Title 11. DUAL NATURE OF RADIATION AND MATTER.pdf ✅

From this graph, it is found that at a frequency, the value of the stopping potential is zero. This frequency is known as the threshold frequency for the photo metal used. The photoelectric effect occurs above this frequency and ceases below it. Therefore, threshold frequency is defined as the minimum frequency of incident radiation, below which the photoelectric emission is not possible completely. The threshold frequency is different for different metals. Laws Of Photoelectric Emission The experimental observations on photoelectric effect may be summarized as follows, which are known as the fundamental laws of photoelectric emission. (i) For a given photo sensitive material, there is a minimum frequency called the threshold frequency, below which emission of photoelectrons stops completely, however great the intensity may be. (ii) For a given photosensitive material, the photo electric current is directly proportional to the intensity of the incident radiation, provided the frequency is greater than the threshold frequency. (iii) The photoelectric emission is an instantaneous process. i.e. there is no time lag between the incidence of radiation and the emission of photo electrons. (iv) The maximum kinetic energy of the photo electrons is directly proportional to the frequency of incident radiation, but is independent of its intensity. Wave theory fails to explain the photoelectric effect (1) According to the wave theory of light, energy and intensity of wave depend on its amplitude. Hence intense radiation has higher energy and on increasing intensity, energy of photoelectrons should increase. But experimental results show that photoelectric effect is independent of intensity of light, but depends on the frequency of light. According to wave theory of light, energy of light has nothing to do with frequency. Hence change in energy of photoelectrons with change in frequency cannot be explained (2) Photons are emitted immediately (within 10-9 s) on making light incident on metal surface. Since the free electrons within metal are withheld under the effect of certain forces, and to bring them out, energy must be supplied Now if the incident energy is showing wave nature, free electrons in metal get energy gradually and when accumulates energy at least equal to work function then they escape from metal. Thus, electrons get emitted only after some time (3) According to wave theory of light, less intense light is ‘weak’ in terms of energy. To liberate photoelectron with such light one has to wait long till electron gather sufficient energy. Whereas experimental result shows that phenomenon depends on frequency and for low intensity light of appropriate frequency photoelectrons are emitted instantly Light waves and photons the electromagnetic theory of light proposed by Maxwell could not explain photoelectric effect. But Max Planck’s quantum theory successfully explains photoelectric effect. According to Planck’s quantum theory, light is emitted in the form of discrete packets of energy called ‘quanta’ or photon. The energy of each photon is E = hν, where h is Planck’s constant. Photon is neither a particle nor a wave. In the phenomena like interference, diffraction, polarization, the photon behaves like a wave. Energy of n photon E = n hν In the phenomena like emission, absorption and interaction with matter (photo electric effect) photon behaves as a particle. Hence light photon has a dual nature. Q. Let an electron requires 5 × 10−19 joule energy to just escape from the irradiated metal. If photoelectron is emitted after 10−9 s of the incident light, calculate the rate of absorption of energy. If this process is considered classically, the light energy is assumed to be continuously distributed over the wave front. Now, the electron can only absorb the light incident within a small area, say 10−19 m2 . Find the intensity of illumination in order to see the photoelectric effect Sol. Rate of absorption of energy is power P = E t = 5×10−19 10−9 = 5 × 10−10 J s From the definition of intensity of light I = Power Area = 5×10−10 10−19 = 5 × 109 J s. m2 Since, practically it is impossibly high energy, which suggest that explanation of photoelectric effect in classical term is not possible Q. Work function is 2eV. Light of intensity 10−5 W m−2 is incident on 2 cm2 area of it. If 1017 electrons of these metals absorb the light, in how much time does the photo electric effect start? Consider the waveform of incident light Sol. Intensity of incident light is 10−5 W m−2 Now intensity I = E A⋅t E = 10−5 × 2 × 10−4 × 1 = 2 × 10−9 J This energy is absorbed by 1017 electrons Average energy absorbed by each electron = 2×10-9 /1017 = 2×10-26 J Now, electron may get emitted when it absorbs energy equal to the work function of its metal = 2eV = 3.6×10-19 J Thus time required to absorb energy = 3.6×10-19 J / 2×10-26 J = 1.6×107 s
