UNIT 1: PHOTONS AND QUANTIZED ENERGY SF7 1 1.1 Planck s Quantum Theory The foundation of the Planck s quantum theory is a theory of black body radiation. Black body is defined as an ideal system that absorbs all the radiation incident on it. The electromagnetic radiation emitted by the black body is called black body radiation. The spectrum of electromagnetic radiation emitted by the black body (experimental result) is shown in figure 1.1a. Experimental result Rayleigh - Jeans theory Classical physics Wien s theory From the fig. 1.1a, the Rayleigh-Jeans and Wien s theories failed to fit the experimental curve because this two theories based on classical ideas. The classical ideas are Energy of the e.m. radiation is not depend on its frequency or wavelength. Energy of the e.m. radiation is continuously. SF7 Fig. 1.1a
In 19, Max Planck proposed his theory that is fit with the experimental curve in fig. 1.1a at all wavelengths known as Planck s quantum theory. The assumptions made by Planck in his theory are : The e.m. radiation emitted by the black body is a discrete (separate) packets of energy known as quanta. This means the energy of e.m. radiation is quantised. The energy size of the radiation depended on its frequency. According to this assumptions, the quantum E of the energy for radiation of frequency f is given by where E = hf h : Planck constant 34 = 6. 63 1 J s (1.1a) Planck s quantum theory c = fλ Since the speed of electromagnetic wave in a vacuum is, then eq. (1.1a) can be written as hc E = λ (1.1b) From the eq. (1.1b), the quantum E of the energy for radiation is inversely proportional to its wavelength. SF7 3 It is convenient to express many quantum energies in electronvolts. The electronvolt (ev) is a unit of energy that can be defined as the kinetic energy gained by an electron in being accelerated by a potential difference (voltage) of 1 volt. Unit conversion : 19 1eV = 1. 6 1 In 195, Albert Einstein extended Planck s idea by proposing that electromagnetic radiation is also quantised. It consists of particle like packets (bundles) of energy called photons of electromagnetic radiation. 1. Photons and Electromagnetic Waves Energy Photon is defined as a particle with zero mass consisting of a quantum of electromagnetic radiation where its energy is concentrated. A photon may also be regarded as a unit of energy equal to hf. Photons travel at the speed of light in a vacuum. They are required to explain the photoelectric effect and other phenomena that require light to have particle property. J SF7 4
Table 1.a shows the differences between the photon and electromagnetic wave. E.M. Wave Energy of the e.m. wave depends on the intensity of the wave. Intensity of the wave is proportional to the squared of its amplitude where I A Its energy is continuously and spread out through the medium as shown in figure 1.a. Photon Energy of a photon is proportional to the frequency of the e.m.w. where E f E = hf Its energy is discrete as shown in figure 1.b. Fig. 1.a Table 1.a Photon Fig. 1.b SF7 5 Example 1 : A photon of green light has a wavelength of 5 nm. Calculate the photon s frequency and energy (in joules and electronvolts). (Given the speed of photon in the vacuum, c = 3. x 1 8 m s -1 and Planck constant, h = 6.63 x 1-34 J s) Solution: λ=5x1-9 m By applying the equation relates c, f and λ, thus the photon s frequency is c = λf 14 f = 5. 77 1 Hz By using the equation of Planck s quantum theory, thus the photon s energy is E = hf 19 In electronvolt : E = 3. 83 1 19 3. 83 1 E = 19 1. 6 1 Example : (exercise) For waves propagating in air, calculate the energy of a photon in electronvolts of a. gamma rays of wavelength 4.61 x 1-14 m. b. visible light of wavelength 5.1 x 1-7 m. Ans. :.7 x 1 7 ev,.39 ev SF7 6 J E =. 39 ev
1.3 The Photoelectric Effect Definition is defined as the emission of electron from the surface of a metal when the e.m.. radiation (light) of higher frequency strikes its surface. Figure 1.3a shows the emission of the electron from the surface of the metal after shining by the light. light - photoelectron - - - - - - - - - - Metal Fig. 1.3a Free electrons Photoelectron is defined as an electron emitted from the surface of the metal when light strikes its surface. The photoelectric effect can be studied through the experiment made by Hertz in 1887. SF7 7 1.3.1 Experiment of Photoelectric Effect Figure 1.3b shows a schematic diagram of an experimental arrangement for studying the photoelectric effect. cathode photoelectron - - - vacuum V power supply e.m.. radiation (light) anode glass G rheostat The set-up as follows : Fig. 1.3b Two conducting electrodes, the anode (positive electric potential) and the cathode (negative electric potential) are encased in an evacuated tube (vacuum). The monochromatic light of known frequency and intensity are incident on the cathode. SF7 8
