Photoelectric Effect & Einstein's Equation Photoelectric Effect & Einstein's Equation Photoelectric Effect Graphs Graph Relation X-Axis / Y-Axis Shape/Slope Intercept Meaning Physical Insight Intensity governs the Count (current), Frequency governs the Punch (energy), and the Slope of V 0- is the Ratio h/e . Intensity ( I ) vs. Photoelectric Current ( i ) Straight line passing through origin Intercept at (0,0) The number of photoelectrons emitted per second ( i ) is directly proportional to the incident intensity ( I ). Frequency ( u ) vs. Max Kinetic Energy ( K max ) Straight line; Slope = Planck's constant ( h ) x -intercept is threshold frequency ( u 0 ); y -intercept is work function ( - ) Einstein's equation: K max = h - . The slope h is universal for all metals. Frequency ( u ) vs. Stopping Potential ( V 0 ) Straight line; Slope = h/e x -intercept is u 0 ; y -intercept is - /e Stopping potential depends linearly on frequency but is independent of intensity. Collector Potential ( V ) vs. Current ( i ) [Fixed , Variable I ] Sigmoid curves saturating at different levels Common x -intercept ( V 0 ) Saturation current increases with intensity ( I 3 > I 2 > I 1 ), but stopping potential ( V 0 ) remains same. Collector Potential ( V ) vs. Current ( i ) [Fixed I , Variable ] Sigmoid curves with different V 0 Different x -intercepts; Common saturation current Stopping potential magnitude increases with frequency ( |V 03 | > |V 02 | > |V 01 | ). Same I yields same saturation. Reciprocal Wavelength ( 1/ ) vs. Stopping Potential ( V 0 ) Straight line; Slope = hc/e x -intercept is 1/ 0 (Threshold wavelength) Derived from eV 0 = hc(1/ ) - hc(1/ 0) . Slope hc/e is constant. Intensity ( I ) vs. Stopping Potential ( V 0 ) Horizontal straight line (Slope = 0 ) Constant y -value The energy of the emitted photoelectrons does not depend on the number of incident photons. Intensity ( I ) vs. Max Kinetic Energy ( K max ) Horizontal straight line (Slope = 0 ) Constant y -value Evidence against classical wave theory; energy per photon is fixed for a specific frequency. Frequency ( u ) vs. Photoelectric Current ( i ) Step-like: Zero for < 0 , constant saturation for > 0 Discontinuity at 0 No photoemission occurs below the threshold frequency, regardless of intensity. (i) vs. (I) Straight line with slope = 1 Intercept represents quantum efficiency Confirms the linear power-law relationship between current and light intensity. photoelectric effect graphs Shining light on certain metals can eject electrons from their surface. This is the photoelectric effect. A simple lamp, a zinc plate, and a sensitive ammeter reveal something deep: increasing the brightness of red light, no matter how much, may produce no electrons, while even a dim violet beam can instantly liberate them. Classical wave theory predicted that energy delivery depends on intensity alone and that electrons might need a time to accumulate energy, but experiments showed immediate emission with a sharp threshold in frequency. Einstein solved the puzzle by treating light as packets of energy called photons. Each photon has energy E = h . An electron on the surface needs a minimum energy (work function) to escape. If h exceeds , the excess appears as kinetic energy of the photoelectron. This explains why frequency, not intensity, controls whether emission occurs and how energetic the emitted electrons are. Intensity only controls how many photons hit per second, so it affects current, not the maximum kinetic energy. Analogy: Think of a locked door that opens only with a coin of at least a certain denomination. Many 50-paise coins (higher intensity) cannot open a Rs. 1 lock. A single Rs. 2 coin (higher frequency) opens it and leaves change (kinetic energy) for the electron. remember Photon A quantum of light carrying energy E = h and momentum p = h/ . Work Function ( ) Minimum energy required to liberate an electron from a metal surface. Depends on the material and its surface condition. Minimum light frequency required to eject electrons from a given metal. It satisfies h 0 = . Threshold Frequency ( 0 ) Stopping Potential ( V 0 ) The retarding potential just sufficient to stop the most energetic photoelectrons from reaching the collector, so current falls to zero. The maximum photocurrent attained when all emitted electrons are collected, even if the accelerating potential is increased further. Saturation Current Photoelectron An electron ejected from a material due to absorption of a photon in the photoelectric effect. Historically, Hertz and Lenard observed that ultraviolet light falling on a negatively charged metal surface causes it to emit electrons and lose its charge. Key