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Dual Nature of Matter and Radiation – JEE Main Physics Formula Sheet | Photoelectric Effect, de Broglie Wavelength, Work Function, Stopping Potential, Einstein's Equation & All Formulas
This is the complete JEE Main Physics Formula Sheet and Class 12 Formula Sheet for Dual Nature of Matter and Radiation — Chapter 16 from the Aakash Rapid Revision & Formula Bank. This chapter is the gateway to modern physics, establishing that light behaves as particles (photons) and particles behave as waves. It covers: Photon Properties — energy E=hf=hc/λ, momentum p=h/λ=E/c, mass equivalent m=h/cλ=E/c²; Photoelectric Effect — threshold frequency f₀=φ/h, threshold wavelength λ₀=hc/φ, Einstein's photoelectric equation KE_max=hf–φ=h(f–f₀), stopping potential eV₀=KE_max=hf–φ, saturation current ∝ intensity, KE independent of intensity; Work Function — φ=hf₀=hc/λ₀ (in eV); de Broglie Hypothesis — λ=h/p=h/mv; de Broglie Wavelength Formulas — λ=h/√(2mKE), λ=h/√(2meV) (accelerated particle), λ=h/√(3mkT) (thermal particle); Davisson-Germer Experiment — confirmed wave nature of electrons; Heisenberg's Uncertainty Principle — Δx·Δp ≥ h/4π, ΔE·Δt ≥ h/4π; and Photoelectric effect laws — effect of frequency, intensity, and stopping potential. Dual Nature of Matter and Radiation contributes 2–3 questions in every JEE Main session. Download the Free PDF for all formulas in one JEE Main exam-ready reference.
Topics Covered in This Dual Nature of Matter Formula Sheet
Dual Nature of Matter and Radiation JEE Main Formula Sheet PDF Preview
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Introduction: Why Dual Nature of Matter and Radiation Is the Modern Physics Gateway Chapter in JEE Main
Dual Nature of Matter and Radiation is the pivot point between classical physics (waves, particles as distinct things) and quantum mechanics (wave-particle duality for both light and matter). Two revolutionary insights define this chapter: (1) Einstein's 1905 explanation of the photoelectric effect established that light, long known as a wave, can also behave as discrete packets of energy called photons — particle behaviour of light; (2) de Broglie's 1924 hypothesis proposed that matter particles (electrons, protons, atoms) also have wave properties — wave behaviour of matter. These two ideas together constitute wave-particle duality, the foundation of all quantum mechanics.
For JEE Main physics, Dual Nature of Matter and Radiation contributes 2–3 questions per session. Questions test: photon energy E=hf=hc/λ (using hc=1240 eV·nm), Einstein's photoelectric equation KE_max=hf–φ, stopping potential eV₀=KE_max, threshold frequency/wavelength (f₀=φ/h, λ₀=hc/φ), de Broglie wavelength λ=h/√(2mKE)=h/√(2meV), electron wavelength λ=1.226/√V nm, and the qualitative laws of the photoelectric effect (KE_max independent of intensity, saturation current proportional to intensity, no emission below threshold frequency).
Download the Free PDF for Dual Nature of Matter and Radiation to access all photon properties, photoelectric effect formulas, stopping potential, work function, de Broglie wavelength (all forms), Davisson-Germer, Heisenberg uncertainty, and Compton effect formulas in one structured JEE Main physics revision reference.
Key Concepts and Formulas in Dual Nature of Matter and Radiation
Photon — Energy, Momentum, and Key Constants
Why E=hf=hc/λ and hc=1240 eV·nm Are the Most-Used Dual Nature JEE Main Photon Formulas
Planck's Quantum Theory and Photon (from Aakash PDF — Dual Nature of Matter JEE Main):
Light consists of discrete energy packets called photons. Each photon has:
Energy: E = hf = hc/λ
Planck's constant: h = 6.626×10⁻³⁴ J·s = 4.136×10⁻¹⁵ eV·s
Speed: c = 3×10⁸ m/s. Frequency f and wavelength λ related by: c = fλ.
Useful shortcut: hc = 1240 eV·nm = 12400 eV·Å
This means for λ in nm: E(eV) = 1240/λ(nm). For λ=500nm (green light): E=1240/500=2.48 eV.
Unit conversion: 1 eV = 1.6×10⁻¹⁹ J.
Photon Momentum and Mass Equivalent (from Aakash PDF — Dual Nature JEE Main):
Momentum: p = h/λ = E/c = hf/c
Equivalent mass: m = E/c² = h/cλ = hf/c²
Rest mass of photon = 0 (photon has no rest mass, but has momentum and energy).
