#### Exercise

**Question 1. Find the(a) maximum frequency, and(b) minimum wavelength of X-rays produced by 30 kV electrons.**

**Question 2. The work function of caesium metal is 2.14 eV. When light of frequency 6 x 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?**

**Note:** Stopping potential is sufficient to repel even the most energetic electrons and so, photoelectric current will be zero.

**Question 3. The photoelectric cut-off voltage in a certain experiment is 1.5 V. What is the maximum kinetic energy of photoelectrons emitted?**

**Note**: Refer Key concept (3)

**Question 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 crossÂ 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?**

**Question 5. The energy flux of sunlight reaching the surface of the earth is 1.388 x 103 W/m2. How many photons (nearly) per square metre are incident on the Earth per second? Assume that the photons in the sunlight have an average wavelength of 550 nm.**

**Question 6. In an experiment on photoelectric effect, the slope of the cut-off voltage versus frequency of incident light is found to be 4.12 x 10-15v s.Calculate the value of Planck’s constant.**

**Question 7. A 100 W sodium lamp radiates energy uniformly in all directions. The lamp is located at the centre of a large sphere that absorbs all the sodium light which is incident on it. The wavelength of the sodium light is 589 nm.**

**(a) What is the energy per photon associated with the sodium light?****(b) At what rate are the photons delivered to the sphere?**

**Question 8. The threshold frequency for a certain metal is 3.3 x 1014 Hz. If light of frequency 8.2 x 1014 Hz is incident on the metal, predict the cut-off voltage for the photoelectric emission.**

**Question 9. The work function for a certain metal is 4.2 eV. Will this metal give photoelectric emission for incident radiation of wavelength 330 nm?**

**Note:** Refer Chapter at a Glance (1)

**Question 10. Light of frequency 7.21 x 1014 Hz is incident on a metal surface. Electrons with a maximum speed of 6.0 x 105 mis are ejected from the surface. What is the threshold frequency for photo emission of electrons?**

**Question 11. 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.**

**Question 12. Calculate the(a) momentum, and(b) de Broglie wavelength of the electrons accelerated through a potential difference of 56 V.**

**Question 13. What is the (a) momentum, (b) speed, and (c) de Broglie wavelength of an electron with kinetic energy of 120 eV.**

**Question 14. The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which**

**(a) an electron, and(b) a neutron, would have the same de Broglie wavelength.**

**Question 15. What is the de Broglie wavelength of(a) a bullet of mass 0.040 kg travelling at the speed of 1.0 km ls,(b) a ball of mass 0.060 kg moving at a speed of 1.0 mis, and(c) a dust particle of mass 1.0 x 10-9 kg drifting with a speed of 2.2 m is**

**Note:** De- broglie wavelength is very small in case of macroscopic objects, commonly encountered in everyday life.

**Question 16. An electron and a photon each have a wavelength of l.00 nm. Find(a) their momenta,(b) the energy of the photon, and(c) the kinetic energy of electron.**

**Question 17. (a) For what kinetic energy of a neutron will the associated de Broglie wavelength be 1.40 x 10-10 m?(b) Also find the de Broglie wavelength of a neutron, in thermal equilibrium with matter, having an average kinetic energy of (3/2) kT at 300K.**

**Question 18. Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).**

**Question 19. What is the de Broglie wavelength of a nitrogen molecule in air at 300 K? Assume that the molecule is moving with the root-mean-square speed of molecules at this temperature. (Atomic mass of nitrogen= 14.0076 u, K = 1.38 x 10-23 Jk-1)**

##### ADDITIONAL NCERT EXERCISES

**Question 20. (a) Estimate the speed with which electrons emitted from a heated emitter of an evacuated tube impinge on the collector maintained at a potential difference of 500 V with respect to the emitter. Ignore the small initial speeds of the electrons. The specific charge of the electron, i.e., its e/ mis given to be 1.76 x 1011 C kg-1.**

**(b) Use the same formula you employ in (a) to obtain electron speed for collector potential of 10 MV. Do you see what is wrong? In what way is the formula to be modified?**

**Question 21. A monoenergetic electron beam with electron speed of S.20 x 106 m s-1 is subject to a magnetic field of l.30 x 10-4 T normal to the beam velocity. What is the radius of the circle traced by the beam, given e/m for electron equals 1.76 x 1011C kg-1.**

**(b) Is the formula you employ in (a) valid for calculating radius of the path ofa 20 MeV electron beam? If not, in what way is it modified?**

**Question 22. An electron gun with its collector at a potential of l00 V fires out electrons in a spherical bulb containing hydrogen gas at low pressure (~10 ^{-2} mm of Hg).A magnetic field of 2.83 x 10^{-4} T curves the path of the electrons in a circular orbit of radius 12.0 cm.**

(The path can be viewed because the gas ions in the path focus the beam by attracting electrons, and emitting light by electron capture; this method is known as the fine beam tube’ method.)

