Dual Nature of Radiation and Matter

2.1 Core concepts

By the end of the nineteenth century, electrons had been identified as constituents of all matter, with a universal charge-to-mass ratio independent of the parent material (NCERT §11.1, p. 274). The next question is: how do these electrons get out of a metal? Inside the metal the free electrons drift around freely, but at the surface they are pulled back by the residual positive-ion attraction; an electron trying to escape has to do work against this attraction. The minimum energy needed to do so is the work function φ₀, measured in electron-volts (1 eV = 1.602 × 10⁻¹⁹ J). The work function depends both on the metal and on the state of its surface — even small contamination changes φ₀ (NCERT §11.2, p. 275). Typical values: caesium 2.14 eV, sodium 2.75 eV, copper 4.7 eV, platinum 5.65 eV.

There are three principal ways the required energy can be supplied to free an electron from a metal surface (NCERT §11.2, p. 275–276): thermionic emission, where the metal is heated so that some electrons acquire enough thermal energy (used in vacuum tubes); field emission, where a very strong external electric field of the order of 10⁸ V m⁻¹ is applied to literally pull electrons out of the surface (this is what happens at the tip of a spark plug); and photoelectric emission, where electromagnetic radiation of suitable frequency is absorbed and the energy is delivered to an electron, which then escapes. The third is the focus here because it shaped quantum theory.

Discovery (NCERT §11.3, p. 276). Heinrich Hertz, while studying his spark-gap apparatus in 1887, noticed that the high-voltage sparks at the detector loop were enhanced when the emitter plate was illuminated with ultraviolet light — the first observation of photoelectric emission. In 1888 Wilhelm Hallwachs showed that a freshly polished zinc plate, negatively charged, lost its charge under UV; an uncharged plate became positively charged; and a positively charged plate gained more positive charge — proving the ejected particles were negatively charged (electrons). Philipp Lenard (1862–1947) then built the modern photoelectric experimental cell: an evacuated glass/quartz tube containing two metal plates, with monochromatic UV falling on the emitter. Current flowed instantly when the light was switched on and stopped instantly when it was switched off; current was observed only above a certain threshold frequency ν₀ characteristic of the emitter material.

Experimental study (NCERT §11.4, p. 277). The standard cell uses an evacuated glass/quartz tube with photosensitive emitter C and collector A; an external battery and commutator set the C-to-A potential difference V (either accelerating or retarding), a voltmeter reads V, and a sensitive microammeter measures the photocurrent. Monochromatic light enters through a quartz window W (quartz transmits UV; ordinary glass does not). The experiment maps out three dependencies.

(a) Effect of intensity (§11.4.1, p. 278). Keep frequency ν above threshold and the accelerating potential fixed, vary only the intensity I of incident light. The photocurrent rises linearly with I — the photoelectric current is directly proportional to intensity. Physically, more intense light delivers more energy per second to the surface, ejecting more photoelectrons per second.

(b) Effect of collector potential (§11.4.2, p. 278–279). At fixed frequency and intensity, increase the positive collector potential. The current rises and then saturates at a maximum value — the saturation current, reached when every photoelectron is collected. If instead the collector is made negative (retarding), the current falls because the field now decelerates electrons. At a sharply defined value of the negative potential — the stopping potential V₀ — the current drops to zero: only those electrons with kinetic energy ≥ eV₀ can reach the collector, so the maximum kinetic energy of photoelectrons is eV₀. Repeating with brighter light gives a higher saturation current but the same stopping potential — V₀ is independent of intensity.

(c) Effect of frequency (§11.4.3, p. 279–280). Keep intensity fixed, vary frequency above threshold. The saturation current is essentially unchanged for the same intensity, but the stopping potential V₀ becomes more and more negative as ν is increased — i.e., Kmax = eV₀ grows linearly with ν. Plotting V₀ versus ν gives a straight line of slope h/e with a positive x-intercept at ν = ν₀, the threshold frequency. Below ν₀ no photoemission occurs, no matter how intense or how prolonged the illumination.

This experimental package yields the four laws of photoelectric emission (NCERT §11.4.3, p. 280): (i) photocurrent is proportional to intensity (above threshold); (ii) saturation current is proportional to intensity, but stopping potential is independent of intensity; (iii) a threshold frequency ν₀ exists for each metal, below which no emission occurs, and above ν₀ the maximum kinetic energy of photoelectrons varies linearly with ν; (iv) photoemission is essentially instantaneous — within ~10⁻⁹ s of switching the light on, even at intensities so low that classical theory predicts hours-long energy build-up.

Why classical wave theory fails (NCERT §11.5, p. 280–281). In the wave picture, light delivers energy continuously over the entire wavefront. Intensity → energy per second per unit area, so a brighter beam should give photoelectrons of greater Kmax; with very dim light the time needed to accumulate enough energy at a single electron should be measurable in minutes or hours; and any frequency, given enough time, should eventually liberate electrons — there should be no threshold frequency. Every one of these predictions is contradicted by experiment.

Einstein's photon hypothesis (NCERT §11.6, p. 281–283). In 1905 Einstein proposed that light of frequency ν is exchanged with matter in discrete energy packets — photons or light quanta — each of energy hν. Photoelectric emission is then a one-to-one process: a single photon is absorbed by a single electron, gives up all its energy hν, of which φ₀ is spent in liberating the electron from the surface and the remainder appears as kinetic energy:

Kmax = hν − φ₀ = eV₀ (Einstein's photoelectric equation, Eq. 11.2, p. 281).

