

E = mc² = Δm × c²

This expression was proposed by Albert Einstein. [NDA 2021-II] Not Rutherford, not Bohr, not Heisenberg.

NDA 2021-II: “Nuclear energy is generated by nuclear fission and its expression was proposed by Einstein.” Only this option is correct.

20. Solar Energy: Nuclear Fusion in the Sun

The Sun generates energy through nuclear fusion. The process (the proton-proton chain):

Step	Process
1. Start	Hydrogen nuclei (protons) are fused at extremely high temperature (~1.5 × 10⁷ K) and pressure at the Sun’s core
2. Conversion	Hydrogen (H) is converted to helium (He); the product mass is slightly less than the reactant mass
3. Energy release	Mass defect (Δm) is converted to energy by E = mc². Nuclear fusion generates a vast quantity of energy
4. Transport	Energy finds its way to the Sun’s surface (radiative and convective zones)
5. Constancy	Despite enormous energy output, the Sun maintains nearly constant surface temperature (~6000°C) because fusion continuously replenishes the energy

Sequence for NDA 2013-I (arrange in order starting from the cycle beginning): 1-3-2 → Hydrogen converted to helium (1) → nuclear fusion generates energy (3) → energy finds way to surface (2).   [NDA 2013-I]

Solar energy source = fusion (NOT fission, NOT radioactivity). [NDA 2010-II | NDA 2012-II]

Required conditions for fusion: very high temperature AND very high pressure. [NDA 2012-II] Not low temperature. Not high temperature and low pressure. Both conditions must be satisfied.

21. Nuclear Reactors: Components and Principle

A nuclear reactor generates electricity using controlled nuclear fission. [NDA 2019-I] Not uncontrolled fission (that is an atomic bomb). Not fusion (not yet commercially available).

1. Reaction chamber (core): Where fission occurs. Fuel rods (uranium-235 or plutonium-239) are arranged here.

2. Moderator: Slows down neutrons to increase probability of fission (e.g., heavy water, graphite).

3. Control rods: Absorb neutrons to regulate the rate of the chain reaction (e.g., boron or cadmium rods). Inserting rods more deeply slows the reaction; withdrawing them allows it to speed up.

4. Coolant: Carries heat away from the core (e.g., water, heavy water, CO₂ gas, liquid sodium).

5. Heat exchanger: Transfers heat from the coolant to water, producing steam.

6. Turbine: Steam drives a turbine to produce electricity.

7. Biological shield: Concrete shielding prevents radiation from escaping.

Does NOT belong to a nuclear reactor: A mechanism specifically designed to reduce CO₂ emissions. [NDA 2024-I] A nuclear reactor produces no CO₂ during operation, but the question tests whether CO₂ reduction equipment is a reactor component. It is not.

22. Nuclear Fuel: Pitchblende and Uranium

The fuel in nuclear power stations is uranium, specifically uranium-235 (fissile) and/or plutonium-239. [NDA 2019-I] Uranium ore is found in the mineral pitchblende (also called uraninite, chemical formula UO₂). Pitchblende is a naturally occurring uranium oxide and is the primary ore from which uranium is extracted.

Common confusion: Bauxite = aluminium ore. Quartz = SiO₂ (silicon dioxide). Feldspar = silicate mineral. None of these contains uranium.

23. Radioactivity and the GM Counter

Radioactivity is the spontaneous emission of radiation from the nuclei of unstable isotopes. The emitted radiation can be alpha particles, beta particles, or gamma rays.

Radioactivity is measured using a Geiger-Müller (GM) Counter. [NDA 2017-II] The GM counter detects ionising radiation: radiation entering the tube ionises the gas inside, creating a brief electrical pulse that is counted. NOT a polarimeter (measures rotation of polarised light). NOT a calorimeter (measures heat). NOT a colorimeter (measures light absorption in solution).

24. Radioactive Decay: Alpha, Beta, Gamma

Radioactive decay is the spontaneous change in an unstable nucleus, emitting radiation to become more stable.

Radiation	What It Is	Charge	Mass	Penetrating Power	Ionising Power
Alpha (α)	Helium nucleus (₂He⁴: 2 protons + 2 neutrons)	+2e	~4 u	Lowest (stopped by paper)	Highest
Beta (β)	Electron (emitted from nucleus when neutron → proton + electron + antineutrino)	-e	~0 (electron mass)	Medium (stopped by thin Al foil)	Medium
Gamma (γ)	Electromagnetic radiation (very high energy photon)	0 (neutral)	0	Highest (stopped by lead/thick concrete)	Lowest

Conservation Laws in Decay: In every nuclear decay: mass number (A) is conserved AND atomic number (Z) is conserved (charge is conserved).

