Class 12 Physics · NCERT chapter 11 · 6 topics · free to read
Photoelectric Effect
When light of sufficient frequency hits a metal surface, electrons are ejected instantly. This phenomenon — which classical wave theory could NOT explain — became the first direct evidence that light carries energy in discrete packets (photons). Einstein won his Nobel Prize for this, not relativity!
KE_ = h -
Hertz & Lenard Experiments
Hertz accidentally discovered that UV light helps spark discharges. Lenard then systematically studied the ejected electrons and found results that shattered classical physics.
In Lenard's experiment, light falls on a metal plate (emitter) inside an evacuated tube. Ejected electrons travel to the collector plate, creating a measurable photocurrent. By applying a reverse voltage (retarding potential), the maximum kinetic energy of the electrons can be measured.
Hertz (1887) — discovered that UV light assists spark discharge across metal electrodes
Lenard (1900) — observed that ejected electrons (photoelectrons) have kinetic energy that depends on frequency, not intensity
Hallwachs (1888) — showed that a negatively charged zinc plate loses charge when exposed to UV light
The emitter is called the photosensitive plate or cathode; the collector is the anode
Laws of Photoelectric Emission
Five key experimental observations define the photoelectric effect. Classical wave theory could explain NONE of them properly — this forced physicists to accept the particle nature of light.
Law 1: For a given metal, photoelectric emission occurs ONLY if the frequency _0 (threshold frequency). Below _0 , no emission — no matter how intense the light
Law 2: The emission is instantaneous — no time lag (within 10^ -9 s), even with very dim light above _0
Law 3: The number of photoelectrons emitted per second (photocurrent) is proportional to the intensity of light
Law 4: The maximum kinetic energy of photoelectrons depends on frequency, NOT on intensity
Law 5: The stopping potential V_0 is independent of intensity but increases linearly with frequency
Higher intensity → more KE
Higher intensity → more electrons, same KE
Any frequency should work (given enough time)
Below threshold _0 : zero emission, period
Time lag expected for dim light to accumulate energy
Emission is instantaneous (no lag)
KE depends on intensity
KE depends only on frequency
Stopping Potential
By applying a negative voltage to the collector, we can measure the maximum kinetic energy of ejected electrons. The voltage that reduces the photocurrent to zero is the stopping potential V₀.
When a retarding (negative) potential is applied to the collector, photoelectrons are slowed down. The stopping potential V_0 is the minimum retarding potential that stops even the most energetic photoelectrons from reaching the collector — reducing photocurrent to zero.
KE_ = eV_0
1 2 mv_ ^2 = eV_0
V_0 is independent of intensity — doubling light intensity does NOT change V_0
V_0 increases linearly with frequency above _0
At V_0 , photocurrent = 0 (even the fastest electrons are turned back)
At zero applied voltage, photocurrent is NOT maximum — some electrons miss the collector
Saturation current occurs at high positive voltage when ALL emitted electrons are collected
Einstein's Photoelectric Equation
Einstein's brilliant insight: one photon transfers ALL its energy hv to ONE electron. Part of that energy is spent escaping the metal (work function phi), and the rest becomes kinetic energy. This single equation explained every puzzling observation and won him the 1921 Nobel Prize.
KE_ = h -
Einstein's Equation
Einstein applied energy conservation to the photon-electron interaction: photon energy = work function + maximum kinetic energy. This beautifully explained all five laws of photoelectric emission.
KE_ = h -
Here h is the energy of the incident photon, is the minimum energy needed to free an electron from the metal surface (work function), and KE_ is the maximum kinetic energy of the emitted photoelectron.
h = Planck's constant = 6.63 10^ -34 J·s
= work function = minimum energy to extract an electron from the surface
KE_ = h - — this is the energy conservation equation
One photon interacts with one electron — no energy sharing between multiple electrons
If h < : no emission (photon doesn't have enough energy)
If h = : electron just barely escapes with zero KE
Stopping Potential Form
Since KE_max = eV₀, Einstein's equation can be rewritten in terms of stopping potential — the form most useful for graph-based and numerical NEET problems.