This single equation accounts for every experimental fact. The linear Kmax–ν relation is built in; the threshold frequency is ν₀ = φ₀/h (below which Kmax would be negative, i.e., no emission); the saturation current is proportional to intensity because intensity now means number of photons per second; and emission is instantaneous because each absorption event is itself an instantaneous process. Combining Kmax = eV₀ with the photoelectric equation gives V₀ = (h/e) ν − φ₀/e, a straight line of slope h/e independent of the metal (NCERT Eq. 11.4, p. 282). Millikan (1906–1916), who initially disbelieved the photon picture, painstakingly verified this prediction for several alkali metals and used the slope to measure Planck's constant as h ≈ 6.63 × 10⁻³⁴ J s — bringing him a Nobel Prize and Einstein the 1921 Physics Nobel for the photoelectric law.

The photon (NCERT §11.7, p. 283). Photons have energy E = hν, momentum p = hν/c = h/λ, speed c, are electrically neutral and are not deflected by electric or magnetic fields. Their rest mass is zero (they exist only at speed c). In a photon–particle collision, total energy and total momentum are conserved, but the number of photons need not be — a photon may be absorbed, emitted or created. An intense light beam consists of many photons travelling together; "intensity" is energy per second per unit area = (number of photons per second per unit area) × hν.

de Broglie hypothesis (NCERT §11.8, p. 285). In 1924 Louis de Broglie argued for the converse of Einstein's idea: if light (wave) has particle aspects, then matter (particle) should have wave aspects. He proposed that to every particle of momentum p is associated a wave of wavelength λ = h/p = h/(mv) — the de Broglie wavelength. The relation reduces correctly to a photon's λ = c/ν, so it is consistent with what is already known about light. For macroscopic objects the wavelength is unimaginably tiny — a 0.12-kg cricket ball at 20 m/s has λ ≈ 2.76 × 10⁻³⁴ m, far below any conceivable experimental resolution. But for an electron the same momentum gives λ comparable to the spacings between atomic planes in a crystal, which is what made Davisson–Germer's 1927 electron-diffraction experiment possible.

Accelerated electron (NCERT §11.8 and Summary point 9, p. 285–287). A common laboratory situation is an electron starting from rest and accelerated through potential difference V volts. Its kinetic energy is eV = p²/(2m), so p = √(2meV). Substituting into λ = h/p gives λ = h/√(2meV). Plugging in numerical values (h = 6.63 × 10⁻³⁴ J s, m = 9.11 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C) yields the workhorse formula λ = 12.27/√V Å when V is in volts and λ in ångströms. A 100-V electron has λ ≈ 1.23 Å — comparable to inter-atomic distances, hence its diffraction by crystals.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 11.1 (p. 277): Experimental arrangement for photoelectric effect — evacuated tube, monochromatic source S, quartz window W (UV-transparent), emitter C, collector A, commutator to reverse polarity, voltmeter V, microammeter mA. The whole cell is in series with a variable battery.
• Fig. 11.2 (p. 278): Photoelectric current vs intensity of incident light — straight line through the origin, illustrating I_photo ∝ I_light.
• Fig. 11.3 (p. 278): Photocurrent vs collector plate potential for three intensities I₁ < I₂ < I₃ at fixed frequency — three curves saturate at progressively higher levels but all hit zero current at the same stopping potential V₀ on the negative-V side.
• Fig. 11.4 (p. 279): Photocurrent vs collector potential for three frequencies ν₁ < ν₂ < ν₃ at fixed intensity — same saturation current but the stopping potentials shift V₀₁, V₀₂, V₀₃ further into the negative-V region with increasing ν.
• Fig. 11.5 (p. 279): Stopping potential V₀ vs frequency ν — straight line of slope h/e with x-intercept ν₀ (threshold). Two metals A and B give parallel lines (same slope) but different x-intercepts (different φ₀).
• The qualitative picture of one-photon-one-electron absorption inside the metal: incoming photon hν → liberates one electron at cost φ₀ → leftover energy hν − φ₀ appears as Kmax.

2.4 Common confusions / NTA trap points

• Stopping potential vs work function: V₀ depends on frequency and metal but is independent of intensity; φ₀ depends only on the metal. Students often write V₀ ∝ intensity — wrong.
• Saturation current vs Kmax: saturation current depends on intensity (number of photoelectrons per second) but NOT on frequency; Kmax depends on frequency but NOT on intensity. NTA likes to swap these.
• Threshold condition: below ν₀, no emission whatsoever; "very intense" or "long exposure" does not help — a direct contradiction of wave theory that examiners exploit.
• Photon momentum: p = hν/c = h/λ. Beware the wrong "p = mv" form — photons have zero rest mass. Photon energy E = hν and E = pc are the correct relations.
• de Broglie for accelerated electron: the working formula is λ = 12.27/√V Å only when V is in volts and λ is in ångströms; if V is in kilovolts or λ asked in nm, do not blindly apply the constant.
• Slope of V₀–ν line: the slope is h/e and is the same for all metals; only the x-intercept ν₀ changes from metal to metal.
• Photon number conservation: in collisions, energy and momentum are conserved but photon number is not.
• Quartz vs glass: the photoelectric cell uses a quartz window because ordinary glass blocks UV. A common conceptual trap.
• Instantaneous emission: the wave theory's predicted "build-up time" is contradicted; emission occurs within ~10⁻⁹ s.
• Field-emission threshold: 10⁸ V m⁻¹ is the textbook signature; lower fields do not eject electrons by this mechanism.
• Negative Kmax is unphysical: if hν < φ₀, the photoelectric equation would give Kmax < 0 — instead, no emission occurs.
• Wavelength of a macroscopic object: for cricket balls etc. λ is so tiny it cannot be measured; matter-wave effects matter only for subatomic particles.

2.5 Key formulas table