DHS 19 Quantum Physics (Tutorial Solutions)

DHS Physics H2 Year 6 / 2025 Topic 19 Quantum Physics Tutorial Suggested Answers 1 For Internal Use Only 21-May-25 Photoelectric Effect 1. Estimate the rate of emission of photons emitted by a 4.00 mW red laser (632 nm). 3 9 16 1 34 8 4.00 10 632 10 1.27 10 s6.63 10 3.00 10 Nhf NhcP t t N P t hc                  2. Light of wavelength 3.82 × 10−7 m is incident on a substance and electrons are emitted with a maximum speed of 6.87 × 105 m s −1. Calculate the work function1 energy of the substance. [N07 H2 P3 Q7c] Using Einstein’s photoelectric equation,   max max 34 8 31 5 2 7 19 6.63 10 3.00 10 1 9.11 10 (6.87 10 )23.82 10 3.06 10 J hf KE hc KE                     3 When light of wavelength 350 nm falls on a potassium surface, electrons are emitted that have a maximum kinetic energy of 1.31 e V. Determine (a) the work function of potassium, (b) the cut-off wavelength, (c) the frequency corresponding to the cut-off wavelength. (a) Using Einstein’s photoelectric equation,   34 8 19 19 max max 9 6.63 10 3.00 10 1.31 1.60 10 3.59 10 J 350 10 hc hc KE KE                          (b) At cut-off wavelength c , KEmax = 0 34 8 9 19 6.63 10 3.00 10 554 10 m 554 nm 3.59 10 c c hc hc                (c) Threshold frequency 8 14 9 3.00 10 5.42 10 Hz 554 10 c c cf        1 There are conduction electrons in a metal which are not attached to a particular atom. Removing thes e electrons from a metal surface using a photon involves a different amount of energy from that for ionizing a single metal atom. Thus, the work function energy of a metal differs from the energy required to remove the outer electron from an isolated atom. There is a simple connection between the work function energy Φ of a solid metal and the first ionization energy (Ei)1 of its corresponding gaseous atoms: Φ ~ ½(Ei)1.

DHS Physics H2 Year 6 / 2025 Topic 19 Quantum Physics Tutorial Suggested Answers 2 For Internal Use Only 21-May-25 4 In an experiment on photoelectric effect, a beam of light is used and the voltage across the electrodes required to stop the photocurrent is 1.5 V. (a) Determine the maximum speed of the photoelectrons. (b) Sketch a graph to illustrate how the photocurrent varies with the voltage across the electrodes and label it (b). (c) With reference to photoelectrons, explain the significance of the sloping section of your graph for negative values of potential difference. (d) The metal is replaced by another with a greater work function. Add a new line to the graph above to indicate how the photocurrent now varies with the voltage and label it (d). (e) With this new metal, the intensity of the light is now doubled. Add another line to the graph to indicate how the photocurrent now varies with the voltage and label it (e). (a) 2 max 19 5 1 max 31 1 2 2 2(1.60 10 )(1.5) 7.26 10 m s9.11 10 s s eV mv eVv m          (b) (d), (e) (c) It shows that electrons are emitted with a range2 of kinetic energy. 5 In a photoelectric emission experiment, ultraviolet radiation of wavelength 254 nm and intensity 210 Wm−2, was incident on a silver surface in an evacuated tube, so that an area of 12 mm 2 was illuminated. A photocurrent of 4.8 × 10−10 A was collected at an adjacent electrode. (a) What was the rate of incidence of photons on the silver surface? (b) What was the rate of emission of electrons? (c) The photoelectric quantum yield is defined as the ratio of number of photoelectrons emitted per sec ond number of photons incident per second (i) Find the quantum yield of this silver surface at wavelength of 254 nm. 2 This shows that while some photoelectrons are able to overcome the negative potential difference applied across the two plates to contribute to the current, less energetic electrons (electrons with less kinetic energy) are stopped by the negative potential difference. As the potential difference becomes more negative, more electrons are prevented from reaching the collector, resulting in less photocurrent. (e) voltage / V I 2I −1.5 (b) (d) Vs (new) photocurrent, i − Vs

DHS Physics H2 Year 6 / 2025 Topic 19 Quantum Physics Tutorial Suggested Answers 3 For Internal Use Only 21-May-25 (ii) Give two reasons why this value might be expected to be much less than one. (d) When the experiment was repeated with the radiation of wavelength 313 nm, no photoelectron was emitted. Explain this observation. [J83 P1 Q16 part] [3.22 1015, 3 109, 9.3 10−7] (a) Intensity p pN hf N hc At At  I 9 6 15 34 8 (210)(254 10 )(12 10 ) 3.22 10 photons per second(6.63 10 )(3.00 10 ) pN A t hc            I (b) Using the photocurrent, 10 9 19 (4.8 10 ) 3.0 10 electrons per second(1.60 10 ) e e N eQi t t N i t e          (c)(i) 9 7 15 (3.0 10 )Quantum Yield 9.3 10(3.22 10 )     (ii) 1. Most of the photons are reflected from the metal surface. 2. Atoms are made up of mostly empty space and thus the probability of a photon hitting a surface electron is very small. Instead, photons may hit electrons which are deeper below the surface, and these electrons may lose all its kinetic energy on its way up to the surface (due to collisions with other free electrons ) before they are emitted. 3. Electrons emitted may travel at an angle away from the normal and may not be collected at the anode. (d) With a longer wavelength, the energy of the incoming photons is lower. If it is lower than the work function energy of the silver surface, then no photoemission occurs. ------------------------------------------- Energy distribution of photoelectron energy Photo-emission from a metal surface is a multistep process. When the photon hits the metal surface it excites a photoelectron with a very high quantum yield, but that photoelectron is travelling in the same direction as the photon i.e. down into the metal. For an electron to be emitted from the surface the initial photoelectron has either to ricochet back out of the metal, or more likely to transfer energy to other electrons so one of the other electrons has enough energy to leave the surface. This process is essentially random and consequently has a very low quantum yield, so the overall quantum yield for photoemission is around 10−5 to 10−6.

DHS Physics H2 Year 6 / 2025 Topic 19 Quantum Physics Tutorial Suggested Answers 4 For Internal Use Only 21-May-25 The scattering of the initial photoelectron occurs via inelastic collisions, so whether it's the original electron scattered backwards, or other electrons scattered by the initial photoelectron, the energy of the electron leaving the surface is less that the energy of the photon. The balance of the energy goes into lattice vibrations of the metal i.e. internal energy. 6 The diagram shows a circuit used for photoelectric emission experiments. The two electrodes E and F are made of diff