CBSE Class 12 Physics Dual Nature of Radiation and Matter Notes Set 03

1. Electron Emission

A metal contains free electrons that move about inside it but cannot normally leave the surface, because the positive ions pull them back. An electron can escape only if it is given a certain minimum energy.

Work function (φ₀): The minimum energy that must be supplied to an electron to just pull it out of a metal surface. It depends on the nature of the metal and the condition of its surface. Caesium (≈ 2.1 eV) has one of the lowest values; platinum (≈ 5.6 eV) has one of the highest.

Electron volt (eV): The energy gained by an electron when it is accelerated through a potential difference of 1 volt.
1 eV = 1.6 × 10⁻¹⁹ J

The energy needed to free electrons can be supplied in three ways:
(i) Thermionic emission: by heating the metal, so that electrons gain enough thermal energy to escape.
(ii) Field emission: by applying a very strong electric field (of the order of 10⁸ V m⁻¹) to the metal, which pulls electrons out.
(iii) Photoelectric emission: by shining light of suitable frequency on the metal surface. The electrons emitted are called photoelectrons.

2. Photoelectric Effect

The emission of electrons from a metal surface when light (electromagnetic radiation) of suitable frequency falls on it is called the photoelectric effect.
1. Heinrich Hertz first noticed it in 1887: sparks across a detector gap jumped more easily when the gap was lit by ultraviolet light.
2. Hallwachs and Lenard studied it in detail. A negatively charged zinc plate lost its charge when exposed to ultraviolet light, while an uncharged plate became positive. This showed that negatively charged particles (later identified as electrons) were being emitted.
3. For every metal there is a minimum frequency, called the threshold frequency ν₀, below which no electrons are emitted, however intense the light.
4. Zinc, cadmium and magnesium respond only to ultraviolet light. Alkali metals such as lithium, sodium, potassium, caesium and rubidium have low work functions and emit electrons even with visible light.

3. Experimental Study of the Photoelectric Effect

An evacuated glass tube contains a photosensitive plate C (emitter) and a metal plate A (collector). Light passes through a quartz window (quartz lets ultraviolet through) and falls on C. A battery whose voltage and polarity can be changed maintains a potential of A with respect to C, measured by the voltmeter. The photoelectrons that reach A produce a current, called the photoelectric current, measured by the microammeter.
Three quantities can be varied one at a time: the intensity of the light, the potential of A, and the frequency of the light.

4. Effect of Intensity and of Potential

(i) Effect of intensity: For a fixed frequency (above ν₀) and fixed accelerating potential, the photocurrent is directly proportional to the intensity of light. More intense light ejects more electrons per second.

(ii) Accelerating potential: As A is made more positive, the current rises until all the emitted electrons are collected. After this it stays constant; this maximum value is the saturation current. The saturation current is larger for greater intensity.

(iii) Retarding potential and stopping potential: When A is made negative with respect to C, electrons are repelled and the current falls. The minimum negative potential of A for which the current becomes exactly zero is the stopping potential (cut-off potential) V₀. It stops even the fastest electrons, so
K_max = \( \frac{1}{2} m v_{\max}^2 \) = eV₀
The graph shows that, for a given frequency, V₀ is the same for all intensities. So the maximum kinetic energy of photoelectrons does not depend on intensity.

5. Effect of Frequency of Incident Light

1. For light of the same intensity but different frequencies, the saturation current is the same, but the stopping potential is larger for higher frequency (V₀₃ > V₀₂ > V₀₁ when ν₃ > ν₂ > ν₁).
2. The stopping potential, and hence K_max, varies linearly with the frequency of the incident light.
3. The graph of V₀ against ν is a straight line which cuts the frequency axis at the threshold frequency ν₀. For frequencies below ν₀, no emission occurs at all.

Summary of experimental results:
1. For a given metal and frequency (ν ≥ ν₀), the photocurrent is proportional to the intensity of light.
2. The saturation current depends on intensity, but the stopping potential does not.
3. Above ν₀, the stopping potential (and K_max) increases linearly with frequency and is independent of intensity.
4. Below ν₀, no photoelectrons are emitted, however intense the light.
5. Emission is practically instantaneous (within about 10⁻⁹ s), even for very dim light.