A photon always travels at speed c in vacuum regardless of its frequency or energy.
Number of photons in radiation of total energy E: N = E/hf = E/(hc/λ) = Eλ/hc.
Number of photons emitted per second by source of power P at wavelength λ: n = P/hf = Pλ/hc.
Download the Free PDF for Dual Nature of Matter for all photon property examples for JEE Main.
Photoelectric Effect — Laws, Einstein's Equation, Stopping Potential, Work Function
Why Einstein's Equation KE_max=hf–φ and Stopping Potential eV₀=KE_max Are the Most-Tested Dual Nature JEE Main Formulas
Photoelectric Effect (from Aakash PDF — Dual Nature JEE Main):
When light of sufficient frequency shines on a metal surface, electrons are emitted from the surface. This phenomenon is called the photoelectric effect. Emitted electrons are called photoelectrons.
Work Function φ (from Aakash PDF — Dual Nature JEE Main):
The minimum energy required to eject an electron from the metal surface.
φ = hf₀ = hc/λ₀
Threshold frequency: f₀ = φ/h (minimum frequency for photoelectric emission)
Threshold wavelength: λ₀ = hc/φ (maximum wavelength; λ₀ in nm = 1240/φ(eV))
No photoelectric effect if f < f₀ (or λ > λ₀), regardless of intensity. Different metals have different φ and f₀ values. Cs: φ≈2.0eV; Na: φ≈2.3eV; K: φ≈2.25eV; Ag: φ≈4.7eV; Pt: φ≈5.6eV.
Einstein's Photoelectric Equation (from Aakash PDF — Dual Nature JEE Main):
Energy of incident photon = Work function + Maximum kinetic energy of emitted photoelectron:
KE_max = hf – φ = h(f – f₀)
Also written as: KE_max = hc/λ – hc/λ₀ = hc(1/λ – 1/λ₀)
KE_max = ½mv²_max = eV₀ (where V₀ = stopping potential)
KE_max > 0 only when f > f₀ (or λ < λ₀). KE_max increases linearly with frequency f.
Stopping Potential V₀ (from Aakash PDF — Dual Nature JEE Main):
The negative potential applied to the collector that just stops the most energetic photoelectrons:
eV₀ = KE_max = hf – φ
V₀ = (hf – φ)/e = (h/e)(f – f₀)
V₀ vs f graph: straight line with slope h/e (Planck's constant/electron charge). Intercept on f-axis = f₀. V₀ is independent of intensity.
Laws of Photoelectric Effect (from Aakash PDF — Dual Nature JEE Main):
(1) Threshold frequency: Emission occurs only if f ≥ f₀ (no emission for any intensity if f < f₀).
(2) KE_max is independent of intensity (depends only on frequency). Intensity only determines the number of emitted electrons.
(3) Saturation current (photoelectric current) ∝ intensity of incident light (more photons → more electrons).
(4) KE_max ∝ (f – f₀) — increases linearly with frequency.
(5) Stopping potential V₀ ∝ f (and independent of intensity).
(6) Emission is instantaneous — no time lag between illumination and emission (even at very low intensities).
Classical wave theory failed to explain: (i) existence of threshold frequency (wave theory predicts emission for all frequencies); (ii) independence of KE from intensity (wave theory: more intensity = more energy = higher KE); (iii) instantaneous emission (wave theory: need time to accumulate energy). Download the Free PDF for Dual Nature for all photoelectric effect examples for JEE Main.
de Broglie Wavelength — All Forms for Different Situations
Why λ=h/√(2meV) for Accelerated Electrons and λ=1.226/√V nm Are the Most-Tested de Broglie Formulas in JEE Main
de Broglie Hypothesis (from Aakash PDF — Dual Nature JEE Main):
Louis de Broglie (1924): If light (a wave) can behave as particles (photons), then particles (electrons, protons, etc.) should behave as waves. The wavelength associated with a moving particle:
λ = h/p = h/(mv)
where p = momentum, m = mass, v = speed. This is called the de Broglie wavelength.