Explanation of the experiment : When a monochromatic light of suitable frequency (or wavelength) shines on the cathode, photoelectrons are emitted. These photoelectrons are attracted to the anode and give rise to a photoelectric current or photocurrent I which is detected by the galvanometer. When the positive voltage (potential difference) is increased, more photoelectrons reach the anode, hence the photoelectric current also increase. As positive voltage becomes sufficiently large, the photoelectric current reaches a maximum constant value I m, called saturation current. Saturation current is defined as the maximum constant value of photocurrent when all the photoelectrons have reached the anode. If the positive voltage is gradually decreased, the photoelectric current I also decreases slowly. Even at zero voltage there are still some photoelectrons with sufficient energy reach the anode and the photoelectric current flows is I. SF7 9 Finally, when the voltage is made negative by reversing the power supply terminal as shown in figure 1.3c, the photoelectric current decreases even further to very low values since most photoelectrons are repelled by anode which is now negative electric potential. e.m.. radiation (light) cathode photoelectron - - - vacuum V power supply anode glass G rheostat Fig. 1.3c : reversing power supply terminal As the potential of the anode becomes more negative, less photoelectrons reach the anode thus the photoelectric current drops until its value equals zero which the electric potential at this moment is called stopping potential (voltage) V s. Stopping potential is defined as the minimum value of negative voltage when there are no photoelectrons reaching the anode. SF7 1
The potential energy U due to this retarding voltage V s now equals the maximum kinetic energy K max of the photoelectron. U = K max 1 ev s = mv (1.3a) The variation of photoelectric current I as a function of the voltage V can be shown through the graph in figure 1.3d. Photoelect ric current, I I m I V s Voltage,V After Fig. 1.3d Before reversing the terminal SF7 11 1.3. Einstein s theory of Photoelectric Effect A photon is a packet of electromagnetic radiation with particle-like like characteristic and carries energy E given by E = hf and this energy is not spread out through the medium. Work function W of a metal Is defined as the minimum energy of e.m.. radiation required to emit an electron from the surface of the metal. It depends on the metal used. Equation : and E min = hf W = E min W f = h where f is called threshold frequency and is defined as the minimum frequency of e.m.. radiation required to emit an electron from the surface of the metal. Since then c = fλ λ = (1.3b) SF7 1 c f (1.3c)
where λ is called threshold wavelength and is defined as the maximum wavelength of e.m.. radiation required to emit an electron from the surface of the metal. Table 1.3a shows the work functions of several elements. Table 1.3a Element Aluminum Sodium Copper Gold Silver Einstein s photoelectric equation : Work function (ev( ev) In photoelectric effect, Einstein summarizes that some of the energy E imparted by a photon is actually used to release an electron from the surface of a metal (i.e. to overcome the binding force) and that the rest appears as the maximum kinetic energy of the emitted electron (photoelectron). It given by 1 E = K max + W where E = hf and K max = mv 1 hf mv + W = (1.3d) (1.3d) SF7 13 4.3.7 4.7 5.1 4.3 Einstein s photoelectric eq. Since s then eq. (1.3d) can be written as Note : 1 mv = ev hf = ev s + W (1.3e) V s where : stopping voltage e : magnitude of the electron charge First case : hf >W or f >f Second case : hf=w or f =f hf - v max K max hf - v= W Metal - Electron is emitted with maximum kinetic energy. Metal - W K max = Electron is emitted but maximum kinetic energy is zero. Third case : hf<w or f <f hf W SF7 Metal 14 No electron is emitted. -