6. Photoelectric Effect and the Wave Theory of Light

In the wave picture, light energy is spread continuously over the wavefront, and an electron absorbs energy gradually. This picture fails to explain the observations:
1. A more intense wave carries more energy, so K_max should increase with intensity. Experiment shows it does not.
2. Any frequency, if intense enough, should eventually supply enough energy to free electrons, so there should be no threshold frequency. Experiment shows a sharp threshold.
3. With dim light an electron would need a long time (hours, by estimate) to collect enough energy, so emission should be delayed. Experiment shows it is instantaneous.

7. Einstein's Photoelectric Equation

In 1905 Einstein proposed that light energy is not spread out but is carried in discrete packets called quanta (photons), each of energy hν. In photoelectric emission an electron absorbs one whole photon. Part of this energy is used to overcome the work function and the rest appears as kinetic energy of the electron.
K_max = hν − φ₀
Since K_max = eV₀ and φ₀ = hν₀:
eV₀ = hν − φ₀ = h(ν − ν₀)  and  V₀ = \( \frac{h}{e} \)ν − \( \frac{\varphi_0}{e} \)
Threshold wavelength: λ₀ = \( \frac{c}{\nu_0} \) = \( \frac{hc}{\varphi_0} \). Light of wavelength longer than λ₀ cannot cause emission.
Useful value: hc ≈ 1240 eV nm, so E (in eV) ≈ \( \frac{1240}{\lambda \text{ (in nm)}} \).

How the equation explains the observations:
1. K_max depends linearly on ν and not on intensity.
2. If hν < φ₀, no electron can escape, which gives the threshold frequency.
3. Greater intensity means more photons per second, so more electrons per second and a larger current, but each electron still gets only hν.
4. Absorption of a photon by an electron is a single, immediate event, so emission is instantaneous.
5. The slope of the V₀–ν graph is h/e, the same for all metals. Millikan used this to measure Planck's constant (h ≈ 6.63 × 10⁻³⁴ J s) and confirmed Einstein's equation.
6. Electrons below the surface lose some energy before escaping, so photoelectrons come out with kinetic energies ranging from zero up to K_max.

8. Particle Nature of Light: The Photon

The photoelectric effect shows that, in its interaction with matter, light behaves as if made of particles called photons.
1. Every photon of light of frequency ν has energy E = hν = \( \frac{hc}{\lambda} \) and momentum p = \( \frac{h\nu}{c} \) = \( \frac{h}{\lambda} \).
2. All photons travel at speed c in vacuum, whatever the frequency.
3. Photons are electrically neutral; they are not deflected by electric or magnetic fields.
4. The rest mass of a photon is zero.
5. In a photon–particle collision (such as photon–electron), total energy and total momentum are conserved, but the number of photons need not be: photons may be absorbed or created.
6. For light of a given frequency, a higher intensity means more photons crossing unit area per second, not more energy per photon.
Number of photons emitted per second by a source of power P: n = \( \frac{P}{h\nu} \) = \( \frac{P\lambda}{hc} \)

9. Wave Nature of Matter: de Broglie Hypothesis

Light shows wave properties (interference, diffraction, polarisation) and particle properties (photoelectric effect). In 1924 Louis de Broglie suggested that nature is symmetric, and that moving material particles should also have wave properties.
The wavelength associated with a particle of mass m moving with velocity v (momentum p) is the de Broglie wavelength:
λ = \( \frac{h}{p} \) = \( \frac{h}{mv} \)
1. The relation links a wave property (λ) to a particle property (p) through Planck's constant.
2. In terms of kinetic energy K: λ = \( \frac{h}{\sqrt{2mK}} \)
3. For a particle of charge q accelerated from rest through a potential difference V, K = qV, so λ = \( \frac{h}{\sqrt{2mqV}} \). For an electron this result gives λ = \( \frac{1.227}{\sqrt{V}} \) nm (V in volt).
4. The lighter the particle and the smaller its speed, the longer its wavelength. For a cricket ball or a moving car, λ is so small (about 10⁻³⁴ m) that no wave effect can ever be noticed. For electrons, λ is comparable to the spacing of atoms in a crystal, so electron waves can be diffracted; this is the principle of the electron microscope.
5. A photon and a particle with the same de Broglie wavelength have the same momentum.
6. Light and matter both show dual nature: the wave picture describes propagation and the particle picture describes interaction (emission, absorption). The two never show up in full in the same measurement.

10. Quick Formula Recap