λ decreases as momentum increases: faster or heavier particles have smaller wavelength (less wave-like behaviour).
de Broglie Wavelength in Terms of KE (from Aakash PDF — Dual Nature JEE Main):
KE = p²/2m → p = √(2m×KE)
λ = h/√(2m×KE) = h/p
de Broglie Wavelength for Particle Accelerated Through Potential V (from Aakash PDF — Dual Nature JEE Main):
A particle of charge q accelerated through potential difference V gains KE = qV:
p = √(2m×qV)
λ = h/√(2mqV)
For an electron (m = 9.11×10⁻³¹ kg, q = e = 1.6×10⁻¹⁹ C) accelerated through V volts:
λ = 1.226/√V nm (V in volts, λ in nm)
Or: λ ≈ 12.26/√V Å
At V=100V: λ=0.1226nm=1.226Å. At V=54V: λ≈0.167nm (Davisson-Germer result).
de Broglie Wavelength for Thermal Particle (from Aakash PDF — Dual Nature JEE Main):
For a particle in thermal equilibrium at temperature T: KE = (3/2)kT (or (1/2)kT per DoF):
Using KE = p²/2m = (3/2)kT → p = √(3mkT)
λ = h/√(3mkT)
Also written as thermal de Broglie wavelength: λ_th = h/√(2πmkT) (exact quantum statistical form).
Comparison of de Broglie Wavelengths (from Aakash PDF — Dual Nature JEE Main):
For same KE: λ ∝ 1/√m → lighter particle has larger λ (electron has larger λ than proton at same KE).
For same V: λ ∝ 1/√(mq) → electron (small mass) has much larger λ than heavy ion.
For same speed v: λ = h/mv ∝ 1/m → lighter particle has larger λ.
At room temperature (T=300K) for electron: λ_th = h/√(3m_e kT) = 6.626×10⁻³⁴/√(3×9.11×10⁻³¹×1.38×10⁻²³×300) ≈ 6.3 nm (significant quantum effects).
Download the Free PDF for Dual Nature of Matter for all de Broglie examples for JEE Main.
Davisson-Germer Experiment, Heisenberg Uncertainty, and Compton Effect
Why Davisson-Germer Confirmed de Broglie and Heisenberg Δx·Δp≥h/4π Is the Key Quantum Principle
Davisson-Germer Experiment (from Aakash PDF — Dual Nature JEE Main):
Davisson and Germer (1927) demonstrated the wave nature of electrons by observing diffraction of electrons scattered from a nickel crystal.
Setup: electrons accelerated through 54V, directed at nickel crystal at angle. A detector measured scattered electrons at various angles.
Observation: strong peak (constructive interference) at scattering angle 50° from grazing — consistent with diffraction maxima.
de Broglie wavelength for 54V electrons: λ = 1.226/√54 ≈ 0.167 nm = 1.67 Å.
From Bragg diffraction condition (for Ni crystal d=0.215nm, 2d sinθ=nλ): predicted λ ≈ 0.165 nm ≈ 0.167 nm ✓
This perfect agreement confirmed de Broglie's hypothesis experimentally — electrons have wave properties. This was the first direct evidence of the wave nature of matter.
Heisenberg's Uncertainty Principle (from Aakash PDF — Dual Nature JEE Main):
It is fundamentally impossible to simultaneously know both the exact position and exact momentum (or velocity) of a particle:
Δx · Δp ≥ h/(4π) = ℏ/2
where Δx = uncertainty in position, Δp = uncertainty in momentum. ℏ = h/2π = 1.055×10⁻³⁴ J·s.
Energy-time uncertainty: ΔE · Δt ≥ h/(4π) = ℏ/2
The uncertainty principle is NOT due to limitations of measurement instruments — it is a fundamental property of quantum systems. Smaller Δx (more precise position) → larger Δp (less precise momentum).
Application: minimum energy of a quantum particle confined to a box of size L: from Δx≈L → Δp≥h/4πL → KE_min ≈ (Δp)²/2m ≥ h²/(32π²mL²).
Compton Effect (from Aakash PDF — Dual Nature JEE Main):
When X-rays scatter off electrons (Compton scattering), the scattered X-ray has a longer wavelength than the incident X-ray. This proves photon-electron collision (particle nature of X-rays).
Compton shift: Δλ = λ' – λ = (h/m_e c)(1 – cosφ)
where φ = scattering angle of X-ray from original direction.
Compton wavelength: λ_c = h/m_e c = 0.00243 nm = 2.43 pm
At φ=0°: Δλ=0 (forward scatter, no change). At φ=90°: Δλ=λ_c=0.00243nm. At φ=180°: Δλ=2λ_c=0.00486nm (maximum).
Download the Free PDF for Dual Nature of Matter for all Davisson-Germer and Compton examples for JEE Main.
Wave-Particle Duality — Summary, Graphs, and Key Distinctions
Why Wave-Particle Duality and Photoelectric Effect Graphs Are Essential Dual Nature JEE Main Concepts
Wave-Particle Duality (from Aakash PDF — Dual Nature JEE Main):
Nature exhibits duality: light and matter both exhibit wave AND particle properties, depending on the experiment.
