Photoelectric Effect

11.3 Photoelectric Effect

The photoelectric effect is the emission of electrons from a metal surface when light of suitable frequency falls on it. This discovery, which could not be explained by classical wave theory, became one of the cornerstones of quantum mechanics.

11.3.1 Hertz's Observations

In 1887, Heinrich Hertz was conducting experiments to produce and detect electromagnetic waves using spark discharges. He noticed an unexpected phenomenon: when ultraviolet light from an arc lamp fell on the emitter plate of his detector loop, the sparks across the detector gap became much more vigorous.

Hertz had accidentally discovered that light could help electrons escape from a metal surface. The ultraviolet light was providing energy to free electrons near the metal surface, enabling them to overcome the attractive forces holding them within the metal. Once these electrons gained sufficient energy from the incident light, they escaped into the surrounding space.

11.3.2 Hallwachs' and Lenard's Detailed Investigations

Wilhelm Hallwachs and Philipp Lenard carried out systematic studies of this phenomenon between 1886 and 1902, using an evacuated glass tube containing two metal electrodes.

The basic experimental setup: An evacuated tube contains an emitter plate (C) and a collector plate (A). When ultraviolet radiation falls on the emitter plate, a current flows in the external circuit. The moment the ultraviolet light is stopped, the current stops. This shows that light falling on the emitter causes electrons to be ejected, which are then attracted to the positive collector plate by the electric field, producing a current.

Hallwachs, in 1888, performed a simpler experiment: he connected a negatively charged zinc plate to an electroscope. When he illuminated the zinc plate with ultraviolet light, the electroscope showed that the plate lost its negative charge. An uncharged zinc plate became positively charged when irradiated. A positively charged zinc plate became even more positively charged. The only consistent explanation was that negatively charged particles were being emitted from the zinc plate under the action of ultraviolet light.

After J.J. Thomson discovered the electron in 1897, it became clear that these emitted particles were electrons. The incident light causes electrons to be emitted from the emitter plate. Because of their negative charge, these photoelectrons are pushed toward the positive collector plate by the electric field.

The crucial discovery of threshold frequency: Hallwachs and Lenard observed that for a given metal, no electrons were emitted at all if the frequency of the incident light was below a certain minimum value. This minimum frequency is called the threshold frequency ($\nu_0$), and it depends on the nature of the metal.

Different metals have different sensitivities:

• Metals like zinc, cadmium, and magnesium respond only to ultraviolet light (short wavelength, high frequency).
• Alkali metals such as lithium, sodium, potassium, caesium, and rubidium are sensitive even to visible light.

All these photosensitive substances emit electrons when illuminated by light of sufficient frequency. The emitted electrons are called photoelectrons, and the phenomenon is called the photoelectric effect.

Key Experimental Observations from Hallwachs and Lenard

The investigators studied how the photocurrent varied with three factors:

1. The potential difference between the emitter and collector plates
2. The frequency of the incident light
3. The intensity of the incident light

These systematic studies revealed several properties that classical physics could not explain.

The Photoelectric Equation and Einstein's Explanation

The complete theoretical explanation came from Albert Einstein in 1905. He proposed that light consists of discrete packets of energy called photons, each with energy $E = h\nu$, where $h$ is Planck's constant ($6.63 \times 10^{-34} \text{ J s}$) and $\nu$ is the frequency of light.

When a photon strikes a metal surface, its entire energy is transferred to a single electron. The electron uses part of this energy to overcome the attractive forces holding it in the metal — this minimum energy required to remove an electron from the metal surface is called the work function ($\phi_0$ or $W_0$).

Where:

• $h\nu$ = energy of the incident photon
• $\phi_0$ = work function of the metal (minimum energy to remove an electron)
• $K_{\text{max}}$ = maximum kinetic energy of the emitted photoelectron

The maximum kinetic energy of the emitted photoelectron is:

This equation explains all the experimental observations:

1. Threshold frequency: If $h\nu < \phi_0$, no electron can be emitted. The threshold frequency $\nu_0$ is given by:

2. Effect of intensity: Increasing the intensity of light (at a fixed frequency above threshold) increases the number of photons per second, which increases the number of photoelectrons emitted per second — hence the photocurrent increases. But each photon still has the same energy $h\nu$, so the maximum kinetic energy of individual electrons does not change.

3. Instantaneous emission: Energy transfer from a photon to an electron is a single quantum event, occurring in about $10^{-9}$ seconds — there is no need for energy to accumulate over time.

Properties of Photoelectric Emission (Complete Derivation)

The section lists several properties that follow from Einstein's photoelectric equation. Each is derived below.

Property I: For a given photosensitive material and frequency of incident radiation (above threshold), the photoelectric current is directly proportional to the intensity of incident light.

Proof: The number of photoelectrons emitted per second is proportional to the number of photons incident per second. The number of photons per second is proportional to the intensity of light (since intensity = energy per unit area per second = $N h\nu$, where $N$ is the number of photons per second per unit area). Therefore, photocurrent $\propto$ intensity.

Property II: For a given photosensitive material and frequency of incident radiation, the stopping potential $V_0$ is independent of intensity. …