CBSE Class 12 Physics Notes: Dual Nature of Radiation and Matter

Key Concepts: Must Remember

• Photoelectric Effect: Emission of electrons from a metal surface when electromagnetic radiation of suitable frequency falls on it.
• Work Function (Φ₀): Minimum energy required for an electron to escape from a metal surface. Characteristic property of the metal.
• Threshold Frequency (ν₀): Minimum frequency of incident radiation below which no photoelectric emission occurs, regardless of intensity.
• Einstein's Photoelectric Equation: hν = Φ₀ + K_max, where K_max is the maximum kinetic energy of emitted photoelectrons.
• Stopping Potential (V₀): Minimum negative anode potential required to stop the photoelectric current. K_max = eV₀.
• Photon: Quantum of electromagnetic radiation. Energy E = hν = hc/λ, momentum p = h/λ.
• Wave Nature of Matter (de Broglie Hypothesis): All material particles in motion exhibit wave-like properties. Wavelength λ = h/p = h/mv.
• Davisson-Germer Experiment: Experimentally verified the wave nature of electrons through diffraction patterns.
• Intensity of light affects the number of emitted photoelectrons (photoelectric current), not their kinetic energy (for ν > ν₀).
• Frequency of light affects the kinetic energy of emitted photoelectrons (for ν > ν₀) and determines if emission occurs.

Essential Definitions

Photoelectric Effect

The phenomenon of emission of electrons from a metal surface when light of a sufficiently high frequency (or low wavelength) strikes it.

Work Function (Φ₀)

The minimum amount of energy required to remove an electron from the surface of a given metal, specific to each material.

Threshold Frequency (ν₀)

The minimum frequency of incident radiation below which no photoelectric emission takes place, irrespective of the intensity of the incident radiation.

Stopping Potential (V₀)

The minimum negative (retarding) potential applied to the anode with respect to the cathode, for which the photoelectric current becomes zero.

Photon

A quantum of electromagnetic energy, considered as an elementary particle, having zero rest mass and carrying energy hν and momentum h/λ.

de Broglie Wavelength

The wavelength associated with a moving particle, as proposed by Louis de Broglie, given by λ = h/p, where h is Planck's constant and p is the momentum of the particle.

Intensity of Radiation

The power incident per unit area. In the context of photons, it is proportional to the number of photons per unit area per unit time.

Understanding the Photoelectric Effect and De Broglie Hypothesis

The chapter "Dual Nature of Radiation and Matter" fundamentally challenges the classical understanding of light and particles. Historically, light was considered a wave, explaining phenomena like interference and diffraction. However, the photoelectric effect revealed limitations in this wave theory. Experiments showed that electron emission from a metal surface depends on the frequency of light, not its intensity. Below a certain threshold frequency (ν₀), no electrons are emitted, regardless of how bright the light is. Above this frequency, the number of emitted electrons (photoelectric current) is proportional to the intensity, and their maximum kinetic energy increases with frequency.

Albert Einstein explained this by proposing that light consists of discrete energy packets called photons. Each photon carries energy E = hν, where h is Planck's constant and ν is the frequency. When a photon strikes a metal, it transfers its entire energy to an electron. If this energy hν is greater than the work function (Φ₀) of the metal (the minimum energy to liberate an electron), the electron is emitted. The excess energy (hν - Φ₀) becomes the maximum kinetic energy (K_max) of the photoelectron. This particle nature of light explains all aspects of the photoelectric effect.

Conversely, Louis de Broglie hypothesized that if light can behave as both a wave and a particle, then matter particles (like electrons, protons, etc.) should also exhibit wave-like properties. He proposed that every moving particle has an associated wavelength, called the de Broglie wavelength, given by λ = h/p, where p is the momentum of the particle. This was a revolutionary idea, suggesting a fundamental wave-particle duality for all entities in the universe. While the wave nature of macroscopic objects is negligible due to their large momentum, it becomes significant for microscopic particles like electrons. The Davisson-Germer experiment later provided crucial experimental evidence for the wave nature of electrons, confirming de Broglie's hypothesis through electron diffraction patterns, similar to X-ray diffraction patterns.