observations emerged: there is practically no time lag between shining suitable light and emission; no electrons are emitted below a certain frequency however intense the light; the maximum kinetic energy of the emitted electrons increases linearly with the frequency of incident light; and the photocurrent at a given frequency increases with intensity. These observations could not be reconciled with classical waves distributing energy continuously across the wavefront. A localized, one-photon-to-one-electron interaction naturally explains the immediate response and the strict frequency threshold. Photon Energy 17.1 Energy carried by a single photon of frequency or wavelength . Use the photon energy for the energy budget at the surface. An electron bound at the surface requires at least the work function to escape. If the photon energy equals , emission is just possible with zero kinetic energy. If it exceeds , the extra energy appears as kinetic energy of the fastest electrons. Not all emitted electrons come out with the same energy because some lose energy in collisions inside the metal before escaping. 17.2 Excess photon energy overcomes binding and provides kinetic energy to the most energetic photoelectrons. Einstein's Photoelectric Equation A practical way to measure K is by applying a retarding potential V between the emitter and collector. As V becomes more negative at the collector, fewer electrons arrive. The stopping potential V 0 is where the photocurrent falls to zero; the most energetic electrons are just stopped. This gives eV 0 = K = h - . Measuring V 0 at different gives a straight line whose slope is h/e and whose intercept on the -axis is 0 = /h . 17.3 Retarding potential V 0 required to stop the most energetic photoelectrons. Stopping Potential Relation tip Applicability: Clean metal surface, one-photon one-electron interaction, non-relativistic photoelectrons, and negligible space-charge effects at low to moderate intensities. Surface contamination alters and hence 0 . By Planck and Einstein, energy is quantized in photons of frequency . Kinematic relation of electromagnetic waves in vacuum. Substitute frequency in terms of wavelength. E = h = hc Planck quantization of energy exchange Light in vacuum travels at speed c Frequency-wavelength relation c = Energy of a photon E = h = hc Einstein's photoelectric equation K = h - One photon interacts with one electron Energy is conserved at the metal surface Electrons may lose some energy internally; K refers to the fastest K = h - Photon energy available for the process. Minimum energy to liberate the electron from the surface. Conservation of energy at the instant of emission. Final relation connecting frequency and maximum kinetic energy. Photon energy scales linearly with frequency and inversely with wavelength. Excess photon energy over work function appears as the maximum kinetic energy of photoelectrons. The threshold frequency 0 is defined by K = 0 . Setting h 0 - = 0 gives 0 = /h . It is often useful to express this as a cutoff wavelength 0 = hc/ . Light with > 0 , even if very intense, cannot eject electrons. Light with < 0 can eject electrons, and their maximum kinetic energy grows as decreases further. Minimum frequency and corresponding maximum wavelength that just cause emission. 17.4 Threshold and Cutoff This value represents the minimum energy required from incident photons to eject electrons from the metal surface. Current-voltage characteristics capture how many electrons are collected as we vary the potential between emitter and collector. For a fixed frequency above threshold, increasing intensity raises the saturation current because more photons mean more emitted electrons. The stopping potential V 0 , however, remains unchanged by intensity because it depends only on the maximum kinetic energy, which is set by frequency and work function, not by the photon count. Photocurrent vs potential at fixed frequency: intensity changes saturation current, not stopping potential. iv Frequency above threshold control control Intensity dependent Current Collector potential V -V0 Stopping potential V0 (same for both) positive V Saturation current (higher for higher intensity) I s Two I-V curves at the same frequency: higher intensity curve saturates at a larger current but both have the same stopping potential intercept on the negative V-axis. arbitrary Photocurrent I Plotting K or eV 0 against frequency yields a straight line. The slope equals h for K vs and equals h/e for V 0 vs . The x -intercept gives 0 = /h , and the negative y -intercept on a K plot gives - . This linearity provides an experimental way to determine Planck’s