Photoelectric Effect

Take a clean strip of zinc, connect it to a sensitive electrometer, and shine a desk lamp on it. Nothing happens. Swap the desk lamp for a mercury vapour lamp that emits ultraviolet. The electrometer needle swings immediately — electrons are leaving the zinc. Now dim the ultraviolet lamp until it is a hundred times fainter. The needle still swings, just more slowly. Turn the desk lamp back on at full brightness. Still nothing.

Something is wrong with the intuition that "more light means more energy." A faint source of one colour can knock electrons out of a metal that a brilliant source of another colour cannot touch. Light is doing something to the electrons that does not scale with how bright the light is — only with which light it is.

The puzzle is deeper than it looks. Before 1905, every physicist knew that light was a wave. Young's double-slit experiment, Maxwell's equations, the entire machinery of electromagnetism — all of it said light was a continuous oscillation of electric and magnetic fields. A brighter wave carries more energy. Give a metal enough time under a dim wave and eventually the electrons should soak up enough energy to escape.

That is not what happens. And the reason it does not happen is that light, at its most fundamental, is not a continuous wave. It is a stream of discrete energy packets, each of a fixed size set by the frequency. That is the claim that won Einstein the Nobel Prize in Physics in 1921.

The experiment — light on a metal surface

The setup is almost embarrassingly simple, which is part of why the result was so shocking.

Inside the evacuated tube, the cathode is a clean plate of the metal you are studying — sodium, potassium, caesium, zinc. Light enters through a quartz window (ordinary glass blocks ultraviolet). If the light has the right colour, electrons — called photoelectrons — are ejected from the cathode, cross the vacuum, and land on the anode. A microammeter in the external circuit reads the resulting current, called the photocurrent.

You can vary three knobs independently:

• The frequency f of the light (change the source or insert a filter).
• The intensity I of the light (dim or brighten without changing colour).
• The voltage V across the tube (battery with a reversing switch).

And you measure two quantities:

• The photocurrent (how many electrons per second cross the tube).
• The stopping voltage V_s — the reverse voltage that just stops the most energetic electrons from reaching the anode.

That last one is the most important. If you reverse the battery so the anode is negative with respect to the cathode, electrons now have to climb an electrical hill to reach the anode. Only the electrons whose kinetic energy exceeds eV can make it across. When you ramp up the reverse voltage until the photocurrent drops to exactly zero, you have found the voltage V_s that just barely stops the fastest electron the metal emitted. Energy conservation then tells you the maximum kinetic energy of the emitted electrons:

Why: an electron of charge e climbing a potential difference V_s loses energy eV_s. The fastest electron just barely makes it, which means its entire kinetic energy converted to electrical potential energy. Measure V_s, multiply by the electron charge, and you have read off K_\text{max} directly.

This is the experimental handle that lets you extract a clean number — K_\text{max} in joules — from a tabletop apparatus.

The observations — four stubborn facts

When Philipp Lenard did this experiment carefully in the 1890s and early 1900s, four facts emerged. Each one contradicts the wave theory of light.

Fact 1: There is a threshold frequency

For every metal, there is a threshold frequency f_0 below which no photoelectrons are emitted at all, no matter how intense the light or how long you wait. Shine the brightest red laser you can buy on a sheet of zinc, and zero electrons come off — forever. Shine a faint ultraviolet beam on the same zinc and electrons come off immediately. The cutoff is sharp. For zinc, f_0 corresponds to a wavelength of about 290 nm, well into the ultraviolet; for caesium, it corresponds to about 660 nm, deep in the visible red — which is why caesium cells are used in old photomultiplier tubes and solar-triggered street lights.

Fact 2: K_\text{max} depends on frequency, not intensity

Above the threshold, if you increase the frequency, the maximum kinetic energy of the photoelectrons increases. But if you hold the frequency fixed and increase the intensity, K_\text{max} does not budge. More intense light releases more electrons (the photocurrent goes up), but each electron leaves with the same energy distribution.

A graph of stopping voltage V_s versus frequency f is a straight line. Below f_0 there is no photocurrent at all, so V_s is undefined; above f_0, the line climbs with a slope that is the same for every metal — it depends only on a universal constant.

Fact 3: Emission is instantaneous

Turn the light on, and electrons come off immediately — within nanoseconds, which was already "instantaneous" by the standards of 1905 instruments, and we now know is within 10^{-9} s. There is no delay, no build-up, no lag. Even at intensities so low that a single electron's worth of energy should take minutes to accumulate on a classical wave picture, emission is still immediate.

