Physics · Quantum Mechanics · Wave Mechanics
Photoelectric Effect Calculator
Calculates the maximum kinetic energy of emitted photoelectrons, stopping potential, and threshold frequency using Einstein's photoelectric equation.
Calculator
Formula
KE_{\max} is the maximum kinetic energy of the emitted photoelectron (J or eV); h = 6.626 \times 10^{-34} \text{ J·s} is Planck's constant; f is the frequency of the incident photon (Hz); \phi is the work function of the metal (J or eV), representing the minimum energy required to eject an electron; f_0 = \phi / h is the threshold frequency below which no electrons are emitted. The stopping potential V_s satisfies eV_s = KE_{\max}, where e = 1.602 \times 10^{-19} \text{ C}.
Source: Einstein, A. (1905). 'Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.' Annalen der Physik, 17, 132–148. NIST CODATA 2018 values for physical constants.
How it works
The photoelectric effect occurs when photons of sufficiently high frequency strike a metal surface and transfer enough energy to free electrons from the metal's surface. Classical wave theory predicted that any frequency of light, given enough intensity, should eject electrons — but experiments showed that below a certain threshold frequency, no electrons are emitted regardless of light intensity. Einstein resolved this contradiction by proposing that light comes in discrete energy packets called photons, each carrying energy E = hf, where h is Planck's constant and f is the photon's frequency.
Einstein's photoelectric equation states that the maximum kinetic energy of an emitted photoelectron equals the incoming photon energy minus the work function of the metal: KEmax = hf − φ. The work function φ represents the binding energy of the most loosely held surface electron — the minimum energy needed to liberate it. The threshold frequency f₀ = φ/h is the lowest frequency capable of ejecting any electron; photons below this frequency lack sufficient energy regardless of beam intensity. The stopping potential Vs is the minimum reverse voltage required to halt even the fastest photoelectrons, related by eVs = KEmax, making it directly measurable in experiment.
Practical applications of the photoelectric effect and its governing equations span an enormous range: photodiodes and photomultiplier tubes in scientific instruments, solar cell design and optimization, X-ray photoelectron spectroscopy (XPS) for material surface analysis, night-vision devices, and automatic door sensors. Understanding the relationship between photon frequency, work function, and emitted electron energy is fundamental to designing efficient light-sensitive devices and interpreting spectroscopic data.
Worked example
Suppose ultraviolet light of frequency f = 1.20 × 10¹⁵ Hz strikes a copper surface with a work function of φ = 4.70 eV.
Step 1 — Calculate photon energy:
E = hf = (6.626 × 10⁻³⁴ J·s)(1.20 × 10¹⁵ Hz) = 7.951 × 10⁻¹⁹ J
Converting to eV: E = 7.951 × 10⁻¹⁹ / 1.602 × 10⁻¹⁹ = 4.963 eV
Step 2 — Find threshold frequency:
f₀ = φ/h = (4.70 × 1.602 × 10⁻¹⁹) / 6.626 × 10⁻³⁴ = 7.527 × 10⁻¹⁹ / 6.626 × 10⁻³⁴ ≈ 1.136 × 10¹⁵ Hz
Since f = 1.20 × 10¹⁵ Hz > f₀, electrons will be emitted.
Step 3 — Calculate maximum kinetic energy:
KEmax = hf − φ = 4.963 eV − 4.70 eV = 0.263 eV
In joules: KEmax = 0.263 × 1.602 × 10⁻¹⁹ = 4.21 × 10⁻²⁰ J
Step 4 — Determine stopping potential:
Vs = KEmax / e = 0.263 eV / e = 0.263 V
A reverse voltage of 0.263 V applied across the photoelectric cell would completely stop the emitted electron current.
Limitations & notes
This calculator uses Einstein's idealized photoelectric equation and several important limitations apply. First, the formula gives the maximum kinetic energy — electrons emitted from deeper within the metal lose additional energy through collisions and emerge with less than KEmax; the result here represents only the most energetic electrons from the outermost surface layer. Second, the work function is treated as a fixed constant, but in reality it depends on crystal face orientation, surface contamination, temperature, and surface roughness — tabulated values are approximate averages for polycrystalline samples under ideal conditions. Third, the equation assumes a sharp threshold frequency, whereas real metals exhibit a gradual onset due to the Fermi-Dirac energy distribution of electrons at non-zero temperatures; thermal electrons slightly broaden the effective threshold. Fourth, relativistic effects are negligible for photon energies in the UV and visible range but become significant for hard X-rays and gamma rays, where a relativistic treatment is required. Fifth, this model does not account for quantum yield (the probability that a photon actually ejects an electron), which is typically much less than 1 and depends on material and wavelength. Finally, for extremely high-intensity laser pulses, multiphoton photoelectric effects can occur, where two or more photons combine to eject a single electron — a nonlinear regime not described by this equation.
Frequently asked questions
What is the photoelectric effect in simple terms?
The photoelectric effect is the emission of electrons from a metal surface when light shines on it. Crucially, emission depends on the light's frequency rather than its intensity — if the photon frequency is below a threshold value, no electrons are emitted no matter how bright the light. Einstein explained this by treating light as a stream of discrete energy packets called photons.
Why does the kinetic energy depend on frequency, not intensity?
Each photon interacts with only one electron in a one-to-one quantum event. A single photon carries energy hf — if this energy exceeds the work function, the electron is freed with the remaining energy as kinetic energy. Higher intensity means more photons per second (more electrons emitted) but each individual photon still carries the same energy hf, so KE<sub>max</sub> is unchanged. Only increasing frequency increases the energy per photon.
What is the work function and where can I find values for common metals?
The work function φ is the minimum energy required to remove an electron from the surface of a metal to a point just outside, overcoming the electrostatic attraction of the lattice. Common values include cesium (2.1 eV), sodium (2.3 eV), aluminum (4.1 eV), copper (4.7 eV), and platinum (5.6 eV). Reliable values are tabulated by NIST and in the CRC Handbook of Chemistry and Physics, though values vary with surface condition.
What is the stopping potential and how is it measured?
The stopping potential V<sub>s</sub> is the magnitude of the reverse voltage applied in a photoelectric experiment just sufficient to reduce the photocurrent to zero by decelerating even the fastest emitted electrons. It is measured by connecting a variable voltage source in opposition to the photocurrent and finding the voltage at which the current reaches zero. Since eV<sub>s</sub> = KE<sub>max</sub>, measuring V<sub>s</sub> gives a direct experimental determination of maximum kinetic energy.
Can the photoelectric effect occur with visible light?
Yes, but only for metals with low work functions. For example, cesium (φ ≈ 2.1 eV) can be ejected by visible red light at around 590 nm (f ≈ 5.1 × 10¹⁴ Hz). Most common metals like copper or iron require ultraviolet light because their work functions (4–5 eV) exceed the energy of visible photons (1.8–3.1 eV). This is why photoelectric sensors often use specially selected materials or UV light sources.
Last updated: 2025-01-15 · Formula verified against primary sources.