Worked Examples & Formulas

• {"title":"Example 1: Photoelectric Effect","description":"Question: The work function of a metal is 3.0 eV. What is the threshold frequency for photoelectric emission?\n\nSolution: \nGiven, Work function Φ₀ = 3.0 eV. We need to convert this to Joules: Φ₀ = 3.0 × 1.602 × 10⁻¹⁹ J = 4.806 × 10⁻¹⁹ J.\nUsing the formula Φ₀ = hν₀, where h = 6.626 × 10⁻³⁴ Js.\nν₀ = Φ₀ / h = (4.806 × 10⁻¹⁹ J) / (6.626 × 10⁻³⁴ Js) ≈ 7.25 × 10¹⁴ Hz.\nThus, the threshold frequency is approximately 7.25 × 10¹⁴ Hz."}
• {"title":"Example 2: de Broglie Wavelength","description":"Question: Calculate the de Broglie wavelength of an electron moving with a speed of 1.0 × 10⁶ m/s.\n(Mass of electron m_e = 9.1 × 10⁻³¹ kg, Planck's constant h = 6.626 × 10⁻³⁴ Js).\n\nSolution: \nGiven, v = 1.0 × 10⁶ m/s, m_e = 9.1 × 10⁻³¹ kg.\nMomentum p = m_e v = (9.1 × 10⁻³¹ kg) × (1.0 × 10⁶ m/s) = 9.1 × 10⁻²⁵ kg m/s.\nUsing de Broglie wavelength formula λ = h/p.\nλ = (6.626 × 10⁻³⁴ Js) / (9.1 × 10⁻²⁵ kg m/s) ≈ 7.28 × 10⁻¹⁰ m.\nThe de Broglie wavelength is approximately 0.728 nm."}

Davisson-Germer Experiment: Verifying Electron Wave Nature

1. — Electrons are produced by a heated tungsten filament and accelerated through a potential difference V to a desired energy. This forms a fine beam of electrons.
2. — The electron beam is directed onto a nickel crystal. The crystal acts as a diffraction grating due to its regular atomic arrangement.
3. — A movable detector is used to measure the intensity of the scattered electrons at different scattering angles (φ) from the crystal surface.
4. — It was observed that the intensity of scattered electrons showed a strong peak at a specific angle (e.g., 50° for 54V accelerating potential). This peak corresponds to constructive interference, characteristic of a wave phenomenon.
5. — The observed diffraction peak was consistent with the Bragg's law for X-ray diffraction, if the electrons were assumed to have a de Broglie wavelength calculated by λ = h / √(2m_e eV). This agreement confirmed the wave nature of electrons.

Section 6

When solving numerical problems related to the photoelectric effect, always pay attention to the units. Energy is often given in electron-volts (eV), but Planck's constant h is in Joules-seconds (Js). Remember to convert eV to Joules (1 eV = 1.602 × 10⁻¹⁹ J) for consistency. Also, distinguish between the effect of intensity and frequency on photoelectric current and stopping potential – a common conceptual trap in exams. Intensity affects the number of photons, thus current; frequency affects photon energy, thus kinetic energy and stopping potential.

Practice Questions with Solutions

• Q: What is the main difference between the classical wave theory of light and Einstein's photon theory regarding the photoelectric effect? A: Classical wave theory predicts that the energy of emitted electrons should depend on light intensity and emission should occur at any frequency given sufficient intensity. Einstein's photon theory explains that emission depends on the frequency (photon energy) and only above a threshold frequency, with intensity affecting the number of emitted electrons.
• Q: Why is the de Broglie wavelength of a moving cricket ball not observed? A: The de Broglie wavelength is inversely proportional to momentum (λ = h/mv). For a macroscopic object like a cricket ball, its mass and velocity are relatively large, resulting in an extremely small de Broglie wavelength, which is practically impossible to detect or observe through diffraction effects.
• Q: What happens to the stopping potential if the intensity of incident light increases, keeping the frequency constant (and above threshold)? A: The stopping potential remains unchanged. Stopping potential depends on the maximum kinetic energy of the photoelectrons, which in turn depends only on the frequency of incident light (and work function), not its intensity.
• Q: Name the experiment that established the wave nature of electrons. A: The Davisson-Germer experiment.

Frequently Asked Questions