Modern Physics

Concept Core

Planck's quanta, photoelectric effect, atomic spectra, radioactivity — the quantum revolution.

Photoelectric Effect — Einstein's Equation

Light incident on a metal surface ejects electrons if frequency ν ≥ threshold frequency ν₀:

E = hν = KE_max + φ = hν₀ + KE_max KE_max = hν − φ = h(ν − ν₀) = eV₀

φ = work function (energy to eject an electron). V₀ = stopping potential. h = 6.626×10⁻³⁴ J·s.

Key facts: KE_max is independent of intensity. Stopping potential V₀ = KE_max/e. Rate of emission ∝ intensity (number of photons).

Photon Properties

Energy: E = hν = hc/λ Momentum: p = h/λ = E/c Rest mass: 0 (always moves at c) de Broglie (matter wave): λ = h/p = h/mv

Photons travel at c always. No rest mass. Their "mass-equivalent" from E = mc² gives effective inertia for momentum purposes.

Bohr's Atomic Model (Recap)

Eₙ = −13.6/n² eV (H atom) rₙ = 0.529n² Å Number of spectral lines from orbit n: n(n−1)/2

Energy levels are negative (bound state). Ionisation energy = |E₁| = 13.6 eV. Emission spectrum: electron falls to lower orbit.

Radioactivity — Decay Laws

N = N₀ e^(−λt) = N₀ (1/2)^(t/t₁/₂) Activity A = λN = A₀ e^(−λt) t₁/₂ = 0.693/λ = ln2/λ Mean life τ = 1/λ = t₁/₂/0.693

λ = decay constant. Activity in Becquerel (1 Bq = 1 decay/s) or Curie (1 Ci = 3.7×10¹⁰ Bq).

Nuclear Reactions — Mass Defect & Binding Energy

Mass defect: Δm = Zm_p + (A−Z)m_n − M_nucleus Binding energy: BE = Δm × c² = Δm × 931.5 MeV/u BE per nucleon = BE/A

Higher BE/nucleon → more stable nucleus. Iron (Fe-56) is most stable. Fission (heavy nuclei split) and fusion (light nuclei merge) both release energy by moving toward Fe-56.

Alpha, Beta, Gamma Decay

Decay	Emission	Effect on (Z,A)
α decay	₂He⁴	Z−2, A−4
β⁻ decay	electron + antineutrino	Z+1, A unchanged
β⁺ decay	positron + neutrino	Z−1, A unchanged
γ decay	photon	No change in Z, A

Formula Vault

Modern physics formulas — photoelectric, photon, radioactivity, and nuclear.

Einstein's Photoelectric

KE_max = hν − φ = eV₀

φ = work function; V₀ = stopping potential

Threshold Frequency

ν₀ = φ/h

hν₀ = φ; minimum frequency

Photon Energy

E = hν = hc/λ

h = 6.626×10⁻³⁴ J·s

Photon Momentum

p = h/λ = E/c

Zero rest mass, p ≠ 0

de Broglie

λ = h/mv = h/p

All matter has wavelength

Radioactive Decay

N = N₀ e^(−λt)

λ = decay constant

Half-Life

t₁/₂ = 0.693/λ

After n half-lives: N = N₀(½)ⁿ

Activity

A = λN (Becquerel)

1 Ci = 3.7×10¹⁰ Bq

Binding Energy

BE = Δm × 931.5 MeV/u

Δm in atomic mass units (u)

Q-value of Reaction

Q = (m_reactants − m_products)c²

Q > 0: exothermic; Q < 0: endothermic

Worked Examples

5 problems — photoelectric, half-life, mass-energy, nuclear decay, and a trap.

EasyFind KE of photoelectrons: ν=1.5×10¹⁵ Hz, φ=3.0 eV▾

Light of frequency 1.5×10¹⁵ Hz strikes a metal with work function 3.0 eV. Find max KE. (h=4.14×10⁻¹⁵ eV·s)

E = hν = 4.14×10⁻¹⁵ × 1.5×10¹⁵ = 6.21 eV

KE_max = hν − φ = 6.21 − 3.0 = 3.21 eV

✓ KE_max = 3.21 eV

EasyAfter 3 half-lives, what fraction of a radioactive sample remains?▾

A radioactive sample has half-life 2 years. What fraction remains after 6 years?

n = 6/2 = 3 half-lives

N/N₀ = (1/2)³ = 1/8

✓ Fraction remaining = 1/8

MediumAlpha decay of Ra-226: write the daughter nucleus▾

Radium-226 (₈₈Ra²²⁶) undergoes α decay. Write the daughter nucleus.

