What is the photoelectric effect?
Metals have many free electrons. They are held inside the metal by a small energy 'wall'. When light of the right kind falls on a clean metal surface, some electrons get enough energy to jump over this wall and fly out. This is the photoelectric effect. The electrons that come out are called photoelectrons. Their flow makes a photocurrent.
Work function (φ): the smallest energy needed to pull one electron out of a metal surface. Each metal has its own value, for example about 2.3 eV for sodium and about 4.7 eV for copper. 1 eV = 1.6 × 10⁻¹⁹ J.
Hertz and Lenard: the first clues
In 1887 Heinrich Hertz was making sparks jump across a gap between two metal balls. He noticed that the spark jumped more easily when ultraviolet light fell on the balls. Light was helping charges escape from the metal.
Around 1902 Philipp Lenard studied this carefully in a vacuum tube. He found:
- Ultraviolet light made current flow between two plates in vacuum; with no light there was no current.
- The particles coming out were negative (later known to be electrons).
- For some metals, even visible light worked; for others only UV worked.
- The speed of the electrons depended on the colour (frequency) of the light, not on how bright it was.
Wallace Hallwachs also saw that a negatively charged zinc plate lost its charge when UV light fell on it.
Experimental study of the photoelectric effect
Set-up: an evacuated glass tube with a quartz window (quartz lets UV pass). Inside: an emitter plate C (photosensitive metal) and a collector plate A. A battery with a potential divider sets the voltage of A, and a commutator lets us make A positive or negative. A microammeter measures the current.
1. Effect of intensity
Keep frequency and voltage fixed. The photocurrent is directly proportional to the intensity of light. Double the brightness → double the current. Reason: more photons per second knock out more electrons per second.
2. Effect of voltage (I–V graph)
Make A more and more positive: the current rises and then becomes constant. This constant value is the saturation current: every electron that comes out is being collected. Now make A negative: the current falls. At one voltage −V₀ it becomes zero. V₀ is the stopping potential (cut-off potential). Here eV₀ = KEmax.
For the same frequency but different intensities, the curves have different saturation currents but the same stopping potential. So KEmax does not depend on intensity.
3. Effect of frequency
For the same intensity but higher frequency, the stopping potential is larger (more negative). Saturation current stays the same. So KEmax grows with frequency.
4. Threshold frequency
Plot V₀ against ν. You get a straight line. It cuts the ν-axis at ν₀, the threshold frequency. Below ν₀ no electrons come out, however bright the light. Its slope is h/e, the same for every metal.
5. No time lag
Electrons come out within about 10⁻⁹ s of switching on the light, even for very dim light.
Why the wave idea of light fails
If light were only a spread-out wave:
- Bright light of any colour should, after some time, give electrons enough energy. But red light never works on sodium, however bright. ✗
- Brighter light should make faster electrons. But KEmax does not change with brightness. ✗
- Dim light should need time to 'fill up' an electron with energy. But there is no delay. ✗
So the wave picture cannot explain this experiment.
Einstein's photoelectric equation
In 1905 Albert Einstein said: light energy comes in small packets called quanta or photons. A photon of frequency ν carries energy E = hν, where h = 6.63 × 10⁻³⁴ J s (Planck's constant).
One photon is absorbed by one electron, all at once. The electron spends φ to get out. The rest is its kinetic energy:
KEmax = hν − φ, or hν = φ + ½mv²max
Since KEmax = eV₀: eV₀ = hν − φ, so V₀ = (h/e)ν − φ/e. This is the straight line of the V₀–ν graph.
Put KEmax = 0: φ = hν₀. So ν₀ = φ/h and threshold wavelength λ₀ = hc/φ.
Einstein's equation explains everything: threshold (photon energy must beat φ), KEmax depends on ν only (one photon per electron), current depends on intensity (number of photons), and no time lag (a single collision).
Handy shortcut: E (in eV) = 1240 / λ (in nm).
Particle nature of light: the photon
A photon is a tiny packet of light energy. Its properties:
- Energy E = hν = hc/λ.
- Momentum p = hν/c = h/λ.
- It always moves at speed c = 3 × 10⁸ m/s in vacuum. It has no rest mass.
- It has no charge, so electric and magnetic fields do not bend it.
- Brighter light of the same colour means more photons per second, not bigger photons.
- Photons can be created or absorbed, but in a collision total energy and momentum are kept (conserved).
Light spreads and interferes like a wave, but gives and takes energy like a particle. This is the dual nature of radiation.
Try it: predict, then check
- In the 3D, set frequency to 5 × 10¹⁴ Hz. Predict: will bright light (intensity 10) give current? Check.
- Set frequency to 9 × 10¹⁴ Hz, voltage +1 V. Change the intensity from 2 to 8. What happens to the current? To KEmax?
- Slowly make the voltage negative. Note the voltage where the green dot on the graph reaches zero. Compare it with KEmax in eV.
- At home: a solar garden light or a calculator's solar cell works in sunlight and even under a bright white LED, but not well under a dim red night lamp. Try it and think about photon energy.
