Photoelectric Effect: Work Function, Threshold Frequency and Stopping Potential

The photoelectric effect is the ejection of electrons from a metal surface when light shines on it. It matters because classical wave theory predicted the wrong behavior, and the correct explanation treats light as particles called photons, each carrying energy E = hf.

The key equation

KE_max = hf - φ

Here h is Planck's constant (6.63 × 10^-34 J·s), f is the light frequency, and φ is the work function, the minimum energy needed to free an electron from that metal. One photon gives all its energy to one electron. If hf is less than φ, no electron escapes, no matter how many photons arrive.

The threshold frequency is f₀ = φ/h. Below it nothing is emitted. Above it, the extra energy becomes kinetic energy of the electron. Using wavelength, E = hc/λ, with hc about 1240 eV·nm.

What the experiment showed

The most tested distinction: intensity changes the number of electrons (the current); frequency changes their maximum kinetic energy.

Stopping potential

Reverse the voltage between the plates until the current drops to zero. That voltage is the stopping potential, V_s, and it satisfies

eV_s = KE_max

A graph of KE_max against frequency is a straight line. Its slope is h, its x-intercept is f₀, and its y-intercept is -φ. The slope is the same for every metal, and only the intercept changes.

Worked example

Sodium has a work function of about 2.3 eV. Light of wavelength 400 nm hits it.

1. Photon energy: 1240 / 400 = 3.1 eV.
2. KE_max = 3.1 - 2.3 = 0.8 eV.
3. Stopping potential: 0.8 V, since the electron charge is one elementary charge.
4. Threshold wavelength: 1240 / 2.3, about 540 nm. Light of longer wavelength, such as red, ejects nothing.

If the intensity doubles, the current doubles but KE_max and V_s stay 0.8 eV and 0.8 V.

Common traps

• Thinking brighter light raises electron energy. Only frequency does.
• Mixing units. Use eV with 1240 eV·nm, or joules with h and f, but not both.
• Forgetting that KE_max is a maximum. Electrons freed from below the surface lose extra energy.
• Assuming a lower work function lowers the slope. The slope is always h.

One-minute recap

Photons carry E = hf. An electron leaves only if hf is at least φ, and then KE_max = hf - φ. Frequency sets electron energy, intensity sets electron count, and eV_s = KE_max. The KE versus f line has slope h and x-intercept f₀.