Light as WAVE: interference (Young's double slit), diffraction, polarisation.
Light as PARTICLE: photoelectric effect, Compton scattering, radiation pressure.
Electrons as PARTICLES: definite mass, charge, deflection by fields.
Electrons as WAVES: diffraction (Davisson-Germer), electron microscopy.
Wave and particle natures are complementary — they are never both apparent simultaneously in the same experiment (Bohr's complementarity principle).
Key Graphs in Photoelectric Effect (from Aakash PDF — Dual Nature JEE Main):
(1) KE_max vs frequency (f): straight line, slope = h, x-intercept = f₀, y-intercept = –φ. Different metals → parallel lines (same slope h, different f₀).
(2) Stopping potential (V₀) vs frequency (f): straight line, slope = h/e, x-intercept = f₀. Independent of metal? No — different metals have different f₀ intercepts but the same slope h/e.
(3) Photoelectric current vs voltage (V): current increases with positive V until saturation; stopping potential at negative V; for higher intensity: higher saturation current, same V₀.
(4) Saturation current vs intensity: linear (direct proportion).
(5) KE_max vs intensity: flat (horizontal line) — KE_max is INDEPENDENT of intensity.
Key Values and Constants (from Aakash PDF — Dual Nature JEE Main):
h = 6.626×10⁻³⁴ J·s; ℏ = h/2π = 1.055×10⁻³⁴ J·s
m_e = 9.11×10⁻³¹ kg; m_p = 1.67×10⁻²⁷ kg; m_p/m_e ≈ 1836
e = 1.6×10⁻¹⁹ C; 1 eV = 1.6×10⁻¹⁹ J
c = 3×10⁸ m/s; k = 1.38×10⁻²³ J/K
Bohr's Condition as de Broglie Wave:
Bohr's postulate mvr = nℏ = nh/2π is equivalent to saying that an electron's de Broglie wavelength (λ=h/mv) must fit an integral number of times around the circular orbit: nλ = 2πr → n(h/mv) = 2πr → mvr = nh/2π ✓. This gives a quantum mechanical justification for Bohr's angular momentum quantisation postulate. Download the Free PDF for Dual Nature of Matter for all wave-particle duality examples for JEE Main.
Download Free PDF — Dual Nature of Matter and Radiation JEE Main Formula Sheet
All Dual Nature of Matter and Radiation formulas from the Aakash Rapid Revision PDF: E=hf=hc/λ; h=6.626×10⁻³⁴ J·s=4.136×10⁻¹⁵ eV·s; hc=1240 eV·nm=12400 eV·Å; p=h/λ=E/c; m_eq=E/c²=h/cλ; rest mass photon=0; speed photon=c; 1eV=1.6×10⁻¹⁹J; n=P/hf=Pλ/hc (photons/sec); φ=hf₀=hc/λ₀; f₀=φ/h; λ₀=hc/φ=1240/φ(eV)nm; Einstein KE_max=hf–φ=h(f–f₀)=hc(1/λ–1/λ₀); eV₀=KE_max=hf–φ; V₀=(hf–φ)/e=(h/e)(f–f₀); laws: (1)f
Why Dual Nature of Matter and Radiation Is a Conceptually Rich JEE Main Chapter
The shortcut hc=1240 eV·nm is the most time-saving Dual Nature formula in JEE Main. It converts any wavelength directly to photon energy in eV without knowing h and c separately. For λ=400nm (violet): E=1240/400=3.1eV. For λ=700nm (red): E=1240/700=1.77eV. This same shortcut applies to finding threshold wavelength from work function: λ₀(nm)=1240/φ(eV). For cesium (φ=2.0eV): λ₀=620nm (visible red — Cs is photosensitive to visible light). For silver (φ=4.7eV): λ₀=264nm (UV — Ag needs UV to emit).
The graph of stopping potential V₀ vs frequency f is the most-tested Dual Nature graph in JEE Main. It's a straight line with slope h/e (the same for all metals) and x-intercept at f₀ (different for different metals). The significance: slope = h/e gives an experimental measurement of Planck's constant from the graph. The y-axis value at any frequency gives the stopping potential. If two different metals are tested with the same frequency, they have different V₀ values (since different φ values) but both follow lines of the same slope h/e.
The electron de Broglie wavelength λ=1.226/√V nm is the fastest calculation in Dual Nature JEE Main. Derived from λ=h/√(2meV) with m=m_e, q=e — all the constants combine to 1.226 when V is in volts and λ in nm. At V=100V: λ=0.1226nm. At V=10000V (10kV): λ=0.01226nm. This is why electron microscopes (which use 50–100kV electrons) can resolve features at atomic scale (λ ≈ 0.004nm), far beyond visible light microscopes (λ≈400–700nm). Download the Free PDF for Dual Nature of Matter to have all formulas ready.
