Is it correct to say: Assuming photoelectric emission depends on light intensity?

Emission depends on *frequency* (must exceed \nu_0). Intensity affects only the *number* of ejected electrons, not their KE.

Is it correct to say: Using KE_max = h\nu (missing the work function \phi)?

Correct form: KE_max = h\nu − \phi. Otherwise, KE would be non-zero even for \nu < \nu_0, which contradicts experiments.

Is it correct to say: Writing energy of Bohr level as positive?

E_n = −13.6/n² eV. Sign is negative — bound state. Energy = 0 means the electron just escapes.

Is it correct to say: Confusing decay constant \lambda with wavelength \lambda?

Same symbol, different quantity. In decay: units of s⁻¹. In waves: units of m. Context decides.

Is it correct to say: Using T_{1/2} = 1/\lambda for half-life?

T_{1/2} = 0.693/\lambda (with ln 2). \tau = 1/\lambda is *mean* life. Different!

Is it correct to say: Forgetting 1 u = 931.5 MeV/c² in mass-defect problems?

Binding energy = \Delta m × 931.5 MeV when \Delta m is in u. Common source of factor-931 errors.

Dual Nature of Radiation and Matter / Atoms & Nuclei

Q: Is it correct to say: Writing energy of Bohr level as positive?

E_n = −13.6/n² eV. Sign is negative — bound state. Energy = 0 means the electron just escapes.

Q: Is it correct to say: Confusing decay constant \lambda with wavelength \lambda?

Same symbol, different quantity. In decay: units of s⁻¹. In waves: units of m. Context decides.

Q: Is it correct to say: Using T_{1/2} = 1/\lambda for half-life?

T_{1/2} = 0.693/\lambda (with ln 2). \tau = 1/\lambda is *mean* life. Different!

Q: Is it correct to say: Forgetting 1 u = 931.5 MeV/c² in mass-defect problems?

Binding energy = \Delta m × 931.5 MeV when \Delta m is in u. Common source of factor-931 errors.

Photoelectric effect, de Broglie wavelength, Bohr model of hydrogen, radioactive decay — NCERT Class 12 Physics Ch 11, 12, 13

📐 14 formulas✏️ 3 examples🎯 6 practice⚖️ 7 marks🏫 CBSE📚 Class 12✓ 2025–26 syllabus

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Board Exam Tips

→Einstein's photoelectric equation derivation + graph (stopping potential vs frequency) is a repeat 3-mark question every year.
→Bohr's postulates + derivation of E_n = −13.6/n² eV is a 5-mark HOTS favourite.
→de Broglie wavelength calculation for electron and comparison with photon of same energy — classic 2-mark question.
→Half-life and mean-life formulas — do NOT confuse T_{1/2} = 0.693/\lambda with \tau = 1/\lambda.
→Energy conversions: 1 eV = 1.6×10⁻¹⁹ J; hc ≈ 1240 eV·nm — memorise both.

📊 Diagram

Photon striking a metal surface ejecting a photoelectron

📐 Formulas(14)

Photon Energy★ Board fav

E = h ν = \frac{h c}{λ}

Symbol	Meaning
$h$	Planck's constant = 6.626×10⁻³⁴ J·s
$ν$	Frequency (Hz)
$λ$	Wavelength (m)
$c$	Speed of light = 3×10⁸ m/s

Einstein's Photoelectric Equation★ Board fav

K E_{ma x} = h ν - ϕ = h (ν - ν_{0})

Symbol	Meaning
$ϕ$	Work function of metal (J or eV)
$ν_{0}$	Threshold frequency (Hz)

Stopping Potential

e V_{0} = K E_{ma x} = h (ν - ν_{0})

de Broglie Wavelength★ Board fav

λ = \frac{h}{p} = \frac{h}{m v}

de Broglie Wavelength of Electron via Voltage

λ = \frac{h}{2 m e V} \approx \frac{1.227}{V} nm

Bohr's Quantisation of Angular Momentum★ Board fav

m v r = \frac{nh}{2 π}

Radius of nth Bohr Orbit

r_{n} = \frac{n ^{2} h ^{2} ε _{0}}{π m e ^{2}} = 0.529 n^{2} A ˚

Energy of nth Bohr Level★ Board fav

E_{n} = - \frac{13.6}{n ^{2}} eV

Photon Frequency for Transition

h ν = E_{n_{2}} - E_{n_{1}}

Rydberg Formula

\frac{1}{λ} = R (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})

Mass–Energy Equivalence

E = m c^{2}

Radioactive Decay Law

N = N_{0} e^{- λ t}

Half-Life and Mean Life

T_{1/2} = \frac{l n 2}{λ} = \frac{0.693}{λ}, τ = \frac{1}{λ}

Activity

A = λ N = A_{0} e^{- λ t}

✏️ Solved Examples

1Solved Exampleeasy3 steps

Light of wavelength 400 nm falls on a metal of work function 2.0 eV. Find (a) energy of one photon in eV, (b) maximum KE of photoelectrons, (c) stopping potential.