Key formulas and definitions
- E = hν = hc/λ; E(eV) = 1240/λ(nm)
- hν = φ + KEmax (Einstein's equation)
- KEmax = ½mv²max = eV₀
- φ = hν₀ = hc/λ₀
- V₀ = (h/e)ν − φ/e (slope h/e)
- Number of photons per second n = P/(hν); photon momentum p = h/λ
Worked examples
1. Find the energy of a photon of wavelength 500 nm in joules and in eV.
Step 1: E = hc/λ = (6.63 × 10⁻³⁴ × 3 × 10⁸) / (500 × 10⁻⁹). Step 2: E = 1.989 × 10⁻²⁵ / 5 × 10⁻⁷ = 3.98 × 10⁻¹⁹ J. Step 3: In eV: 3.98 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ ≈ 2.49 eV. Check with the shortcut: 1240/500 = 2.48 eV.
2. The work function of a metal is 2.0 eV. Find its threshold frequency and threshold wavelength.
Step 1: φ = 2.0 × 1.6 × 10⁻¹⁹ = 3.2 × 10⁻¹⁹ J. Step 2: ν₀ = φ/h = 3.2 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ ≈ 4.83 × 10¹⁴ Hz. Step 3: λ₀ = c/ν₀ = 3 × 10⁸ / 4.83 × 10¹⁴ ≈ 6.2 × 10⁻⁷ m = 620 nm (orange-red light).
3. Light of energy 4.0 eV per photon falls on a metal with φ = 2.5 eV. Find KEmax and the stopping potential.
Step 1: KEmax = hν − φ = 4.0 − 2.5 = 1.5 eV. Step 2: eV₀ = KEmax, so V₀ = 1.5 V. Step 3: In joules, KEmax = 1.5 × 1.6 × 10⁻¹⁹ = 2.4 × 10⁻¹⁹ J.
4. Light of wavelength 310 nm falls on a metal with φ = 2.2 eV. Find the maximum speed of the photoelectrons (m = 9.1 × 10⁻³¹ kg).
Step 1: Photon energy = 1240/310 = 4.0 eV. Step 2: KEmax = 4.0 − 2.2 = 1.8 eV = 1.8 × 1.6 × 10⁻¹⁹ = 2.88 × 10⁻¹⁹ J. Step 3: ½mv² = KEmax → v² = 2 × 2.88 × 10⁻¹⁹ / 9.1 × 10⁻³¹ = 6.33 × 10¹¹. Step 4: v ≈ 7.96 × 10⁵ m/s.
5. A 10 W lamp gives out light of wavelength 600 nm. How many photons does it send out each second?
Step 1: Energy of one photon E = hc/λ = 1.989 × 10⁻²⁵ / 6 × 10⁻⁷ = 3.315 × 10⁻¹⁹ J. Step 2: n = P/E = 10 / 3.315 × 10⁻¹⁹. Step 3: n ≈ 3.0 × 10¹⁹ photons per second.
6. For a metal the stopping potential is 1.2 V for light of frequency 8 × 10¹⁴ Hz, and 3.2 V for 12.8 × 10¹⁴ Hz. Find Planck's constant and the work function.
Step 1: eV₀ = hν − φ for both. Subtract: e(3.2 − 1.2) = h(12.8 − 8) × 10¹⁴. Step 2: h = 1.6 × 10⁻¹⁹ × 2 / 4.8 × 10¹⁴ ≈ 6.67 × 10⁻³⁴ J s. Step 3: φ = hν − eV₀ = 6.67 × 10⁻³⁴ × 8 × 10¹⁴ − 1.6 × 10⁻¹⁹ × 1.2 = 5.34 × 10⁻¹⁹ − 1.92 × 10⁻¹⁹ = 3.42 × 10⁻¹⁹ J ≈ 2.13 eV.
7. The threshold wavelength of a metal is 540 nm. Light of 360 nm falls on it. Find the stopping potential.
Step 1: φ = 1240/540 ≈ 2.30 eV. Step 2: hν = 1240/360 ≈ 3.44 eV. Step 3: KEmax = 3.44 − 2.30 = 1.14 eV. Step 4: V₀ = 1.14 V.
8. The intensity of light on a photocell is made 4 times and its frequency is kept the same. What happens to (a) saturation current, (b) stopping potential?
Step 1: Intensity = number of photons per second. 4 times intensity → 4 times photons → 4 times electrons. (a) Saturation current becomes 4 times. Step 2: Each photon still has the same energy hν, so KEmax = hν − φ is the same. (b) Stopping potential does not change.
Common mistakes
- Thinking brighter light gives faster electrons. Brightness changes only the number of electrons (current); speed depends on frequency.
- Forgetting to convert eV to joules (× 1.6 × 10⁻¹⁹) before using m or h in SI units.
- Writing hν = φ + KE for every electron. It is KEmax; most electrons lose some energy inside the metal and come out slower.
- Mixing up threshold frequency and threshold wavelength: longer wavelength means lower frequency, so light works only if λ is SHORTER than λ₀.