IB Physics HLTopic 5 — Quantum PhysicsPaper 1 & 2Ek max = hf − Φ~16 min read
The Photoelectric Equation
Last page we met the photoelectric effect — light knocking electrons out of a metal. Now we turn that idea into a proper equation you can calculate with. Einstein realised the whole thing is just energy bookkeeping: a photon arrives carrying energy hf, part of it pays the “fee” to escape the metal (the work function), and whatever’s left over becomes the electron’s speed. One tidy equation ties it all together — and it’s the workhorse for every photoelectric calculation in the exam.
📘 What you need to know
The energy of a single photon: E = hf = hc/λ
Photon energy and frequency are directly proportional — higher frequency means a more energetic photon
The photoelectric equation: hf = Φ + Ek max
Maximum kinetic energy: Ek max = hf − Φ
The work functionΦ is the minimum energy to free an electron; Φ = hf0
At the threshold frequency f0: the photon energy exactly equals Φ, so Ek max = 0
Rearranged as Ek max = hf − Φ, it matches y = mx + c
On an Ek max-vs-f graph: gradient = h, x-intercept = f0, y-intercept = −Φ
All terms must be in the same units — all in eV, or all in joules
The energy of a photon
Einstein’s starting point: light comes in packets called photons, and each photon carries a fixed amount of energy set entirely by its frequency.
Energy of a photonE = hf = hc / λh = Planck’s constant (6.63 × 10−34 J s) • f = frequency (Hz) • λ = wavelength (m)
Higher frequency (bluer light) → more energetic photons. That’s the key: a photon with a greater frequency than the threshold will also carry greater energy than the work function, so it has energy to spare after freeing the electron.
Use E = hf when the question gives you a frequency, and E = hc/λ when it gives you a wavelength. They’re the same equation dressed differently, because c = fλ. Picking the right form up front saves you an unnecessary conversion step.
Splitting the energy
When a photon lands on a surface electron, its energy is divided into exactly two parts. First, some pays the work functionΦ — the “escape fee” to break free of the metal. Whatever remains becomes the electron’s kinetic energy.
At the threshold, all the photon’s energy is spent on the work function (yellow), leaving zero kinetic energy. Above the threshold, the extra energy becomes the electron’s maximum kinetic energy (blue).
Einstein’s photoelectric equationhf = Φ + Ek maxrearranged for the electron’s speed energy: Ek max = hf − Φ
hf photon energy
=
Φ escape fee
+
Ek max leftover speed
WE 1
Potassium has a work function of 2.30 eV. Light of wavelength 400 nm is shone onto a potassium surface. Calculate the maximum kinetic energy of the emitted photoelectrons, in eV. (h = 6.63 × 10−34 J s, c = 3.00 × 108 m s−1, 1 eV = 1.60 × 10−19 J)
Step 1 — photon energy (wavelength given, use hc/λ)E = hc/λ = (6.63 × 10⁻³⁴)(3.00 × 10⁴) / (400 × 10⁻⁹)E = 4.97 × 10⁻¹⁹ J = 3.11 eVStep 2 — photoelectric equationEₖ ₕₓₓ = hf − Φ = 3.11 − 2.30Eₖ ₕₓₓ = 0.81 eVWavelength given, so reach for E = hc/λ straight away. Converting to eV (divide by 1.60 × 10⁻¹⁹) lets you subtract the work function directly, since it’s already in eV.
The graph: reading off h, f0 and Φ
Because Ek max = hf − Φ has the exact shape of y = mx + c, plotting maximum kinetic energy against frequency gives a straight line that hands you three quantities at once.
Straight line: y = mx + c
Photoelectric equation
What you read off
gradient m
h
Planck’s constant
x-intercept
f0
Threshold frequency
y-intercept c
−Φ
Negative of the work function
Read the gradient to get Planck’s constant, the x-intercept to get the threshold frequency, and extend the line back to the y-axis to get −Φ.