constant and the work function for a surface. Linear dependence of maximum kinetic energy on frequency. custom ν0 Threshold frequency h(ν1-ν0) ν1 > ν0 Kmax at ν1 Frequency control Kmax dependent Frequency ν Hz A straight line with slope h, cutting the ν-axis at ν0 with Kmax = 0. eV Maximum kinetic energy Kmax Frequency = 8.0 10 14 , Hz Work function = 2.2 , eV Planck constant h = 6.626 10 -34 , J ,s Electron charge e = 1.6 10 -19 , C Monochromatic light of frequency 8.0 10 14 , Hz falls on a metal with work function = 2.2 , eV . Find K and the stopping potential V 0 . Conceptual numerical typical of NEET single-step Use K = h - and eV 0 = K . Convert units consistently. Maximum kinetic energy K and stopping potential V 0 easy Photon energy in joules. Convert to eV for easy subtraction. Apply Einstein's equation. In eV units, 1 , eV corresponds to 1 , V of stopping potential. eV, V Photocurrent depends on how many electrons are emitted per second and collected. For a given frequency above threshold, doubling intensity doubles the photon arrival rate and, if quantum efficiency is unchanged, nearly doubles the emitted electron rate. Thus the saturation current scales with intensity. But the maximum kinetic energy and hence the stopping potential do not change with intensity because each photon’s energy is fixed by its frequency. medium Compute work function in joules. Convert to eV. Photon energy at 400 nm. Stopping potential equals Kmax in eV. eV, V Cutoff wavelength 0 = 500 , nm Incident wavelength = 400 , nm Planck constant h = 6.626 10 -34 , J ,s Speed of light c = 3.0 10 8 , m/s Electron charge e = 1.6 10 -19 , C A metal has cutoff wavelength 0 = 500 , nm . (a) Find its work function in eV. (b) If light of wavelength 400 , nm is used, find the stopping potential. Uses cutoff wavelength and stopping potential relation Use = hc/ 0 and eV 0 = hc/ - . (a) Work function (eV). (b) Stopping potential V 0 for = 400 , nm Emission is effectively instantaneous. There is no detectable time delay between switching on light above threshold and electron emission, even at very low intensities. This rules out the classical idea of electrons slowly soaking up energy from a continuous wave. A single photon transfers its energy to a single electron in a localized interaction. Intensity increases the number of emitted electrons and hence the photocurrent. Maximum kinetic energy depends only on frequency and work function: K = h - . Increasing light intensity increases the maximum kinetic energy of photoelectrons. Emission is practically instantaneous for 0 . One photon transfers energy to one electron in a single event. Electrons need time to accumulate energy before emission. Unit trap: Do not plug in eV and h in joules into the same expression. Convert everything to either joules or eV first. Also, V 0 changes with frequency, not with intensity. neet-alert K-phi rule: K comes from h nu minus phi. Keep K high by raising nu, not intensity. Remind yourself: K = h - ; intensity ↦ current, frequency ↦ energy. Quantum efficiency or photoelectric yield is the number of photoelectrons emitted per incident photon. It depends on the material, surface cleanliness, and wavelength. Even if the photon energy exceeds the work function, some electrons fail to escape due to internal scattering, so yield is typically less than 1 except for specially prepared photocathodes. Yield influences saturation current for a given intensity. Increase frequency (above threshold) → increases K and V 0 ; saturation current unchanged if intensity fixed. Increase intensity → increases saturation current; K and V 0 unchanged if frequency fixed. Change metal (different ) → shifts threshold frequency and cutoff wavelength; changes V 0 at the same frequency. What changes what? How to solve PE numericals quickly Check threshold: compare with 0 or with 0 . Compute photon energy E = hc/ or h ; convert units if needed. Apply K = E - and then V 0 = K /e if asked. For intensity effects, discuss saturation current only, not K . arbitrary Photocurrent I At higher frequency, the stopping potential magnitude increases, but saturation current remains the same at fixed intensity. Smaller |V0| at lower ν -V0(low ν) Larger |V0| at higher ν -V0(high ν) Frequency control Intensity fixed control Current dependent Collector potential V custom Effect of frequency at constant intensity: V 0 shifts, saturation current does not. Although this chapter centers on the particle nature of light, its logic connects to matter waves. Once photons are accepted as particles with energy and momentum, de Broglie proposed that material particles also have wave-like properties with wavelength = h/p . While