Master Formula Sheet

# Result Formula
1 Electron volt 1 eV = 1.602×10−191.602 \times 10^{-19} J
2 Work function ϕ0\phi_0 = min energy to escape metal surface; unit eV
3 Photon energy E=hν=hcλE = h\nu = \dfrac{hc}{\lambda}; shortcut E(eV)=1240λ(nm)E(\text{eV}) = \dfrac{1240}{\lambda(\text{nm})}
4 Photon momentum p=hνc=hλ=Ecp = \dfrac{h\nu}{c} = \dfrac{h}{\lambda} = \dfrac{E}{c}
5 Photons per second N=PhνN = \dfrac{P}{h\nu}
6 Stopping potential Kmax=eV0K_{max} = eV_0
7 Einstein's equation Kmax=hν−ϕ0=h(ν−ν0)K_{max} = h\nu - \phi_0 = h(\nu - \nu_0)
8 Threshold ν0=ϕ0h\nu_0 = \dfrac{\phi_0}{h}; λ0=hcϕ0\lambda_0 = \dfrac{hc}{\phi_0}
9 V0V_0-ν\nu line V0=heν−ϕ0eV_0 = \dfrac{h}{e}\nu - \dfrac{\phi_0}{e}; slope he\dfrac{h}{e} (universal), x-intercept ν0\nu_0
10 Max speed vmax=2Kmaxmv_{max} = \sqrt{\dfrac{2K_{max}}{m}}
11 de Broglie λ=hp=hmv=h2mK\lambda = \dfrac{h}{p} = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2mK}}
12 Electron through V volts λ=h2meV≈1.227V\lambda = \dfrac{h}{\sqrt{2meV}} \approx \dfrac{1.227}{\sqrt{V}} nm
13 X-ray limit (inverse PE) hνmax=eVh\nu_{max} = eV; λmin=1240V(volts)\lambda_{min} = \dfrac{1240}{V(\text{volts})} nm

Constants to lock in: h = 6.63×10−346.63 \times 10^{-34} J s; e = 1.6×10−191.6 \times 10^{-19} C; me=9.1×10−31m_e = 9.1 \times 10^{-31} kg; e/m = 1.76×10111.76 \times 10^{11} C/kg; hc ≈ 1240 eV nm.

The Two Dials, and Wave vs Photon

The intensity dial vs the frequency dial — the chapter's most-tested distinction:

Change Saturation current Stopping potential / KmaxK_{max}
Intensity ↑ (fixed ν\nu) increases ∝ intensity unchanged
Frequency ↑ (fixed intensity) ~unchanged increases linearly
Below threshold (ν<ν0\nu < \nu_0) zero at ANY intensity —
Change metal (ϕ0\phi_0 ↑) — decreases; threshold rises

Wave theory vs photon picture:

Observation Wave theory says Photon picture says
KmaxK_{max} vs intensity should increase independent ✔ (one photon per electron)
Threshold frequency should not exist ν0=ϕ0/h\nu_0 = \phi_0/h ✔
Time lag (dim light) hours instantaneous (~10−910^{-9} s) ✔ (single-event absorption)
Photocurrent ∝ intensity explained explained ✔ (more quanta, more electrons)

Photon property list (NCERT's five): particle-like in interactions; E = hνh\nu, p = hν/ch\nu/c, speed c; per-photon values fixed by frequency, never intensity; electrically neutral, undeflected by E and B fields; in collisions energy and momentum conserved, photon NUMBER need not be. Rest mass zero.

The discovery cast: Crookes (cathode rays 1870) → Roentgen (X-rays 1895) → Thomson (e/m, electron 1897; Nobel 1906) → Millikan (oil drop 1913, quantisation; photoelectric verification 1916; Nobel 1923) → Hertz (photoemission 1887) → Hallwachs & Lenard (systematics 1886-1902) → Einstein (quanta 1905; Nobel 1921) → Compton (photon momentum 1924) → de Broglie (matter waves 1924; Nobel 1929) → Davisson-Germer (electron diffraction 1927 — JEE/NEET extra).

de Broglie Toolkit & Comparison Rules

The relation: λ=h/p\lambda = h/p — wave attribute left, particle attribute right, h bridging them. Holds for photons too (h/p=c/ν=λh/p = c/\nu = \lambda). Independent of the particle's charge and nature; significant only for sub-atomic masses (electron at 5.4×1065.4 \times 10^6 m/s: 0.135 nm; a 0.12 kg ball at 20 m/s: 2.76×10−342.76 \times 10^{-34} m — hopeless).