Since the maximum kinetic energy equals the work done against the stopping potential:
eV_0 = h -
V_0 = h e - e
This is a straight line equation of the form y = mx + c where V_0 is plotted against . The slope = h/e is the same for ALL metals (universal constant). The y-intercept = - /e varies from metal to metal.
Slope of V_0 vs graph = h/e — same for all metals (universal)
x-intercept = threshold frequency _0 (where V_0 = 0 )
y-intercept = - /e — different for different metals
All metals give parallel lines (same slope) shifted vertically
From the graph: = h _0 and slope gives Planck's constant h = e slope
Quick recall
Two metals A and B have work functions 2 eV and 4 eV. How do their V_0 vs graphs compare?
Both are straight lines with the SAME slope ( h/e ). Metal A's line is shifted LEFT (lower _0 ) and UP (less negative y-intercept). Metal B needs higher frequency to start emission.
How Einstein's Theory Explains the Laws
Each experimental observation that classical theory failed to explain falls naturally out of Einstein's photon model.
Let's see how the photon model explains each law of photoelectric emission:
Threshold frequency: Each photon must carry energy . Since E = h , we need _0 = /h . Below this, no single photon has enough energy → no emission
Instantaneous emission: One photon transfers all energy to one electron in a single collision — no time needed to 'accumulate' energy. Response time 10^ -9 s
Intensity ∝ photocurrent: Higher intensity = more photons per second = more electrons ejected per second = larger photocurrent. But each photon still carries the same energy h
KE independent of intensity: KE_ = h - has no intensity term. More photons don't make individual electrons faster — they just eject more electrons
KE increases with frequency: KE_ = h - is linear in . Higher frequency photons carry more energy, so electrons escape with more KE
Work Function & Threshold
The work function phi is the minimum energy an electron needs to escape a metal surface — like an exit toll. Each metal has its own value. The threshold frequency v_0 = phi/h is the minimum light frequency that can pay this toll. Below it, nothing happens — regardless of intensity.
= h _0 = hc _0
Work Function
Work function is the minimum energy required to remove the most loosely bound electron from a metal surface. It depends on the metal's atomic structure and surface conditions.
The work function of a metal is the minimum energy needed to extract the most loosely bound electron from the metal surface. It is a characteristic property of the metal — determined by its electronic structure and surface conditions.
Symbol: (or sometimes W_0 ). Unit: electron volt (eV) or joule (J)
Conversion: 1 eV = 1.6 10^ -19 J
Typical values: Cs → 2.14 eV, Na → 2.75 eV, Cu → 4.65 eV, Pt → 5.65 eV
Low metals (alkali metals) are most photosensitive — they emit electrons even with visible light
High metals need UV light for photoelectric emission
depends on the metal AND its surface condition (oxidation, roughness)
Cs (2.14 eV), K (2.30 eV), Na (2.75 eV)
Cu (4.65 eV), Ag (4.70 eV), Pt (5.65 eV)
Emit with visible light
Need UV light
Used in photocells, phototubes
Rarely used in photoelectric devices
Alkali metals (loosely held valence e⁻)
Transition/noble metals (tightly held e⁻)
Threshold Frequency & Wavelength
The threshold frequency v₀ is the minimum frequency of light needed to just barely eject an electron (with zero KE). The corresponding threshold wavelength lambda₀ is the maximum wavelength that works.
_0 = h _0 = hc = c _0
At threshold frequency _0 , the photon energy h _0 exactly equals the work function . The electron absorbs this energy, uses all of it to escape, and comes out with zero kinetic energy. For light with < _0 (or > _0 ), no emission occurs.
_0 → emission occurs; < _0 → no emission regardless of intensity
_0 → emission occurs; > _0 → no emission
At = _0 : KE_ = 0 , electrons just barely escape
Threshold frequency depends ONLY on the metal (its work function), not on the light source
Metal X has a threshold wavelength of 400 nm. Will a 500 nm light cause photoelectric emission?