Who Should Use This Dual Nature of Matter Formula Sheet?
Learning Outcomes After Completing Dual Nature of Matter and Radiation
After working through Dual Nature of Matter and Radiation using this formula sheet, a student should accomplish: On photons: compute E=hf=hc/λ using hc=1240 eV·nm; compute p=h/λ=E/c; compute number of photons per second n=P/hf; state rest mass of photon=0 but momentum=h/λ.
On photoelectric effect: compute work function φ=hf₀=hc/λ₀; find threshold frequency f₀=φ/h and wavelength λ₀=1240/φ nm; apply Einstein's equation KE_max=hf–φ=h(f–f₀)=hc(1/λ–1/λ₀); compute stopping potential eV₀=KE_max; state all six photoelectric laws (threshold, KE independent of I, saturation current∝I, KE∝(f–f₀), V₀∝f independent of I, instantaneous emission); interpret KE_max vs f graph (slope=h, x-int=f₀); interpret V₀ vs f graph (slope=h/e, x-int=f₀).
On de Broglie wavelength: apply λ=h/p=h/mv; apply λ=h/√(2mKE); apply λ=h/√(2mqV) for accelerated particle; use electron formula λ=1.226/√V nm; apply thermal λ=h/√(3mkT); compare wavelengths at same KE (λ∝1/√m), same V (λ∝1/√(mq)), same speed (λ∝1/m). On Davisson-Germer: state experimental setup, observation (diffraction peak at 50°), and calculated λ=0.167nm at 54V matching de Broglie prediction. Apply Heisenberg Δx·Δp≥h/4π; ΔE·Δt≥h/4π. Apply Compton shift Δλ=(h/m_ec)(1–cosφ). State Bohr's angular momentum condition as de Broglie standing wave (nλ=2πr). Download the Free PDF for Dual Nature of Matter to test all outcomes before your JEE Main exam.
Get the Free PDF for Dual Nature of Matter — JEE Main Quick Revision
The Aakash Rapid Revision & Formula Bank PDF for Dual Nature of Matter and Radiation contains all photon property formulas, hc=1240 eV·nm shortcut, work function, threshold frequency and wavelength, Einstein's photoelectric equation, stopping potential, all six photoelectric laws, de Broglie wavelength (all three forms), Davisson-Germer, Heisenberg uncertainty, Compton effect, and Bohr's de Broglie connection in one structured JEE Main physics reference.
Conclusion — Dual Nature of Matter: The Modern Physics Turning Point in JEE Main
Dual Nature of Matter and Radiation represents the fundamental shift from classical to quantum physics. Classical physics said: waves are waves, particles are particles. Quantum physics says: everything has both wave and particle properties — the behaviour exhibited depends on the experiment. Einstein's photoelectric equation (KE_max=hf–φ) established the photon. de Broglie's hypothesis (λ=h/mv) extended wave-particle duality to matter. Davisson and Germer confirmed it experimentally. Heisenberg made it fundamental (not just an approximation). Together these ideas form quantum mechanics — the most successful physical theory ever developed.
Five most JEE Main-tested results: (1) hc=1240 eV·nm (universal shortcut — use in every photon energy calculation); (2) KE_max=hf–φ and eV₀=KE_max (Einstein + stopping potential); (3) λ₀=1240/φ(eV) nm (threshold wavelength from work function); (4) λ=1.226/√V nm for electrons (de Broglie of accelerated electron); (5) Laws: KE_max independent of intensity; V₀ independent of intensity; saturation current proportional to intensity. Use this page and the Free PDF Download for Dual Nature of Matter as your complete JEE Main revision foundation.
Frequently Asked Questions — Dual Nature of Matter Formulas
What is the photoelectric effect and what are its laws in Dual Nature of Matter?
In Dual Nature of Matter and Radiation, photoelectric effect from Aakash PDF: when electromagnetic radiation (usually UV or visible) of sufficient frequency falls on a metal surface, electrons are emitted. Six fundamental laws: (1) Threshold frequency: emission occurs ONLY if f≥f₀ (no emission for any intensity below threshold; f₀ is material-dependent). (2) KE_max is independent of intensity: KE_max=hf–φ depends only on frequency, not on how bright the light is. (3) Saturation current (number of emitted electrons per second) ∝ intensity: more photons per second → more electrons per second → higher current. (4) KE_max increases linearly with frequency: KE_max=h(f–f₀)→proportional to (f–f₀). (5) Stopping potential V₀ is proportional to frequency and independent of intensity: eV₀=hf–φ→same V₀ for same f regardless of intensity. (6) Emission is instantaneous: no time lag between illumination and electron emission, even at very low intensities (one photon→one electron, no energy accumulation needed). Classical wave theory failed to explain (1)(2)(6): wave theory predicts all frequencies should eventually emit (given enough time to accumulate energy), intensity should control KE, and there should be a time delay for very low intensities. Quantum theory (Einstein 1905) solved all three simultaneously with the photon concept.