Fact 4: Photocurrent saturates with voltage

At zero bias, some photoelectrons reach the anode. As you increase the forward voltage (anode positive), more are pulled across, and the photocurrent rises. But at some voltage, the current levels off at a saturation value that depends only on the light intensity — not on the voltage. Every single electron ejected from the cathode is now reaching the anode; you cannot collect more than the light released.

Why the wave theory cannot explain this

Here is the wave-theory prediction, sentence by sentence, against the experimental reality.

Wave theory says: energy is spread continuously across the wavefront. A brighter wave delivers more energy per unit area per second. Given enough time, any wave — of any frequency — can pump enough energy into an electron to knock it free.

Experiment says: below f_0, no electron is ever ejected, no matter how bright or how long.

Wave theory says: a brighter wave should give more energetic electrons. The electrons absorbing the wave have more energy available.

Experiment says: brighter light gives more electrons, each with the same maximum energy. Intensity does not affect K_\text{max}.

Wave theory says: at low intensity, it should take a measurable time for an electron to absorb enough energy from a continuous wave to escape. For a faint source, this delay could be minutes.

Experiment says: emission is immediate at any intensity.

Three direct contradictions, from four clean observations. The wave theory of light — triumphant for a century after Young, solidified by Maxwell, seemingly unassailable — cannot describe what happens when light meets a metal surface.

Einstein's 1905 explanation — light as photons

In 1905 — the same year he published special relativity — Einstein proposed something audacious. What if light is not a continuous wave but a stream of discrete packets, each carrying a fixed amount of energy determined by the frequency? He borrowed the idea from Planck's 1900 analysis of blackbody radiation, but he made it physical: Planck had said the oscillators in a hot body exchange energy in quanta; Einstein said the light itself travels in quanta.

Each packet — now called a photon — carries energy

where h = 6.626 \times 10^{-34} J·s is Planck's constant and f is the frequency of the light. A brighter beam of the same colour has more photons per second; it does not have more energetic photons. The colour sets the energy per photon; the intensity sets the number of photons per second.

With that single move, every observation falls into place.

Why a threshold frequency? Each electron inside the metal is held in by an energy barrier called the work function \phi — the minimum energy needed to yank it out. Ejection is a one-photon-one-electron event: a single photon either delivers enough energy (hf \geq \phi) to break the electron free, or it does not. Below f_0 = \phi/h, no single photon has enough punch. Adding more photons does not help — each electron interacts with one photon at a time. The threshold is sharp because it is the atomic-scale energy budget that matters, not the total wave energy.

Why does K_\text{max} depend on frequency, not intensity? The photon gives its energy hf to the electron. Some of that energy (exactly \phi) is spent breaking the electron out of the metal. Whatever is left shows up as kinetic energy:

Higher frequency means more energy per photon, so more energy left over after paying \phi. Increasing intensity means more photons per second — hence more electrons per second (the photocurrent goes up) — but each individual ejection event is governed by the same hf - \phi balance. K_\text{max} is a single-photon property.

Why is emission instantaneous? A photon either hits an electron or it does not. When it does, the whole transaction happens in one step — there is nothing to accumulate, no build-up. Emission is instantaneous by construction.

Why does photocurrent saturate with voltage? Once the forward voltage is high enough that every emitted electron reaches the anode, increasing the voltage further cannot make a difference — you are already collecting all of them. The saturation level is set by the rate at which photons arrive and eject electrons — the light intensity — not by the voltage.

Einstein's equation is a simple statement of energy conservation, applied one photon at a time:

Rewriting in terms of stopping voltage:

Why: substitute K_\text{max} = eV_s on the left, then divide through by e. This is now the equation of a straight line: V_s as a function of f, with slope h/e and intercept -\phi/e. The slope h/e is a universal constant — it must come out the same for every metal. Only the intercept (and hence the threshold f_0 = \phi/h) depends on the metal. This is Millikan's 1916 test, and it is how h was measured to high precision.

The classic K_\text{max} vs f graph

If the photon hypothesis is right, a plot of K_\text{max} (or equivalently eV_s) against frequency f should be a straight line whose slope is Planck's constant — same slope for every metal, with only the intercept shifting. This is the graph that settled the matter experimentally in 1916, and it is the graph that JEE loves to set problems on.

The interactive below lets you dial the frequency of the incoming light and watch the photoelectric equation play out live. The vertical dashed line is the threshold frequency f_0; below it, no electrons come off.