Alpha decay: ₉₂U²³⁸ → ₉₀Th²³⁴ + ₂He⁴ (mass 238 = 234+4 ✓; atomic number 92 = 90+2 )

Beta decay: ₆C¹⁴ → ₇N¹⁴ + ₋₁e⁰ (mass 14 = 14; atomic number 6 = 7−1)

Gamma emission: no change in A or Z. Nucleus loses energy only.

25. Half-Life

Half-life (t₁/₂) is the time taken for half the nuclei in a radioactive sample to decay. After each half-life, the remaining amount halves:

N(t) = N₀ × (1/2)^(t/t₁/₂)

Worked Example: Initial quantity = 80 g. Half-life = 10 years. After 30 years (3 half-lives):

After 10 years: 80 × (1/2) = 40 g

After 20 years: 40 × (1/2) = 20 g

After 30 years: 20 × (1/2) = 10 g

Half-lives vary enormously: Carbon-14 (5,730 years: useful for carbon dating of organic material), Uranium-238 (4.5 × 10⁹ years: comparable to age of Earth), Polonium-214 (1.6 × 10⁻⁴ seconds: almost instantaneous).

26. Binding Energy and Mass Defect

The actual mass of a nucleus is always less than the sum of the masses of its constituent protons and neutrons (measured separately). This difference is the mass defect (Δm). The energy equivalent of the mass defect is the binding energy: the energy that holds the nucleus together:

BE = Δm × c²

Binding energy per nucleon (BE/A) is a measure of nuclear stability. It peaks at iron (Fe-56), the most stable nucleus. Elements lighter than iron release energy by fusion; elements heavier than iron release energy by fission.

27. Bohr Model: Energy Levels

In Bohr’s model, the energy of the electron in the n-th orbit of hydrogen is:

E_n = −13.6 / n²   eV

n = principal quantum number (1, 2, 3, …). E₁ = −13.6 eV (ground state). E₂ = −13.6/4 = −3.4 eV. E₃ = −13.6/9 = −1.51 eV.

The energy is negative, meaning the electron is bound (it takes energy to remove it). The ionisation energy (energy to completely remove the electron from n = 1) = +13.6 eV.

Spectral Series: When an electron jumps from a higher to lower orbit, it emits a photon. Frequency: hf = E_higher − E_lower. Lyman series (n_final = 1): UV region. Balmer series (n_final = 2): visible region. Paschen series (n_final = 3): infrared region.

28. Semiconductors: p-n Junction, LED, Solar Cell

A semiconductor is a material with electrical conductivity between a conductor and an insulator. Common semiconductors: silicon (Si) and germanium (Ge).

Doping: n-type: Silicon doped with a pentavalent impurity (phosphorus, arsenic). Extra electrons; majority carriers are electrons. p-type: Silicon doped with a trivalent impurity (boron, aluminium). Extra holes; majority carriers are holes.

p-n Junction: When p-type and n-type semiconductors are joined, a depletion region forms with an internal electric field. Forward bias: current flows. Reverse bias: no current.

LED: In a forward-biased p-n junction, when electrons recombine with holes, they release energy as photons (light). LED: electricity → light via electroluminescence.

Solar Cell: Solar cell: light → electricity. Photons knock electrons out of bonds in the semiconductor, creating electron-hole pairs. The internal electric field of the p-n junction separates them, driving current. This is the photovoltaic effect.

29. de Broglie Wavelength

Louis de Broglie (1924) proposed that matter (not just light) has wave-particle duality. Every moving particle has an associated wavelength:

λ = h/mv = h/p

λ = de Broglie wavelength (m). h = Planck’s constant. m = mass of particle (kg). v = speed (m s⁻¹). p = momentum (kg m s⁻¹). The de Broglie wavelength is significant only for particles with very small mass (electrons, protons). For macroscopic objects, λ is so tiny it is unobservable. Application: electron diffraction (confirmed by Davisson-Germer experiment 1927). Electron microscopes use the very short de Broglie wavelength of electrons to achieve resolution far beyond optical microscopes.

30. Important Distinctions: Scientist Attributions

The most persistent error in NDA Modern Physics is attributing discoveries and explanations to the wrong scientist. This table is the single most important reference in this chapter:

Who?	What They Did	What They Did NOT Do
J.J. Thomson	Discovered the electron (cathode rays, 1897)	Did NOT discover nucleus; did NOT propose stable orbits
Ernest Rutherford	Discovered the atomic nucleus (α-scattering, 1909–11)	Did NOT discover electron; did NOT propose stable orbits
Niels Bohr	Proposed stable electron orbits (Bohr model, 1913); ionisation energy = 13.6 eV	Did NOT discover nucleus or electron
Albert Einstein	Explained photoelectric effect (1905); proposed E = mc²	Did NOT discover nucleus or electron; did NOT quantise blackbody radiation
Max Planck	Proposed energy quantisation (h = Planck’s constant, 1900)	Did NOT explain photoelectric effect (that was Einstein)

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