What is Einstein's photoelectric equation in Dual Nature of Matter and how is stopping potential related?
In Dual Nature of Matter, Einstein's equation from Aakash PDF: one photon interacts with one electron. Photon energy hf: part goes to releasing electron from metal (work function φ=binding energy), rest becomes kinetic energy. KE_max=hf–φ=h(f–f₀). This is for the most energetic electrons (those at the surface with minimum binding). Electrons from deeper in the metal have less KE (more binding). Stopping potential V₀: the reverse voltage that stops ALL photoelectrons (including most energetic). eV₀=KE_max=hf–φ → V₀=(hf–φ)/e=(h/e)(f–f₀). V₀ vs f graph: slope=h/e (same for all metals — slope gives Planck's constant experimentally); x-intercept=f₀ (threshold, different for each metal). At f=f₀: V₀=0 (just barely emitted with zero KE). For f
What is the de Broglie wavelength formula in Dual Nature of Matter for accelerated particles?
In Dual Nature of Matter, de Broglie wavelength from Aakash PDF: every moving particle has wavelength λ=h/p=h/mv. For particle accelerated through potential V (gaining KE=qV): p=√(2mKE)=√(2mqV). λ=h/√(2mqV). For electron (m=9.11×10⁻³¹kg, q=e=1.6×10⁻¹⁹C): λ=h/√(2m_e eV)=6.626×10⁻³⁴/√(2×9.11×10⁻³¹×1.6×10⁻¹⁹×V)=1.226×10⁻⁹/√V metres=1.226/√V nm. Numerically: V=100V → λ=0.1226nm=1.226Å. V=10000V → λ=0.01226nm. V=1V → λ=1.226nm. For proton (m=1.67×10⁻²⁷kg, q=e): λ=h/√(2m_p eV)=0.0286/√V nm (28.6/√V pm) — much smaller (heavier mass). Comparison at same KE: λ_e/λ_p=√(m_p/m_e)=√1836≈42.8. Comparison at same V: λ_e/λ_p=√(m_p/m_e)=√1836≈42.8 (same result since both charged e). For different charge: λ∝1/√(mq). Thermal particle at T: KE=(3/2)kT → p=√(3mkT) → λ=h/√(3mkT). JEE Main Dual Nature: "electron accelerated through 100V. Find λ" → λ=1.226/√100=1.226/10=0.1226nm≈1.23Å.
What is the Davisson-Germer experiment in Dual Nature of Matter and what did it prove?
In Dual Nature of Matter, Davisson-Germer experiment from Aakash PDF: performed in 1927 at Bell Labs. Purpose: experimentally verify de Broglie's hypothesis that electrons have wave properties. Setup: electron gun accelerates electrons through 54V. Beam directed at nickel crystal (Ni). Detector measures intensity of scattered electrons as a function of angle. Observation: at scattering angle 50° (from incident direction), a strong peak (constructive interference) was observed — exactly like diffraction of X-rays from crystals. Calculation: de Broglie wavelength for 54V electrons: λ_dB=1.226/√54≈0.167nm. Bragg's law for Ni crystal (spacing d=0.215nm): 2d sinθ=nλ → for θ=65° (glancing angle corresponding to 50° scattering): 2×0.215×sin65°=2×0.215×0.906≈0.390nm=2×0.195nm → λ≈0.165nm. Agreement: 0.167nm (de Broglie) ≈ 0.165nm (Bragg) → excellent agreement. Significance: first direct experimental proof that matter (particles) have wave properties. Confirmed de Broglie's hypothesis. This is analogous to how Young's double slit confirmed wave nature of light — Davisson-Germer confirmed wave nature of matter. Led to development of electron microscopy (using matter waves with very short λ to image atomic structures).
What is Heisenberg's uncertainty principle in Dual Nature of Matter?