A few things to read off that graph by eye:

• The slope is h/e in volts per hertz, or h in eV·s. It is the same for sodium, potassium, caesium, zinc, copper — every metal gives the same slope. This is the single most important experimental consequence of the photon hypothesis: a universal constant appears in the data.
• The horizontal intercept is f_0. Set K_\text{max} = 0: hf_0 = \phi, so f_0 = \phi/h. Extrapolate the line down to the f-axis to read off the threshold.
• The vertical intercept is -\phi/e — negative, because the work function is a cost. A metal with a higher work function pushes the whole line to the right.
• Intensity does not appear on this graph at all. That is the whole point: K_\text{max} is a single-photon property, independent of how many photons arrive per second.

Worked examples

Common confusions

• "Intensity increases K_\text{max}." No — intensity increases the number of photoelectrons per second (the photocurrent), not the maximum kinetic energy of each one. Each ejection is a one-photon-one-electron transaction; the photon's energy is set by the frequency, not by how many photons there are.

• "The work function is the ionisation energy of an atom in the metal." Not quite. The ionisation energy removes an electron from an isolated atom to infinity. The work function removes an electron from the metal's surface — that is, from the partially-filled conduction band to just outside the metal. It is typically much less than the atomic ionisation energy, because the electron you are removing is already loosely bound (a conduction electron).

• "Every ejected electron has K_\text{max}." No — K_\text{max} is an upper limit. Most electrons lose some energy to collisions inside the metal on their way out, so they emerge with less than K_\text{max}. Only the electrons that happen to be right at the surface and moving straight outward get the full hf - \phi.

• "If hf is just barely less than \phi, the electron gets a tiny K_\text{max}." No — if hf < \phi, no electron comes out at all. K_\text{max} is not a small positive number; it is undefined. The photoelectric equation K_\text{max} = hf - \phi only applies when hf \geq \phi.

• "A stronger electric field inside the tube changes K_\text{max}." The field between anode and cathode affects which electrons reach the anode (it can slow them down with a reverse bias, or accelerate them with a forward bias), but it cannot change the kinetic energy with which they leave the cathode. K_\text{max} is a property of the light-metal interaction, measured at the instant of emission.

• "Einstein's equation has a \frac{1}{2} somewhere." It does not. Sometimes students confuse it with the formula K = \frac{1}{2}mv^2 and write hf = \phi + \frac{1}{2}mv_\text{max}^2. That is correct — but writing K_\text{max} = hf - \phi is shorter and equivalent. Just remember K_\text{max} is \frac{1}{2}m_e v_\text{max}^2.

Why this matters — ISRO rooftops, photodiodes, solar India

The photoelectric effect is not a laboratory curiosity. It is the physical basis of every technology that turns light into an electrical signal.

• Solar panels on Indian rooftops work on a close cousin — the photovoltaic effect in semiconductors. When a photon with energy greater than the semiconductor's bandgap (silicon's is about 1.1 eV, so any visible photon qualifies) strikes the junction, it promotes an electron across, generating a usable current. The "threshold frequency" for silicon corresponds to \lambda \approx 1100 nm — just past the near infrared. This is why silicon cells respond to visible sunlight but not to the deep infrared.
• Photodiodes and photomultiplier tubes inside the photometers on ISRO's Aditya-L1 solar observatory convert UV and visible photons from the Sun into measurable currents — the same cathode-anode geometry as the tabletop experiment, with caesium-antimony or multi-alkali photocathodes chosen to lower the work function into the visible range.
• Image sensors in every phone camera (CMOS arrays) are matrices of photodiodes. Each pixel reads out a photocurrent proportional to how many photons landed on it during the exposure.
• The night-vision cascade in military-grade goggles amplifies the single-electron photocurrent through millions of stages, each stage itself a photoelectric emission event.

Every photodetector on Earth — and every photodetector in orbit around Earth — is a descendant of the zinc-plate experiment, reading out Einstein's equation one photon at a time.

Where this leads next

• Photons and Wave-Particle Duality of Light — the full story of why light is simultaneously wave and particle, with the double-slit-one-photon-at-a-time experiment as the key evidence.
• de Broglie Hypothesis — if photons of wavelength \lambda carry momentum h/\lambda, then particles of momentum p should have wavelength h/p. The extension of the photon idea to matter.
• Bohr's Model of the Hydrogen Atom — the same E = hf applied to atomic transitions, which explains the spectral lines of hydrogen.
• Hydrogen Spectrum and Spectral Series — what you see when atoms emit photons of specific energies.
• X-Rays — the reverse of the photoelectric effect: fast electrons hit a metal and emit high-energy photons.