WE 2
The straight-line graph of Ek max against frequency for a metal crosses the frequency axis at 4.0 × 1014 Hz. (a) Calculate the work function of the metal, in eV. (b) Light of frequency 1.0 × 1015 Hz is used. Calculate the maximum kinetic energy of the photoelectrons, in eV. (h = 6.63 × 10−34 J s, 1 eV = 1.60 × 10−19 J)
(a) Step 1 — x-intercept is the threshold frequencyΦ = hf₀ = (6.63 × 10⁻³⁴)(4.0 × 10¹⁴)Φ = 2.65 × 10⁻¹⁹ J = 1.66 eVΦ = 1.66 eV(b) Step 2 — photoelectric equation at f = 1.0 × 10¹⁵ Hzhf = (6.63 × 10⁻³⁴)(1.0 × 10¹⁵) = 6.63 × 10⁻¹⁹ J = 4.14 eVEₖ ₕₓₓ = hf − Φ = 4.14 − 1.66Eₖ ₕₓₓ = 2.49 eVThe x-intercept IS f₀, so Φ = hf₀ drops straight out. Then reuse the photoelectric equation with the new frequency. Convert both energies to eV before subtracting.
WE 3
On an Ek max-vs-f graph for a metal, two points on the straight line are (4.0 × 1014 Hz, 0 J) and (1.0 × 1015 Hz, 3.98 × 10−19 J). Use these to find an experimental value for Planck’s constant.
Step 1 — the gradient equals hh = gradient = ΔEₖ / ΔfStep 2 — substitute the two pointsh = (3.98 × 10⁻¹⁹ − 0) / (1.0 × 10¹⁵ − 4.0 × 10¹⁴)h = (3.98 × 10⁻¹⁹) / (6.0 × 10¹⁴)h = 6.63 × 10⁻³⁴ J sThis is exactly how the graph is used to MEASURE Planck’s constant in the lab. Gradient = rise over run, and here the “rise” is energy and the “run” is frequency — so the gradient carries units of J⋅s, which is Planck’s constant.
⚛ Working a photoelectric-equation question
Photon energy? Frequency given → E = hf. Wavelength given → E = hc/λ.
Pick one unit system and stick to it: everything in eV, or everything in J.
Max KE?Ek max = hf − Φ.
Work function from a graph?Φ = hf0, where f0 is the x-intercept.
Planck’s constant from a graph? It’s the gradient of the line.
Speed? Set Ek max = ½mv2 and rearrange for v.
💡 Top tips
Choose E = hf or E = hc/λ based on what the question gives you.
Keep every energy in the same unit before you add or subtract.
On the graph: gradient = h, x-intercept = f0, y-intercept = −Φ.
At the threshold, Ek max = 0, so Φ = hf0 exactly.
The gradient is the same for every metal — it’s always h. Only the intercepts change.
⚠ Common mistakes
Mixing eV and joules in the same equation
Reading the y-intercept as +Φ instead of −Φ
Using E = hf with a wavelength plugged in for f
Forgetting nm → m (400 nm = 400 × 10−9 m)
Thinking the gradient changes between metals — it’s always h
Confusing the threshold frequency f0 (x-intercept) with the work function Φ
Quick recap: A photon carries energy E = hf = hc/λ. When it hits a surface electron, Einstein’s hf = Φ + Ek max splits that energy into the work function escape fee plus the electron’s maximum kinetic energy. Rearranged as Ek max = hf − Φ it plots as a straight line: gradient h, x-intercept f0, y-intercept −Φ. Always keep energies in the same unit before combining them.
You now have the equation that makes the photoelectric effect calculable. But step back and notice what it really claims: energy arrives in indivisible packets, one photon feeding one electron, with no “topping up” allowed. That’s a radical break from the smooth-wave picture of light. Next we look at exactly why this forces us to treat light as a stream of particles — and where the wave model falls apart. Next page: Light as a Particle.
Photoelectric equation tripping you up?
Book a free meeting and we’ll practise hf = Φ + Ek max, unit conversions, and reading h, f0 and Φ off the graph.
