Dual Nature of Radiation and Matter Class 12 Notes

• For a given metal there is a threshold frequency
• ${\nu }_0$
• below which no electrons are emitted, however intense the light.
• The maximum kinetic energy of emitted electrons depends on the frequency, not the intensity, of light.
• The number of emitted electrons per second (the photocurrent) is proportional to the intensity.
• Emission is nearly instantaneous — there is no meaningful time lag.

• Photon energy:
• $E=h\nu =\frac{hc}{\lambda }$
• .
• Photon momentum:
• $p=\frac{h}{\lambda }$
• .
• Above the threshold:
• $K_{\max }=h\nu -{\varphi }_0$
• ; below it no current flows at any intensity.

• Slope of the V₀–ν graph:
• $\frac{h}{e}$
• .
• Different metals give parallel lines with the same slope but different intercepts (work functions).
• The saturation photocurrent rises with intensity at fixed frequency.

• Heavy or fast objects have tiny wavelengths — that is why macroscopic matter shows no observable wave behaviour.
• Increasing the accelerating voltage decreases the electron wavelength.
• Momentum of a photon can also be written
• $p=\frac{h\nu }{c}$
• .

• The effect appears only for certain accelerating voltages, where the de Broglie wavelength satisfies the diffraction condition.
• It confirms the wave nature of matter and completes the particle–wave duality of electrons.

Example: A metal has a work function of 2.0 eV. Light of wavelength 400 nm falls on it. Find the maximum kinetic energy of the emitted electrons.

Solution: Photon energy $E=\frac{hc}{\lambda }=\frac{1240\text{eVnm}}{400\text{nm}}=3.1\text{eV}$. Then $K_{\max }=3.1-2.0=1.1\text{eV}$. The electrons are released, so the photoelectric effect occurs.

Exercises (NCERT Q11.1–Q11.11)

• 11.1) Find the (a) maximum frequency, and (b) minimum wavelength of X-rays produced by 30 kV electrons.
• 11.2) The work function of caesium metal is 2.14 eV. When light of frequency 6 × 10¹⁴ Hz is incident on the metal surface, photoemission of electrons occurs. What is the (a) maximum kinetic energy of the emitted electrons, (b) stopping potential, and (c) maximum speed of the emitted photoelectrons?
• 11.3) The photoelectric cut-off voltage in a certain experiment is 1.5 V. What is the maximum kinetic energy of photoelectrons emitted?
• 11.4) Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. The power emitted is 9.42 mW. (a) Find the energy and momentum of each photon in the light beam, (b) How many photons per second, on the average, arrive at a target irradiated by this beam? (Assume the beam to have uniform cross-section which is less than the target area), and (c) How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?
• 11.5) In an experiment on photoelectric effect, the slope of the cut-off voltage versus frequency of incident light is found to be 4.12 × 10⁻¹⁵ V s. Calculate the value of Planck's constant.
• 11.6) The threshold frequency for a certain metal is 3.3 × 10¹⁴ Hz. If light of frequency 8.2 × 10¹⁴ Hz is incident on the metal, predict the cut-off voltage for the photoelectric emission.
• 11.7) The work function for a certain metal is 4.2 eV. Will this metal give photoelectric emission for incident radiation of wavelength 330 nm?
• 11.8) Light of frequency 7.21 × 10¹⁴ Hz is incident on a metal surface. Electrons with a maximum speed of 6.0 × 10⁵ m/s are ejected from the surface. What is the threshold frequency for photoemission of electrons?
• 11.9) Light of wavelength 488 nm is produced by an argon laser which is used in the photoelectric effect. When light from this spectral line is incident on the emitter, the stopping (cut-off) potential of photoelectrons is 0.38 V. Find the work function of the material from which the emitter is made.
• 11.10) What is the de Broglie wavelength of (a) a bullet of mass 0.040 kg travelling at the speed of 1.0 km/s, (b) a ball of mass 0.060 kg moving at a speed of 1.0 m/s, and (c) a dust particle of mass 1.0 × 10⁻⁹ kg drifting with a speed of 2.2 m/s?
• 11.11) Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).

Answers at a glance

1. 1
   Q11.1 → (a) ν_max = 7.24 × 10¹⁸ Hz. (b) λ_min = 4.14 × 10⁻¹¹ m.
2. 2
   Q11.2 → (a) K_max = 0.346 eV (0.35 eV). (b) V₀ = 0.35 V. (c) v_max = 3.49 × 10⁵ m/s.
3. 3
   Q11.3 → K_max = 1.5 eV = 2.4 × 10⁻¹⁹ J.
4. 4
   Q11.4 → (a) E = 3.14 × 10⁻¹⁹ J (1.96 eV), p = 1.05 × 10⁻²⁷ kg m s⁻¹. (b) 3.0 × 10¹⁶ photons/s. (c) v = 0.63 m/s.
5. 5
   Q11.5 → h = 6.6 × 10⁻³⁴ J s.
6. 6
   Q11.6 → V₀ = 2.03 V.
7. 7
   Q11.7 → No — the photon energy (3.77 eV) is less than the work function (4.2 eV).
8. 8
   Q11.8 → ν₀ = 4.74 × 10¹⁴ Hz.
9. 9
   Q11.9 → φ₀ = 2.17 eV.
10. 10
    Q11.10 → (a) 1.66 × 10⁻³⁵ m. (b) 1.11 × 10⁻³² m. (c) 3.01 × 10⁻²⁵ m.
11. 11
    Q11.11 → λ_EM = h/p = λ_dB, since the photon's momentum is p = h/λ.

MCQ Questions

1 Mark Questions

1. 1
   State the de Broglie relation for the wavelength of a moving particle.
2. 2
   Define the work function of a metal.
3. 3
   On what does the maximum kinetic energy of emitted photoelectrons depend?
4. 4
   Define the stopping potential for a given metal and frequency of incident light.

2 Mark Questions

1. 1
   Calculate the de Broglie wavelength of an electron accelerated through a potential difference of 100 V. (Ans: λ = 12.27/√V Å = 1.227 Å)
2. 2
   The work function of a metal is 2 eV. Find its threshold frequency. (Ans: ν₀ = W₀/h ≈ 4.8 × 10¹⁴ Hz)
3. 3
   Light of frequency 7.5 × 10¹⁴ Hz is incident on a metal of work function 2 eV. Find the maximum kinetic energy of the photoelectrons. (Ans: Kmax = hν − W₀ ≈ 1.1 eV)

3 Mark Questions

1. 1
   State and explain the laws of photoelectric emission, and give the experimental arrangement used to study them.
2. 2
   Derive Einstein's photoelectric equation hν = W₀ + ½mv² and explain the concept of threshold frequency.
3. 3
   Draw the stopping potential versus frequency graph for a metal and explain how its slope and intercept determine h/e and the work function. (Ans: slope = h/e, intercept = W₀/e)

5 Mark Questions

1. 1
   Describe the experimental study of the photoelectric effect (apparatus, observations, graphs of photocurrent versus potential and versus intensity) and explain why the wave theory of light fails to account for it while Einstein's photon picture succeeds.
2. 2
   Derive the de Broglie wavelength of an electron accelerated through a potential V and hence show that matter shows wave–particle duality; account for why wave nature was observed for electrons but not for macroscopic bodies.