No. 500 nm > 400 nm = _0 . Since > _0 , no emission. (Higher wavelength = lower frequency = less energy per photon.) You'd need 400 nm.
Photoelectric Graphs
Three critical graphs define the photoelectric effect: I vs V (for different intensities), I vs V (for different frequencies), and V₀ vs v. Master these and you can answer 80% of graph-based questions.
There are three key graphs you must know cold:
Graph 1: I vs V (different intensities, same ) — Same stopping potential V_0 , different saturation currents. Higher intensity → higher saturation current → curve shifts UP, but crosses x-axis at same -V_0
Graph 2: I vs V (different frequencies, same intensity) — Different stopping potentials. Higher → larger V_0 → curve extends further LEFT. Saturation currents are approximately equal
Graph 3: V_0 vs (for different metals) — Straight lines with slope h/e (same for ALL metals). Different x-intercepts ( _0 ) and y-intercepts ( - /e ). Lines are PARALLEL
In Graph 3 ( V_0 vs ): extrapolating the line to V_0 = 0 gives _0 on the x-axis
The slope h/e = 4.14 10^ -15 V·s — this is how Millikan experimentally determined h
KE_ vs graph has slope h (not h/e ) — different from V_0 vs !
The V_0 vs graph does NOT pass through the origin — it has a negative y-intercept
Millikan's experiment (1916) used the V_0 vs graph to measure Planck's constant h with high precision. Ironically, Millikan set out to DISPROVE Einstein's theory but ended up confirming it — and both won Nobel Prizes for this work.
Photon Theory
Light is not a continuous wave — it travels as discrete packets of energy called photons. Each photon carries energy E = hv and momentum p = h/lambda. Photons have zero rest mass, always travel at speed c, and are neither created nor destroyed — only emitted or absorbed.
E = h = hc
Properties of a Photon
A photon is a quantum of electromagnetic radiation. It has energy, momentum, and zero rest mass. Understanding photon properties is essential for both the photoelectric effect and de Broglie's hypothesis.
Einstein extended Planck's quantum hypothesis to light itself. Light of frequency consists of a stream of photons, each carrying a discrete packet (quantum) of energy.
E = h = hc
Energy of one photon: E = h = hc/
Momentum of one photon: p = h/ = E/c = h /c
Rest mass of a photon: zero ( m_0 = 0 ). Photons exist only in motion at speed c
Dynamic mass (relativistic): m = E/c^2 = h /c^2 = h/( c) — this is NOT rest mass
Photons are electrically neutral — they are not deflected by electric or magnetic fields
Photons travel at c = 3 10^8 m/s in vacuum — regardless of frequency or source motion
In a photon-particle collision, total energy and momentum are conserved
Intensity & Photon Flux
Intensity of light equals energy delivered per unit area per unit time. In the photon picture, it equals the number of photons hitting unit area per second times the energy per photon.
I = nE At = n h At = N h
where n is the total number of photons hitting area A in time t , and N = n/(At) is the photon flux (photons per unit area per unit time). The total power of a beam is P = n t h , so the number of photons emitted per second = P/(h ) .
Number of photons per second emitted by source of power P : n/t = P/(h ) = P /(hc)
Doubling intensity (at same ) → doubles the number of photons/second → doubles photocurrent
Doubling frequency (at same intensity) → halves the number of photons/second (each photon carries double energy) → photocurrent decreases!
This is why intensity ∝ photocurrent, but KE depends only on frequency
Quick recall
Two light beams have the same intensity. Beam A has frequency , Beam B has frequency 2 . Which has more photons per second?
Beam A. Same intensity means same energy per unit area per second. But each photon of B carries twice the energy ( h 2 ), so B needs half as many photons to deliver the same total energy. Beam A has twice the photon count.
Photon Energy Across the Spectrum
Photon energy increases with frequency. Radio photons carry negligible energy; gamma-ray photons carry enough to break atomic nuclei. This explains why different EM waves have such different effects.