Photon energy using hc/\lambda (with hc ≈ 1240 eV·nm)

E = \frac{h c}{λ} = \frac{1240 eV\cdotnm}{400 nm} = 3.1 eV

2Solved Exampleboard4 steps

An electron in the hydrogen atom jumps from n = 3 to n = 2. Find (a) the energy of the emitted photon in eV, (b) its wavelength, (c) identify the spectral series.

Energy levels

E_{2} = - \frac{13.6}{4} = - 3.4 eV, E_{3} = - \frac{13.6}{9} \approx - 1.51 eV

3Solved ExampleHOTS5 steps

A radioactive sample has a half-life of 20 minutes. Initial activity is 8000 disintegrations/s. Find (a) the decay constant, (b) the activity after 1 hour, (c) the number of nuclei present initially.

Decay constant from half-life

λ = \frac{0.693}{T _{1/2}} = \frac{0.693}{20 \times 60} \approx 5.78 \times 1 0^{- 4} s^{- 1}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Assuming photoelectric emission depends on light intensity
✓Emission depends on *frequency* (must exceed \nu_0). Intensity affects only the *number* of ejected electrons, not their KE.
2
Using KE_max = h\nu (missing the work function \phi)
✓Correct form: KE_max = h\nu − \phi. Otherwise, KE would be non-zero even for \nu < \nu_0, which contradicts experiments.
3
Writing energy of Bohr level as positive
✓E_n = −13.6/n² eV. Sign is negative — bound state. Energy = 0 means the electron just escapes.
4
Confusing decay constant \lambda with wavelength \lambda
✓Same symbol, different quantity. In decay: units of s⁻¹. In waves: units of m. Context decides.
5
Using T_{1/2} = 1/\lambda for half-life
✓T_{1/2} = 0.693/\lambda (with ln 2). \tau = 1/\lambda is *mean* life. Different!
6
Forgetting 1 u = 931.5 MeV/c² in mass-defect problems
✓Binding energy = \Delta m × 931.5 MeV when \Delta m is in u. Common source of factor-931 errors.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Threshold frequency of a metal is 5×10¹⁴ Hz. Light of 8×10¹⁴ Hz falls on it. Stopping potential?
Q2
de Broglie wavelength of an electron accelerated through 100 V?
Q3
Wavelength of the H-\alpha line (Balmer, n=3→2)?
Q4
Energy required to ionise a hydrogen atom from ground state?
Q5
A radioactive sample decays to 1/8 of its initial value in 30 minutes. Half-life?
Q6
Mass defect of a nucleus is 0.05 u. Binding energy in MeV?

📝 Notes

Dual Nature, Atoms and Nuclei — key concepts

Three chapters combined: light behaves as photons, matter behaves as waves, atoms have discrete energy levels, and unstable nuclei decay exponentially.

The photon picture

Light is quantised into photons of energy h\nu and momentum h/\lambda.
Photoelectric emission is instantaneous, has a threshold frequency, and stopping potential depends only on \nu — three facts that classical wave theory cannot explain.
Einstein's equation KE_max = h\nu − \phi encapsulates all three.

The matter wave picture

de Broglie: every particle of momentum p has a wavelength \lambda = h/p.
For electrons, \lambda ≈ 1.227/√V nm (V in volts). Confirmed by Davisson–Germer diffraction from a nickel crystal.
Wave nature is significant only when \lambda is comparable to the system size (electrons: yes; cricket ball: no).

Bohr's atomic model — three postulates

Electrons orbit only in certain "allowed" orbits without radiating.
Angular momentum is quantised: mvr = nh/(2\pi).
Radiation is emitted or absorbed only when the electron jumps between orbits: h\nu = E_2 − E_1.

From these, r_n ∝ n² and E_n = −13.6/n² eV for hydrogen.

Spectral series of hydrogen

Lyman (UV): transitions to n = 1.
Balmer (visible): transitions to n = 2. Includes red H-\alpha at 656 nm.
Paschen, Brackett, Pfund (IR): to n = 3, 4, 5.

Radioactivity

N = N_0 e^{−\lambda t}, A = \lambda N.
Half-life T_{1/2} = 0.693/\lambda; mean life \tau = 1/\lambda.
After n half-lives, fraction surviving = 1/2ⁿ.

Quick sanity checks

Photon of visible light (\lambda ≈ 500 nm) has energy ≈ 2.5 eV — comparable to work functions of common metals.
If stopping potential is not linear in \nu, something is wrong with the experiment.
E_n gets less negative as n increases; ionisation from ground state needs 13.6 eV.
Activity is always positive; N always decreases with time. Any answer violating this is wrong.
Units: 1 eV = 1.6×10⁻¹⁹ J; 1 u = 1.66×10⁻²⁷ kg ≈ 931.5 MeV/c².