Example 3 : Sodium has a work function of.3 ev. Calculate a. its threshold frequency, b. the maximum speed of the photoelectrons produced when the sodium is illuminated by light of wavelength 5 nm, c. the stopping potential with light of this wavelength. (Given c = 3. x 1 8 m s -1, h = 6.63 x 1-34 J s, 1 ev=1.6 x 1-19 J, mass of electron m = 9.11 x 1-31 kg, e = 1.6 x 1-19 C) Solution: W =.3 x (1.6x1-19 )= 3.68 x1-19 J a. The threshold frequency, f is given by W = hf f = 5. 55 1 b. Given λ=5 x 1-9 m By using the Einstein s photoelectric equation, hence the maximum speed of the photoelectrons is 1 hf + hc 1 = mv + W λ = mv W and SF7 15 14 f = Hz c λ v =. 56 1 5 m s -1 c. The stopping voltage V s is given by 1 ev s = mv Example 4 : In an experiment of photoelectric effect, no current flows through the circuit when the voltage across the anode and cathode is -1.7 V. Calculate a. the work function, and b. the threshold wavelength of the metal (cathode) if it is illuminated by ultraviolet radiation of frequency 1.7 x 1 15 Hz. (Given c = 3. x 1 8 m s -1, h = 6.63 x 1-34 J s, 1 ev=1.6 x 1-19 J, mass of electron m = 9.11 x 1-31 kg, e = 1.6 x 1-19 C) Solution: V s =1.7 V, f=1.7x1 15 Hz a. By using the Einstein s photoelectric equation, hence the work function is V s hf = ev s + W W = 8. 55 1 =. 187 V 19 J SF7 16
b. The threshold wavelength is W = hf and W λ f hc = λ =. 33 1 Example 5 : (exercise) The energy of a photon from an electromagnetic wave is.5 ev a. Calculate its wavelength. b. If this electromagnetic wave shines on a metal, photoelectrons are emitted with a maximum kinetic energy of 1.1 ev. Calculate the work function of this metal in joules. (Given c = 3. x 1 8 m s -1, h = 6.63 x 1-34 J s, 1 ev=1.6 x 1-19 J, mass of electron m = 9.11 x 1-31 kg, e = 1.6 x 1-19 C) Ans. : 553 nm, 1.84 x 1-19 J c = λ 7 m SF7 17 Example 6 : (exercise) In an experiment on the photoelectric effect, the following data were collected. Wavelength of e.m. Stopping radiation,λ (nm) potential, V s (V) 35 45 a. Calculate the maximum velocity of the photoelectrons when the wavelength of the incident radiation is 35 nm. b. Determine the value of the Planck constant from the above data. (Given c = 3. x 1 8 m s -1, 1 ev=1.6 x 1-19 J, mass of electron m = 9.11 x 1-31 kg, e = 1.6 x 1-19 C) Ans. : 7.73 x 1 5 m s -1, 6.7 x 1-34 J s Example 7 : (exercise) In a photoelectric effect experiment it is observed that no current flows unless the wavelength is less than 57 nm. Calculate a. the work function of this material in electronvolts. b. the stopping voltage required if light of wavelength 4 nm is used. (Given c = 3. x 1 8 m s -1, h = 6.63 x 1-34 J s, 1 ev=1.6 x 1-19 J, mass of electron m = 9.11 x 1-31 kg, e = 1.6 x 1-19 C) (Giancoli,pg. 974,no.15) SF7 18 Ans. :.18 ev,.9 V 1.7.9
1.3.3 Graphs in Photoelectric Effect Variation of photoelectric current I with voltage V for the radiation of different intensities but its frequency is fixed. Photoelect ric current, I I m Intensity x Fig. 1.3e : graph of I against V I m Intensity 1x V s Voltage,V Explanation: From the experiment, the photoelectric current is directly proportional to the intensity of the radiation as shown in figure 1.3f. Photoelect ric current, I I m I m 1 Light intensity SF7 19 For the radiation of different frequencies but its intensity is fixed. Fig. 1.3f : graph of I against V Stopping voltage, V s V s1 W e Photoelect ric current, I f I m f 1 f > f 1 V s V s1 Explanation: From the Einstein s photoelectric equation, V s h W hf = ev s + W Vs = f f f 1 f e = M Voltage,V SF7 Y X e + C frequency, f When V s =, hf = e( ) + W W = hf f
For the different metals of cathode but the intensity and frequency of the radiation are fixed. Fig. 1.3g : graph of I against V W 1 Stopping voltage, hf e V s1 V s Photoelect ric current, I I m W > W 1 W V s1 V s Explanation: From the Einstein s photoelectric equation, V s 1 hf hf = ev s + W Vs = W + e e Y = M X + C Voltage,V E = hf W 1 W W Energy of a photon in e.m.. radiation SF7 1 Variation of stopping voltage V s with frequency f of the radiation for different metals of cathode but the intensity is fixed. Stopping voltage, V s W 1 W W 3 Fig. 1.3h : graph of V s against f W 3 >W > W 1 f 1 f f 3 frequency, f Explanation: Since W =hf then hf = ev s + W W f h s = e Threshold (cut-off) frequency W f e V = M SF7 Y X + C When V s =, hf = e( ) + W W = hf f