Since E = h = hc/ , photon energy increases as we move from radio waves to gamma rays across the electromagnetic spectrum.
Radio waves ( 10^6 Hz): E 10^ -28 J 10^ -9 eV — negligible per photon
Visible light ( 5 10^ 14 Hz): E 2 eV — enough for photoelectric effect in alkali metals
UV light ( 10^ 16 Hz): E 10 eV — enough for photoelectric effect in most metals
X-rays ( 10^ 18 Hz): E 10^4 eV — can ionize atoms, penetrate tissue
Gamma rays ( 10^ 20 Hz): E 10^6 eV — can break nuclear bonds
Solar cells work on the photoelectric effect. Silicon's band gap is 1.1 eV, so visible light photons (1.8–3.1 eV) can liberate electrons in silicon. This is how sunlight becomes electricity — one photon at a time.
de Broglie Hypothesis
If light (a wave) can behave like a particle, can particles behave like waves? de Broglie said YES. Every moving particle has an associated wavelength lambda = h/p. This radical idea was confirmed when electrons were diffracted — just like light waves — in the Davisson-Germer experiment.
= h p = h mv
de Broglie Wavelength
de Broglie proposed that a particle of mass m moving with velocity v has an associated 'matter wave' with wavelength lambda = h/(mv). This wave-particle duality applies to ALL matter, not just light.
= h p = h mv
de Broglie (1924) extended wave-particle duality to ALL matter. Just as a photon of momentum p has wavelength = h/p , a material particle of momentum p = mv has an associated matter wave (or de Broglie wave) with the same relation.
= h/p = h/(mv) — applies to ALL particles (electrons, protons, neutrons, atoms, even baseballs)
Heavier or faster particles → smaller → harder to observe wave behavior
For macroscopic objects: 10^ -34 m — far too small to detect (no observable diffraction)
For electrons: 10^ -10 m — comparable to atomic spacing → diffraction is observable!
de Broglie waves are NOT electromagnetic waves — they are probability waves (quantum mechanical)
The hypothesis was purely theoretical in 1924 — confirmed experimentally by Davisson & Germer in 1927
Davisson-Germer Experiment
Davisson and Germer shot electrons at a nickel crystal and observed a diffraction pattern — exactly as predicted by de Broglie. This was the first experimental proof that matter has wave properties.
In 1927, Davisson and Germer accelerated electrons through a potential V and directed them at a nickel crystal. They measured the intensity of scattered electrons at various angles. At certain angles, they found strong peaks — diffraction maxima — exactly matching the pattern expected from waves of wavelength = h/p .
Electrons accelerated through 54 V → = 1.226/ 54 0.167 nm
Strong diffraction peak observed at scattering angle = 50°
Using Bragg's law: the measured matched de Broglie's prediction within experimental error
This confirmed that electrons have wave nature — matter waves are real
The nickel crystal acts as a 3D diffraction grating (atomic planes reflect electron waves)
Wave-Particle Duality
Every quantum object exhibits both wave and particle behavior, but never both simultaneously. Light shows particle nature in the photoelectric effect and wave nature in interference. Electrons show wave nature in diffraction and particle nature in CRT screens.
Wave-particle duality is the fundamental concept of quantum mechanics: every entity in nature has BOTH wave and particle aspects. Which aspect you observe depends on the experiment you perform.
The two natures are complementary, not contradictory (Bohr's complementarity principle)
You can never observe both natures simultaneously in the same experiment
For NEET: wave nature is prominent when is comparable to the obstacle/slit size; particle nature dominates otherwise
de Broglie Wavelength Applications
The de Broglie wavelength formula takes different forms depending on what information you're given — KE, accelerating voltage, or temperature. The most important NEET formula: for an electron accelerated through V volts, lambda = 1.226/sqrt(V) nm. This one formula appears in 40% of numerical problems from this chapter.