α particle is ₂He⁴. Conservation: A: 226−4=222, Z: 88−2=86

Element with Z=86 is Radon (Rn)

₈₈Ra²²⁶ → ₂He⁴ + ₈₆Rn²²²

✓ Daughter nucleus: ₈₆Rn²²² (Radon-222)

EAPCET LevelFind binding energy per nucleon for He-4 (Δm=0.0304 u)▾

Calculate the binding energy per nucleon of He-4 if the mass defect Δm = 0.0304 u.

BE = Δm × 931.5 MeV/u = 0.0304 × 931.5 = 28.32 MeV

A = 4 nucleons

BE per nucleon = 28.32/4 = 7.08 MeV/nucleon

✓ BE per nucleon = 7.08 MeV/nucleon

Trap QuestionIncreasing light intensity increases the kinetic energy of photoelectrons — True or False?▾

A student claims: 'Brighter light gives photoelectrons more energy.' Evaluate.

The trap: Einstein's photoelectric equation: KE_max = hν − φ. Only ν (frequency) appears — not intensity.

Increasing intensity increases the NUMBER of photons, not their individual energy.

More photons → more electrons ejected (larger photoelectric current).

But each photon has the same energy hν — so KE_max of each electron is unchanged.

✓ False — intensity affects rate of emission (current), not KE. Only frequency (ν) determines KE_max

Mistake DNA

4 modern physics errors that cost marks in EAPCET.

💡

Intensity Affects Kinetic Energy in Photoelectric Effect

In the photoelectric effect, KE_max depends only on frequency. Intensity controls the NUMBER of photoelectrons emitted per second.

❌ Wrong

Brighter light → higher KE_max ✗ (Einstein's equation has no intensity term)

✓ Correct

KE_max = hν − φ ✓ Intensity → more electrons ✓ (larger current, same KE_max)

Einstein: each photon has energy hν. Higher intensity = more photons = more electrons ejected, but each electron still gets the same energy from each photon.

⚛️

β⁻ Decay Changes Mass Number A

In β⁻ decay, a neutron converts to a proton. Z increases by 1 but A stays the same (total nucleons unchanged).

❌ Wrong

β⁻ decay: ¹⁴₆C → ¹⁰₅B + e⁻ ✗ (A changed from 14 to 10!)

✓ Correct

β⁻ decay: Z+1, A same ✓ ¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ ✓ Neutron → proton + electron

β⁻: n → p + e⁻ + antineutrino. Proton count Z increases by 1. Neutron count decreases by 1. Total A = Z + N is unchanged.

🔢

Half-Life vs Mean Life Confusion

t₁/₂ ≠ τ (mean life). Mean life τ = t₁/₂/0.693 = 1/λ. Mean life is longer than half-life.

❌ Wrong

Mean life = half-life ✗ τ = t₁/₂ ✗

✓ Correct

τ = 1/λ = t₁/₂/0.693 ✓ τ = 1.44 × t₁/₂ ✓ Mean life > half-life

After one half-life, N = N₀/2. After one mean life τ, N = N₀/e ≈ 0.368N₀. Half-life is when 50% remains; mean life (longer) is when 36.8% remains.

🌊

de Broglie: Wavelength Depends Only on Speed (Forgetting Mass)

λ = h/mv. Both mass AND velocity matter. A heavier particle moving at the same speed has a shorter wavelength.

❌ Wrong

Same speed v: proton λ = electron λ ✗ (different masses!)

✓ Correct

λ = h/mv ✓ Proton (1836× heavier) has 1836× shorter λ for same speed ✓

de Broglie wavelength λ = h/mv = h/p. Larger momentum → smaller wavelength. This is why macroscopic objects have negligibly small de Broglie wavelengths.

Chapter Intelligence

Modern Physics connects quantum concepts to atomic structure and nuclear energy.

EAPCET Weightage (2019–2024)

Photoelectric effect

Radioactive decay and half-life

Bohr's atomic model / spectra

Nuclear reactions (α,β,γ)

Binding energy / mass defect

de Broglie wavelength

High-Yield PYQ Patterns

KE_max = hν − φ calculationN remaining after n half-livesα/β decay — find daughter nucleusBinding energy per nucleonStopping potential from frequencyde Broglie wavelength of particleActivity from decay constant

Exam Strategy

Photoelectric: KE_max = hν − φ = eV₀. Frequency determines KE; intensity determines current. This conceptual distinction is tested every year.
Radioactivity: N = N₀(½)ⁿ where n = t/t₁/₂. Count half-lives elapsed first, then apply.
Nuclear decay: α removes 4 from A, 2 from Z. β⁻ changes Z+1 but A stays same. γ changes neither A nor Z.
Binding energy: BE = Δm × 931.5 MeV/u. Higher BE/nucleon = more stable. Fe-56 is most stable.
This chapter connects to Atomic Structure (Bohr model, spectra) and to EMI/waves (wave-particle duality).

Concept Core

Formula Vault

Worked Examples

Mistake DNA

Chapter Intelligence

💡 Suggestions & Feedback

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