Revise Photoelectric effect for Cambridge 9702 (22.2)

The photoelectric effect describes the emission of electrons, known as photoelectrons, from a metal surface when electromagnetic radiation (light) of sufficiently high frequency shines on it. It’s a direct interaction that reveals a lot about the quantum nature of light and matter.

To explain this effect, we model light not just as a wave, but as discrete bundles of energy called photons. Each photon carries an amount of energy directly proportional to its frequency. This fundamental relationship is given by Planck's equation.

E = hf

Where: E = energy of a single photon (J) h = Planck's constant (6.63 × 10⁻³⁴ Js) f = frequency of the electromagnetic radiation (Hz)

Since frequency and wavelength are related by c = fλ, we can also express the photon energy in terms of wavelength:

E = hc / λ

Where: c = speed of light (3.00 × 10⁸ m/s) λ = wavelength of the radiation (m)

For an electron to escape from a metal, it needs a minimum amount of energy to overcome the attractive forces holding it within the material. This minimum energy is called the work function (Φ). Each metal has a unique work function, typically measured in Joules (J) or electronvolts (eV).

Because photon energy depends on frequency (E=hf), there must be a minimum frequency of light, called the threshold frequency (f₀), for electrons to be emitted. If the incident light's frequency is below f₀, no electrons will be emitted, no matter how bright or intense the light is. This is because individual photons simply don't have enough energy. The relationship is:

Φ = hf₀

When a photon with sufficient energy (hf > Φ) strikes an electron, it transfers all its energy in a one-to-one interaction. If this energy is greater than the work function, the electron is ejected. Any energy remaining after overcoming the work function becomes the kinetic energy of the emitted photoelectron. This is the principle of conservation of energy.

hf = Φ + KE_max

Where: hf = energy of the incident photon (J) Φ = work function of the metal (J) KE_max = maximum kinetic energy of the emitted photoelectron (J)

KE_max represents the kinetic energy of electrons emitted from the surface of the metal, which require the least energy to escape. Electrons from deeper within the metal may lose more energy on their way out and will have less kinetic energy.

The photoelectric effect provided critical evidence for the particle nature of light because classical wave theory could not explain the following key observations:

We can measure the maximum kinetic energy of photoelectrons using a concept called stopping potential (V_s). By applying a reverse potential difference in the experimental circuit, we can repel the emitted electrons. The stopping potential is the minimum voltage required to stop even the most energetic electrons from reaching the collector, thus reducing the photocurrent to zero.

The work done by this potential on an electron (charge e) is eV_s. To stop the fastest electron, this work done must equal its maximum kinetic energy:

KE_max = eV_s

Where: e = elementary charge (1.60 × 10⁻¹⁹ C) V_s = stopping potential (V)

This allows us to rewrite Einstein's equation in a form that is very useful for interpreting experimental data:

hf = Φ + eV_s

In atomic and subatomic physics, energies are often very small, so we use a more convenient unit called the electronvolt (eV). One electronvolt is defined as the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 volt.

1 eV = 1.60 × 10⁻¹⁹ J