= 1.226 V nm
de Broglie Wavelength from Kinetic Energy
When kinetic energy is given, use lambda = h/sqrt(2mKE). This works for any particle — electrons, protons, neutrons, alpha particles.
Starting from = h/p and KE = p^2/(2m) , we get p = 2mKE , and therefore:
= h 2mKE
Derivation: KE = p^2 2m p = 2mKE = h/ 2mKE
For same KE: heavier particle → smaller (since 1/ m )
For same momentum: is the same regardless of mass! ( = h/p — no m dependence)
For same : all particles have the same momentum, but lighter particles have more KE
This formula uses NON-relativistic KE — valid when v c
Quick recall
An electron and a proton have the same kinetic energy. Which has the larger de Broglie wavelength?
The electron. Since = h/ 2mKE and both have the same KE, the lighter particle (electron, m_e m_p ) has the larger . Specifically, _e/ _p = m_p/m_e 1836 43 .
Electron Accelerated Through Voltage V
When an electron is accelerated from rest through a potential difference V, its KE = eV. Substituting into the de Broglie formula gives the most-used numerical formula in this chapter.
An electron at rest, accelerated through a potential difference V , gains kinetic energy KE = eV . Substituting into = h/ 2mKE :
= h 2meV
Plugging in h = 6.63 10^ -34 J·s, m_e = 9.11 10^ -31 kg, e = 1.6 10^ -19 C:
= 1.226 V nm (V in volts)
= 1.226/ V nm — ONLY for electrons (mass and charge of electron are baked in)
For protons accelerated through V : = h 2m_p eV = 0.0286/ V nm
For alpha particles through V : = h 2m_ (2e)V = 0.0101/ V nm
At V = 54 V: _e = 0.167 nm ← this is the Davisson-Germer value
At V = 150 V: _e = 0.1 nm = 1 angstrom ← a useful benchmark
Quick recall
An electron is accelerated through 100 V. What is its de Broglie wavelength?
= 1.226/ 100 = 1.226/10 = 0.1226 nm 1.226 angstrom. Takes 5 seconds with the shortcut formula.
Thermal de Broglie Wavelength
For particles in thermal equilibrium at temperature T, the average kinetic energy is (3/2)kT. This gives a thermal de Broglie wavelength that tells you when quantum effects become important.
For a particle in thermal equilibrium at temperature T , the average kinetic energy from kinetic theory is KE = 3 2 k_BT . Substituting:
= h 3mk_BT
k_B = 1.38 10^ -23 J/K (Boltzmann constant)
Higher temperature → smaller (faster particles have shorter wavelengths)
Heavier particles → smaller at same temperature
For neutrons at room temperature (300 K): 0.145 nm — comparable to interatomic spacing, enabling neutron diffraction
This formula appears in NEET when problems say 'particle at temperature T' or 'thermal neutrons'
Comparing de Broglie Wavelengths
NEET frequently asks you to compare wavelengths of different particles under different conditions. Master the proportionalities and you can answer these in seconds.
The key to comparison problems is knowing which formula to use and what is held constant:
Same KE: 1/ m → lighter particle has larger . Order: e^- > p >
Same momentum: = h/p → ALL particles have the SAME . Mass doesn't matter!
Same speed: = h/(mv) 1/m → lighter particle has larger . Order: e^- > p >
Same accelerating voltage (for charged particles): 1/ mq → lighter AND less charged particle wins
Same temperature: 1/ m → lighter particle has larger
Same KE
_e > _p > _ (since m_e < m_p < m_ )
Same momentum p
_e = _p = _ = h/p
Same speed v
_e > _p > _ (since 1/m )
Same voltage V
_e > _p > _ (since 1/ mq )
Quick recall
Arrange in decreasing order of de Broglie wavelength: electron, proton, alpha particle — all with the same kinetic energy.
_e > _p > _ . Since = h/ 2mKE and KE is the same, 1/ m . Electron is lightest → largest . Alpha ( 4m_p , charge 2e ) is heaviest → smallest .