In AP Physics 2, the de Broglie wavelength of an ejected electron is inversely related to its kinetic energy because λ = h/p and p = √(2mK), so λ = h/√(2mK); higher-energy photoelectrons have shorter matter wavelengths, which lets you rank wavelengths using frequencies and work functions.
Every moving particle has a de Broglie wavelength given by λ = h/p, where p is momentum. For an electron with kinetic energy K, momentum is p = √(2mK), so the wavelength becomes λ = h/√(2mK). Read that equation carefully. Kinetic energy sits in the denominator, so as K goes up, λ goes down. A faster electron is a "shorter-wavelength" electron.
This matters most in the photoelectric effect (Topic 15.5). When light above the threshold frequency hits a photoactive material, electrons get ejected with kinetic energy K_max = hf − φ, where φ is the work function. Shine higher-frequency light, or use a material with a smaller work function, and the electrons come out with more kinetic energy. By the inverse relationship, those electrons have shorter de Broglie wavelengths. That chain (frequency and work function → kinetic energy → wavelength) is exactly the reasoning the exam wants you to run.
This relationship lives in Unit 15 (Modern Physics) under Topic 15.5, supporting learning objective 15.5.A, which asks you to describe interactions between photons and matter using the photoelectric effect. It's also where wave-particle duality gets quantitative. The photoelectric effect proves light acts like particles (photons), and de Broglie's equation says matter acts like waves. This term ties both ideas into one calculation. On a deeper level, it tests whether you can chain equations instead of memorizing them in isolation, since you have to combine K_max = hf − φ with λ = h/√(2mK) to get anywhere.
Keep studying AP® Physics 2 Unit 15
Maximum kinetic energy (Unit 15)
K_max = hf − φ is the input to this relationship. The exam often gives you frequencies and work functions, expects you to find or compare K_max, and then asks about wavelength. The electron with the largest K_max always has the shortest de Broglie wavelength.
Threshold frequency (Unit 15)
Light exactly at the threshold frequency ejects electrons with essentially zero kinetic energy. Since λ = h/√(2mK), kinetic energy near zero means the de Broglie wavelength blows up toward infinity. The wavelength only gets short once you're well above threshold.
The Photoelectric Effect (Unit 15)
This is the home topic. The photoelectric effect supplies the electrons and their energies; the de Broglie relation tells you what wavelength those electrons carry. Together they show light behaving as particles and electrons behaving as waves in the same experiment.
Momentum and kinetic energy (earlier mechanics foundations)
The whole derivation hinges on p = √(2mK), which is just K = p²/2m rearranged. If you remember that classical link between momentum and kinetic energy, the inverse relationship falls out of λ = h/p in one step.
This shows up almost entirely as a ranking or comparison task. A typical multiple-choice stem gives you two or three light sources with different frequencies (or two materials with different work functions) and asks which ejected electron has the longest or shortest de Broglie wavelength. The move is always the same. First compute or compare K_max = hf − φ, then flip the order, because bigger K means smaller λ. Watch for trap answers that rank wavelengths in the same order as kinetic energy. No released FRQ has used this phrase verbatim, but free-response questions on the photoelectric effect reward exactly this kind of multi-step quantitative reasoning, especially when they ask you to justify a comparison "using equations" rather than just compute a number.
These are two different wavelengths in the same problem, and mixing them up is the classic error. The photon's wavelength describes the light hitting the material, and shorter light wavelength means higher frequency and more photon energy. The de Broglie wavelength describes the ejected electron, a matter wave found from λ = h/√(2mK). The connection runs through energy. Shorter incident light wavelength gives the electron more kinetic energy, which gives the electron a shorter de Broglie wavelength. Same direction, completely different objects.
The de Broglie wavelength of an electron is λ = h/√(2mK), so wavelength decreases as kinetic energy increases.
This comes from combining λ = h/p with p = √(2mK), the rearranged version of K = p²/2m.
In photoelectric effect problems, find K_max = hf − φ first, then rank wavelengths in the opposite order of the kinetic energies.
Higher incident frequency or a smaller work function means more electron kinetic energy and therefore a shorter de Broglie wavelength.
Light at exactly the threshold frequency ejects electrons with nearly zero kinetic energy, so their de Broglie wavelength is extremely long.
The photon's wavelength and the electron's de Broglie wavelength are different quantities; don't swap them mid-problem.
It's the fact that λ = h/√(2mK), so a particle with more kinetic energy has a shorter de Broglie wavelength. In AP Physics 2 it's used to rank the wavelengths of photoelectrons ejected by different light sources or from different materials.
No, it's the opposite. A faster electron has more momentum, and since λ = h/p, more momentum means a shorter de Broglie wavelength. This trips up a lot of people because energy and wavelength feel like they should grow together.
No. The light's wavelength belongs to the incident photons, while the de Broglie wavelength belongs to the ejected electron as a matter wave. Shorter-wavelength light does produce shorter-wavelength electrons, but they're two separate quantities calculated with separate equations.
First find the electron's kinetic energy from K_max = hf − φ using the light's frequency and the material's work function. Then plug into λ = h/√(2mK), where m is the electron mass and h is Planck's constant.
A smaller work function means less photon energy is spent freeing the electron, so more is left over as kinetic energy (K_max = hf − φ). More kinetic energy means more momentum, and λ = h/p makes the wavelength shorter.
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