In Dual Nature of Matter, Heisenberg uncertainty from Aakash PDF: Werner Heisenberg (1927) — it is fundamentally impossible to measure both position and momentum of a particle simultaneously to arbitrary precision. Mathematical form: Δx·Δpₓ≥h/4π=ℏ/2. Δx=uncertainty in position; Δpₓ=uncertainty in x-component of momentum. Energy-time: ΔE·Δt≥h/4π=ℏ/2. ℏ=h/2π=1.055×10⁻³⁴ J·s. Physical origin: not due to measurement clumsiness — it's a fundamental property of quantum systems (wave-particle duality). A particle's position is related to the spatial extent of its wave packet. A precisely defined position (narrow wave packet) requires superposition of many wavelengths → large spread in momentum (Δp large). A precisely defined momentum (single wavelength) → wave packet spread over all space → position completely unknown (Δx→∞). Applications: (1) Minimum energy of confined particle: box size L → Δx≈L → Δp≥h/4πL → KE_min≈(Δp)²/2m≈h²/32π²mL² (zero-point energy). (2) Natural linewidth of spectral lines: excited state lifetime Δt → ΔE≥h/4πΔt → spectral line has minimum width. (3) Why electrons can't exist in nucleus: if Δx≈nuclear size (10⁻¹⁵m) → Δp≥h/4π×10⁻¹⁵ → KE≈330 MeV >> observed beta decay energy. JEE Main Dual Nature: "Δx=10⁻¹⁰m. Find minimum Δp" → Δp≥h/4πΔx=6.626×10⁻³⁴/(4π×10⁻¹⁰)≈5.27×10⁻²⁵ kg·m/s.
What is the Compton effect in Dual Nature of Matter and what is the Compton wavelength?
In Dual Nature of Matter, Compton effect from Aakash PDF: Arthur Compton (1923) discovered that when X-rays scatter off free electrons, the scattered X-ray has a LONGER wavelength than the incident X-ray. This is the Compton effect. Classical wave theory predicts no wavelength change (Thomson scattering) — only quantum photon-particle collision explains the shift. Compton shift: Δλ=λ'–λ=(h/m_ec)(1–cosφ) where φ=angle of scattered X-ray from original direction. Compton wavelength: λ_c=h/m_ec=6.626×10⁻³⁴/(9.11×10⁻³¹×3×10⁸)=2.43×10⁻¹²m=0.00243nm=2.43pm. Angular dependence: φ=0° → Δλ=0 (no change, forward scatter). φ=90° → Δλ=λ_c=0.00243nm. φ=180° → Δλ=2λ_c=0.00486nm (maximum, backscatter). Significance: Compton effect provides definitive proof of photon's particle nature (momentum h/λ) in collision with electrons. Together with photoelectric effect, it established the particle nature of light. Conservation laws used: energy conservation (photon KE + electron KE = constant) and momentum conservation (vector; both x and y components). JEE Main Dual Nature: "X-rays at 90°. Find Compton shift" → Δλ=λ_c=0.00243nm. "X-ray λ=0.05nm scatters at 180°. Find scattered λ" → Δλ=2×0.00243=0.00486nm → λ'=0.05+0.00486=0.05486nm.
How is Bohr's quantisation condition derived from de Broglie in Dual Nature of Matter?
In Dual Nature of Matter, Bohr-de Broglie connection from Aakash PDF: Bohr's angular momentum postulate: mvr=nh/2π (n=1,2,3...). This was introduced by Bohr in 1913 without physical justification — it worked but seemed arbitrary. de Broglie (1924) provided the justification: electron in orbit has wavelength λ=h/mv. For a stable orbit, the electron wave must form a standing wave (must constructively interfere after going around the orbit). Condition for standing wave: circumference = integral multiple of wavelength. 2πr=nλ=n(h/mv). Rearranging: mvr=nh/2π=nℏ. This is exactly Bohr's postulate! The wave-particle duality of the electron naturally explains why only certain orbits are allowed — only orbits where the de Broglie wave closes on itself are stable. Non-integer circumferences would lead to destructive interference → unstable orbit → not observed. Generalisation: this de Broglie approach is the precursor to Schrödinger's wave equation (full quantum mechanics). JEE Main Dual Nature: "for hydrogen n=1 orbit (r=0.529Å=0.0529nm), verify Bohr condition using de Broglie" → v=2.18×10⁶ m/s → λ=h/mv=6.626×10⁻³⁴/(9.11×10⁻³¹×2.18×10⁶)=3.33×10⁻¹⁰m=0.333nm. 2πr=2π×0.0529nm=0.332nm≈λ ✓ (one wavelength fits in n=1 orbit).
What is the work function and threshold wavelength in Dual Nature of Matter?