Einstein’s Photoelectric Equation In 1905, Albert Einstein successfully applied quantum theory of radiation to photoelectric effect. Plank had assumed that emission of radiant energy takes place in the quantized form, the photon, but once emitted it propagate in the form of wave. Einstein further assumed that not only the emission, even the absorption of light takes place in the form of photons. According to Einstein, the emission of photo electron is the result of the interaction between a single photon of the incident radiation and an electron in the metal. When a photon of energy hν is incident on a metal surface, its energy is used up in two ways: (i) A part of the energy of the photon is used in extracting the electron from the surface of metal, since the electrons in the metal are bound to the nucleus. This energy W spent in releasing the photo electron is known as photoelectric work function of the metal. The work function of a photo metal is defined as the minimum amount of energy required to liberate an electron from the metal surface. (ii) The remaining energy of the photon is used to impart kinetic energy to the liberated electron. If m is the mass of an electron and v, its velocity then Energy of the incident photon = Work function + Kinetic energy of the electron hv = φ0 + 1 2 mv 2 If the electron does not lose energy by internal collisions, as it escapes from the metal, the entire energy (hv −φ0 ) will be exhibited as the kinetic energy of the electron. Thus, (hv − φ0 ) represents the maximum kinetic energy of the ejected photo electron. If Vmax is the maximum velocity with which the photoelectron can be ejected, then hv = φ0 + 1 2 mvmax 2 −−(1) This equation is known as Einstein's photoelectric equation. When the frequency (v) of the incident radiation is equal to the threshold frequency (v0 ) of the metal surface, kinetic energy of the electron is zero. Then equation (1) becomes, hvo = φ0 ... (2) Substituting the value of W in equation (1) we get, hv −hv0 = 1 2 mvmax 2 − −(3) Or Kmax = hv− φ0 or eVo = hv− φ0 −−(4) This is another form of Einstein's photoelectric equation. Heisenberg's Uncertainty Principle According to Heisenberg's uncertainty principle, if the uncertainty in the x-coordinate of the position of a particle is Δx and uncertainty in the x-component of momentum is Δp (i.e. in one dimension) them Δx ⋅ Δp ≥ h 2π Similarly ΔE ⋅ Δt ≥ h 2π Particle Nature of Light: The Photon In interaction of radiation with matter, radiation behaves as if it is made up of particles called photons. (ii) Each photon has energy E (=hν) and momentum p (= hν/c), and speed c, the speed of light. (iii) All photons of light of a particular frequency ν, or wavelength λ, have the same energy E (=hν = hc/λ) and momentum p (= hν/c = h/λ), whatever the intensity of radiation may be. By increasing the intensity of light of given wavelength, there is only an increase in the number of photons per second crossing a given area, with each photon having the same energy. Thus, photon energy is independent of intensity of radiation. (iv) Photons are electrically neutral and are not deflected by electric and magnetic fields. (v) In a photon-particle collision (such as photon-electron collision), the total energy and total momentum are conserved. However, the number of photons may not be conserved in a collision. The photon may be absorbed or a new photon may be created. (v) Mass of photon m = E/c2 De Broglie's Wavelength of Matter Waves de Broglie equated the energy equations of Planck (wave) and Einstein (particle). For a wave of frequency v, the energy associated with each photon is given by Planck's relation, E = h ∨ where h is Planck's constant. According to Einstein's mass energy relation, a mass m is equivalent to energy, E = mc 2 where c is the velocity of light. If, hv = mc 2 ∴ hc λ = mc 2 or λ = h mc For a particle moving with a velocity v, if c = v λ = h mv = h p where p = mv, the momentum of the particle. These hypothetical matter waves will have appreciable wavelength only for very light particles. Q. Light of frequency 1.5 times the threshold frequency is incident on a photosensitive material. What will be the photoelectric current if the frequency is halved and intensity is doubled? Sol. Initially, υ = 1.5 υ0 If the frequency is halved, υ′ = υ 2 = 1.5 υ0 2 < υ0 Hence, no photoelectric emission will take place