Comparison rules (electron vs proton vs alpha):

  • Equal speed: λ∝1m\lambda \propto \dfrac{1}{m} → electron longest by factor ~1836 over proton.
  • Equal kinetic energy: λ∝1m\lambda \propto \dfrac{1}{\sqrt{m}} → λe/λp=1836≈43\lambda_e/\lambda_p = \sqrt{1836} \approx 43.
  • Equal momentum: all equal — photon included.
  • Same accelerating potential: λ∝1mq\lambda \propto \dfrac{1}{\sqrt{mq}} → λe>λp>λd>λα\lambda_e > \lambda_p > \lambda_d > \lambda_\alpha (λp/λα=22\lambda_p/\lambda_\alpha = 2\sqrt{2}).

Scaling snaps: V → 4V halves λ\lambda; K → 4K halves λ\lambda; p → 2p halves λ\lambda. To halve λ\lambda, quadruple V.

Matter-wave fine print (Points to Ponder): phase velocity of a matter wave has no physical significance; the group velocity equals the particle velocity. Work function is the LEAST escape energy (electrons have an energy distribution — higher-energy electrons need less help). Absorption in units of hνh\nu is not quite the same claim as 'light is particles' — the stopping-potential observations are the crucial wave/photon discriminator.

[JEE/NEET Extra] Davisson-Germer pegs: nickel crystal, 54 V, peak at 50 degrees, λ\lambda = 0.165 nm vs predicted 0.167 nm, year 1927. Application: electron microscope (resolving power from tiny λ\lambda).

One-Glance Revision Flow

The chapter in eight steps:

  1. Electron discovered: cathode rays (Crookes), crossed-fields e/m = 1.76×10111.76 \times 10^{11} C/kg (Thomson), charge quantised at 1.602×10−191.602 \times 10^{-19} C (Millikan) → mass 9.1×10−319.1 \times 10^{-31} kg.
  2. Work function: minimum escape energy ϕ0\phi_0 (eV); delivered thermally, by ~10810^8 V/m fields, or by light.
  3. Photoemission discovered: Hertz's sparks; Hallwachs' charged zinc plates; Lenard's tube — negative particles, current tracks light.
  4. Four laws: current ∝ intensity; V0V_0 independent of intensity; V0V_0 linear in ν\nu above a material-specific ν0\nu_0; emission instantaneous.
  5. Wave theory fails on 2, 3, 4 — energy spread too thin, no threshold, hour-long delays predicted.
  6. Einstein: Kmax=hν−ϕ0K_{max} = h\nu - \phi_0; slope h/e universal; Millikan's reluctant proof.
  7. Photon: E = hνh\nu, p = h/λh/\lambda, c, neutral, number non-conserved; Compton clinches.
  8. de Broglie: λ=h/p\lambda = h/p for ALL matter; 1.227/V\sqrt{V} nm for electrons; measurable only sub-atomically.

Morning-of-exam checklist: eV ↔ J via 1.6 × 10⁻¹⁹ … 1240/λ\lambda(nm) = E(eV) … intensity moves CURRENT, frequency moves V0V_0 … below ν0\nu_0: zero, at any brightness … V0V_0-ν\nu slope h/e, lines parallel across metals, never through origin … Kmax=eV0K_{max} = eV_0 (volts read as eV) … photon p = E/c; mirror force 2P/c … λ=1.227/V\lambda = 1.227/\sqrt{V} nm … equal-p → equal-λ\lambda; equal-K → 1/m1/\sqrt{m}; equal-v → 1/m1/m; same V → 1/mq1/\sqrt{mq} … phase velocity meaningless, group velocity = particle velocity. Go score.