threshold frequency

Meaning: Light comes in photons, each with energy hf. An electron needs a minimum energy, the work function φ, to escape the metal. One photon gives its energy to one electron, so the photon must carry at least φ. The frequency at which hf exactly equals φ is the threshold frequency. Below it, no electrons are emitted regardless of intensity or exposure time. Above it, emission begins essentially at once. Each metal has its own f₀.

Equation: hf₀ = φ, so f₀ = φ/h. Above threshold, KEmax = hf − φ = h(f − f₀). Here h = 6.63 × 10⁻³⁴ J·s (Planck's constant), f and f₀ are in Hz, φ and KEmax are in J (1 eV = 1.60 × 10⁻¹⁹ J).

On the MCAT: A graph of KEmax versus f is a straight line. Its slope is h, its x-intercept is f₀, and its y-intercept is −φ. Different metals give parallel lines with different intercepts. A lower work function means a lower threshold frequency. Raising intensity at fixed frequency increases the number of electrons (the current) but not KEmax. Raising frequency raises KEmax. Stopping potential questions use eV = KEmax.

Common confusion: Students expect that bright light of low frequency should eventually eject electrons, as a classical wave picture would predict. It does not. Intensity sets the number of photons, not the energy of each one. In wavelength terms, the threshold is a maximum wavelength, λ₀ = c/f₀.