In Dual Nature of Matter, work function φ from Aakash PDF: minimum energy needed to eject an electron from the metal surface (overcome electrostatic attraction of positive ion cores). Unit: eV. φ=hf₀=hc/λ₀ where f₀=threshold frequency and λ₀=threshold wavelength. Threshold wavelength: λ₀=hc/φ. Using hc=1240 eV·nm: λ₀(nm)=1240/φ(eV). Metal work functions: Cs≈2.0eV → λ₀=620nm (red, sensitive to visible); K≈2.25eV → λ₀≈551nm; Na≈2.3eV → λ₀≈539nm; Ca≈3.2eV → λ₀≈388nm (UV); Zn≈4.3eV → λ₀≈288nm; Ag≈4.7eV → λ₀≈264nm; Pt≈5.6eV → λ₀≈221nm (deep UV). For f>f₀ (λ<λ₀): KE_max=hf–φ=hc/λ–hc/λ₀=hc(1/λ–1/λ₀). Also: V₀=(1/e)(hc/λ–φ)=(1/e)(1240/λ–φ) where λ in nm and φ in eV → V₀ in volts. JEE Main Dual Nature: "metal φ=3.1eV. Find threshold λ" → λ₀=1240/3.1=400nm (violet). "Find V₀ when λ=200nm" → E=1240/200=6.2eV → V₀=6.2–3.1=3.1V. "Metal 1 (φ=2eV) and metal 2 (φ=4eV). Light λ=300nm. Which emits? Find KE_max of emitting metal." → E=1240/300≈4.13eV. Metal 1: KE=4.13–2=2.13eV. Metal 2: KE=4.13–4=0.13eV. Both emit.
What are the graphs of the photoelectric effect in Dual Nature of Matter for JEE Main?
In Dual Nature of Matter, photoelectric graphs from Aakash PDF: (1) KE_max vs frequency f: straight line. Slope=h (Planck's constant). X-intercept=f₀ (threshold frequency, metal-specific). Y-intercept=–φ (negative of work function). For different metals: parallel lines (same slope h) with different x-intercepts (different f₀). (2) Stopping potential V₀ vs frequency f: straight line. Slope=h/e (same for all metals). X-intercept=f₀ (metal-specific). For different metals: parallel lines (same slope h/e, different x-intercepts). (3) Photoelectric current I vs applied voltage V: sigmoid shape. For negative V: I decreases; I=0 at V=–V₀ (stopping potential). For positive V: I increases then saturates at I_sat. For higher intensity (same f): I_sat increases; V₀ UNCHANGED. For higher frequency (same intensity): V₀ increases; I_sat UNCHANGED. (4) Saturation current I_sat vs intensity: straight line through origin (I_sat ∝ intensity). (5) KE_max vs intensity: horizontal straight line (KE_max is constant regardless of intensity). Most tested JEE Main graph: V₀ vs f graph — identify: (a) threshold frequency f₀ from x-intercept; (b) work function φ=hf₀; (c) Planck's constant h from slope=h/e multiplied by e. If two metals on same graph: same slope, different f₀ and V₀ intercepts.
What is the de Broglie wavelength for different particles at the same kinetic energy in Dual Nature?
In Dual Nature of Matter, de Broglie comparison from Aakash PDF: λ=h/√(2mKE). At same KE: λ∝1/√m. Heavier particle → smaller λ. Comparisons at same KE: Electron (m_e=9.11×10⁻³¹kg) vs proton (m_p=1.67×10⁻²⁷kg): λ_e/λ_p=√(m_p/m_e)=√(1836)≈42.8. Electron has ~43× larger de Broglie wavelength than proton at same KE. Electron vs alpha particle (m_α=4m_p=6.68×10⁻²⁷kg): λ_e/λ_α=√(m_α/m_e)=√(6.68×10⁻²⁷/9.11×10⁻³¹)=√7332≈85.6. At same momentum p: λ=h/p → same λ for all particles (momentum determines de Broglie wavelength, not mass alone). At same speed v: λ=h/mv → λ∝1/m. Same KE: all about mass. Same momentum: all same. Same speed: all about mass. Same accelerating voltage V (same charge q): λ=h/√(2mqV)∝1/√(m) for same q. Proton and deuteron (same charge, mass ratio 1:2): λ_p/λ_d=√2≈1.41 (proton has √2 larger λ). Real-world scale: baseball (mass=0.145kg, v=30m/s): λ=6.626×10⁻³⁴/(0.145×30)=1.52×10⁻³⁴m (unmeasurably small → no quantum effects). Electron (mass=9.11×10⁻³¹kg, v=10⁶m/s): λ=6.626×10⁻³⁴/(9.11×10⁻³¹×10⁶)=0.73nm (significant → quantum effects important).
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