not needed to explain photoelectric emission itself, these relations are often cross-referenced in problems involving electrons accelerated through a potential before or after photoemission. Photon relations. Photon momentum-energy relation. Solve for momentum in terms of wavelength. Invert to obtain the de Broglie relation. = h p Photons: E = h and E = pc in vacuum Extend = h/p by analogy to matter waves Non-relativistic limit recovers p = mv for massive particles De Broglie wavelength = h/p Universal wave-particle link: wavelength equals Planck’s constant divided by momentum. De Broglie wavelength in terms of kinetic energy = h/ 2mK Non-relativistic particle so that K = p 2 /2m Positive kinetic energy K De Broglie relation holds for matter waves = h 2mK Momentum from kinetic energy. Substitute into the de Broglie formula. Handy for non-relativistic particles when kinetic energy is known. Same relation expressed with kinetic energy; ensure K is in joules for SI h. De Broglie wavelength for a charged particle accelerated through V: = h/ 2mqV Particle starts from rest and is accelerated through potential V Non-relativistic speeds after acceleration Energy conservation: work qV becomes kinetic energy = h 2mqV Work done equals kinetic energy gained. Momentum from kinetic energy. Apply de Broglie relation. Relates accelerating voltage to matter wavelength; valid when v c and initial speed is negligible. ( ) 12.27 V( V ) Start from the charged particle formula with q=e . Plug in constants in SI. Convert meters to angstroms: 1 , = 10 -10 , m . Electron-specific shortcut ( ) 12.27/ V( volt ) Electron accelerated from rest through V Substitute numerical constants for electron mass and charge Non-relativistic regime, typically V 511 kV Quick estimate of electron wavelength in angstroms for a given accelerating voltage. Relativistic mass-energy relation applies: E = mc 2 Photon energy E = hc/ Rest mass of photon is zero; this is an effective or relativistic mass Effective mass of a photon m = E/c 2 = h/(c ) Einstein’s mass-energy equivalence. Photon energy from Planck relation. Solve for effective mass. m = h c Not a rest mass; encodes inertia-equivalent of photon energy. Photon momentum p = h/ leads to observable effects like radiation pressure and Compton scattering. In photoelectric setups, momentum is small and typically does not control emission; the energy condition is dominant. However, understanding that photons carry momentum clarifies why light can exert mechanical effects on surfaces. Two-point determination of h and φ via stopping potential data An experiment measures stopping potentials V 0 = 0.50 , V at = 7.0 10 14 , Hz and V 0 = 1.10 , V at = 9.0 10 14 , Hz . Determine Planck’s constant h and the work function for the surface. eV 0 = h - V 0,1 = 0.50 , V at 1 = 7.0 10 14 , Hz V 0,2 = 1.10 , V at 2 = 9.0 10 14 , Hz Electron charge e = 1.6 10 -19 , C Planck’s constant h and work function φ Use two-point slope on V0 vs ν: slope = h/e. Then back-substitute for φ. Compute slope of V0 vs ν. Estimate of Planck’s constant. Back-substitute using one data point. Compute φ in joules. Convert to eV. hard J s, eV Typical work functions at room temperature for clean surfaces Metal Work function φ (eV) Approximate cutoff λ0 (nm) Cesium (Cs) 1.9 650 Sodium (Na) 2.3 540 Zinc (Zn) 3.7 335 Aluminum (Al) 4.2 295 Copper (Cu) 4.7 265 Edge cases and practical notes: Near threshold 0 , K and V 0 approach zero, so distinguishing signal from noise can be hard. Strong surface oxidation raises , shifting 0 upward and suppressing emission. Temperature changes affect electron distributions slightly but do not eliminate the frequency threshold. At very high intensities, space-charge near the cathode may distort I–V curves; keep beams weak to moderate and use short gaps to minimize these effects. Boundary conditions: The linear K vs holds for single-photon emission from clean, uniform surfaces. For semiconductors, band gaps and surface states modify the effective threshold. In gases, photoionization replaces simple work function ideas. tip Big picture: Frequency decides if electrons can escape and how fast the fastest ones are. Intensity decides how many escape per second. remember Key terms recap light quantum Quantum of light energy E = h Photon Work function Minimum energy to free an electron from the surface surface binding energy Threshold frequency cutoff frequency Minimum frequency 0 for emission Stopping potential cutoff potential Retarding potential V 0 that makes photocurrent zero V 0 Saturation current Maximum photocurrent for given light and geometry Quantum efficiency photoelectric yield Emitted electrons per incident photon