1.4 Quantization of light Table below shows the classical predictions, photoelectric experimental observation and modern theory explanation of experimental observation. Classical predictions Experimental observation Modern theory The higher the intensity, the greater the energy imparted to the metal surface for emission of photoelectrons. When the intensity is low, the energy of the radiation is too small for emission of electrons. Very low intensity but high frequency radiation could emit photoelectrons. The maximum kinetic energy of photoelectrons is independent of light intensity. The intensity of light is the number of photons radiated per unit time on a unit surface area. Based on Einstein s photoelectric equation: Kmax = hf W The maximum kinetic energy of photoelectron depends only on the light frequency and the work function. If the light intensity is doubled, the number of electrons emitted also doubled but the maximum kinetic energy remains unchanged. SF7 3 Classical predictions Emission of photoelectrons occur for all frequencies of light. Energy of light is independent of frequency. Energy of light depends only on amplitude ( or intensity) and not on frequency. Experimental observation Emission of photoelectrons occur only when frequency of the light exceeds the certain frequency which value is characteristic of the material being illuminated. Energy of light depends on frequency. Modern theory When the light frequency is greater than threshold frequency, a higher rate of photons striking the metal surface results in a higher rate of photoelectrons emitted. If it is less than threshold frequency no photoelectrons are emitted. Hence the emission of photoelectrons depend on the light frequency According to Planck s quantum theory which is E=hf Energy of light depends on its frequency. SF7 4
Classical predictions Light energy is spread over the wavefront, the amount of energy incident on any one electron is small. An electron must gather sufficient energy before emission, hence there is time interval between absorption of light energy and emission. Time interval increases if the light intensity is low. Experimental observation Photoelectrons are emitted from the surface of the metal almost instantaneously after the surface is illuminated, even at very low light intensities. Modern theory The transfer of photon s energy to an electron is instantaneous as its energy is absorbed in its entirely, much like a particle to particle collision. The emission of photoelectron is immediate and no time interval between absorption of light energy and emission. Experimental observations deviate from classical predictions based on Maxwell s e.m. theory. Hence the classical physics cannot explain the phenomenon of photoelectric effect. The modern theory based on Einstein s photon theory of light can explain the phenomenon of photoelectric effect. It is because Einstein postulated that light is quantized and light is emitted, transmitted and reabsorbed as photons. SF7 5 Example 8 : In a photoelectric experiments, a graph of the light frequency f is plotted against the maximum kinetic energy K max of the photoelectron as shown in figure below. f 1 14 Hz. K max ( ev ) Based on the graph, for the light frequency of 6. x 1 14 Hz, calculate a. the threshold frequency. b. the maximum kinetic energy of the photoelectron. c. the maximum velocity of the photoelectron. (Given c = 3. x 1 8 m s -1, h = 6.63 x 1-34 J s, 1 ev=1.6 x 1-19 J, mass of electron m = 9.11 x 1-31 kg, e = 1.6 x 1-19 C) SF7 6
Solution: f=6.x1 14 Hz a. By rearranging Einstein s photoelectric equation, hf = K max + W Kmax = hf W 14 f 1 Hz Y = M X + C f=, Kmax = h( ) W = K max When f=, W. K max ( ev ) From the graph, W =(.)(1.6x1-19 )=3.x1-19 J The threshold frequency is W = hf 14 f = 4. 83 1 Hz b. By applying the Einstein s photoelectric equation, thus hf = K max + W Kmax = 7. 78 1 J c. The maximum velocity of the photoelectron is 1 v = 4. 13 1 5-1 K max = mv SF7 7 m s Example 9 : (exercise) A photocell with cathode and anode made of the same metal connected in a circuit as shown in the figure below. Monochromatic light of wavelength 365 nm shines on the cathode and the photocurrent I is measured for various values of voltage V across the cathode and anode. The result is shown in the graph. I(nA) 365 nm G 5 V a. Calculate the maximum kinetic energy of the photoelectron. b. Deduce the work function of the cathode. c. If the experiment is repeated with monochromatic light of wavelength 313 nm, determine the new intercept with the V-axis for the new graph. (Given c = 3. x 1 8 m s -1, h = 6.63 x 1-34 J s, 1 ev=1.6 x 1-19 J, mass of electron m = 9.11 x 1-31 kg, e = 1.6 x 1-19 C) Ans. : 1.6 x 1-19 J, 3.85 x 1-19 J, -1.57 V SF7 8 1 V (V )
THE END Next Unit UNIT 11 : Wave Particle Duality SF7 9