**Question 23. (a) An X-ray tube produces a continuous spectrum of radiation with its short wavelength end at 0.45 A. What is the maximum energy of a photon in the radiation?**

**(b) From your answer to (a), guess what order of accelerating voltage (for electrons) is required in such a tube?**

**Question 24. In an accelerator experiment on high-energy collisions of electrons with positrons, a certain event is interpreted as annihilation of an electronÂ positron pair of total energy 10.2 BeV into two y-rays of equal energy. What is the wavelength associated with each y-ray? (1BeV = 109 eV)**

**Question 25. Estimating the following two numbers should be interesting. The first number will tell you why radio engineers do not need to worry much about photons! The second number tells you why our eye can never ‘count photons’, even in barely detectable light.**

**(a) The number of photons emitted per second by a medium wave transmitter of 10 kW power, emitting radio waves of wavelength 500m.(b) The number of photons entering the pupil of our eye per second corresponding to the minimum intensity of white light that we humans can perceive (~10 ^{-10} W m^{-2}). Take the area of the pupil to be about 0.4 cm2, and the average frequency of white light to be about 6 x 1014 Hz.**

**Question 26. Ultraviolet light of wavelength 2271A from a 100 W mercury source irradiates a photo-cell made of molybdenum metal. H the stopping potential is -1.3 V, estimate the work function of the metal. How would the photo-cell respond to a high intensity (~105 W m ^{-2}) red light of wavelength 6328 A produced by a He-Ne laser?**

**Question 27. Monochromatic radiation of wavelength 640.2 nm (1nm = 10 ^{-9} m) from a neon lamp irradiates photosensitive material made of caesium on tungsten. The stopping voltage is measured to be 0.54 V. The source is replaced by an iron source and its 427.2 nm line irradiates the same photo-cell. Predict the new stopping voltage.**

**Question 28. A mercury lamp is a convenient source for studying frequency dependence of photo-electric emission, since it gives a number of spectral lines ranging from the UV to the red end of the visible spectrum. In our experiment with rubidium photo-cell, the following lines from a mercury source were used:**

[You will notice that to get h from the data, you will need to know e (which you can take to be 1.6 x 10^{-19} C). Experiments of this kind on Na, Li, K, etc. were performed by Millikan, who, using his own value of e (from the oil-drop experiment) confirmed Einstein’s photoelectric equation and at the same time gave an independent estimate of the value of h.]

**Question 29. The work function for the following metals is given: Na: 2.75 eV; K: 2.30 eV; Mo: 4.17 eV; Ni: 5.15 eV. Which of these metals will not give photoelectric emission for a radiation of wavelength 3300 Afrom a He-Cd laser placed 1 m away from the photocell? What happens if the laser is brought nearer and placed 50 cm away?**

**Note:** Number of photoelectrons emitted per second is proportional to the intensity of incident radiation.

**Question 30. Light of intensity 10 ^{-5} W m^{-2} falls on a sodium photo-cell of surface area 2 cm2. Assuming that the top 5 layers of sodium absorb the incident energy, estimate time required for photoelectric emission in the wave-picture of radiation. The work function for the metal is given to be about 2 eV. What is the implication of your answer.**

**Sol.** Wave picture of radiation state that incident energy is uniformly distributed among all the electrons continuously. Let us first calculate the total number of recipient electrons in 5 layers of sodium.Consider each sodium atom has one electron free as conduction electron.

**Question 31. Crystal diffraction experiments can be performed using X-rays, or electrons accelerated through appropriate voltage. Which probe has greater energy? (For quantitative comparison, take the wavelength of a probe equal to 1 A, which is of the order of inter-atomic spacing in the lattice) (me= 9.11 x 10 ^{– 31} kg).**

**Question 32. (a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV an electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equallysuitable? Explain (mn = 1.675 x 10 ^{-27} kg)**

**(b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature (27Â°C). Hence explain why a fast neutron beam needs to be the realized with the environment before it can be used for neutron diffraction experiments.**

As the interatomic spacing (1 A= 10-10 m) is about hundred times greater than this wavelength, so a neutron beam of 150 eV energy is not suitable for diffraction experiment.

Note: Refer Chapter at a Glance (17)

**(b)** de-Broglie wavelength associated with thermal neutron at room temperature 300 K can be also calculated.

Average kinetic energy of a neutron at absolute temperature T is

As this wavelength is comparable to interatomic spacing(= l A) in a crystal, so thermal neutrons can be used for diffraction experiments. So high energy neutron beam should be first thermalised before using it for diffraction.

**Question 33. An electron microscope uses electrons accelerated by a voltage of 50 kV. Determine the de Broglie wavelength associated with the electrons. If other factors (such as numerical aperture, etc.) are taken to be roughly the same, how does the resolving power of an electron microscope compare with that of an optical microscope which uses yellow light?**

**Question 34. The wavelength of a probe is roughly a measure of the size of a structure that it can probe in some detail. The quart structure of protons and neutrons appears at the minute length scale of 10-15 m or less. This structure was first probed in early 1970’s using high energy electron beam produced by a linear accelerator at Standford, USA. Guess what might have been the order of energy of these electron beams. (Rest mass energy of electron= 0.511 MeV).**

**Question 35. Find the typical de-Broglie wavelength associated with a He atom in helium gas at room temperature (27Â°C) and 1 atm pressure; and compare it with the mean separation between two atoms under these conditions.**

**Question 36. Compute the typical de-Broglie wavelength of an electron in a metal at 27Â°C and compare it with the mean separation between two electron in a metal which is given to be about 2 x 10-10 m.**

**[Note: **Exercise 11.35 and 11.36 reveal that while wave-packets associated with gaseous molecules under ordinary conditions are non-overlapping, the electron wave-packets in a metal strongly overlap with one another. This suggests that whereas molecules in an ordinary gas can be distinguished apart, electrons in a metal cannot be distinguished apart from one another. This indistinguishibility has many fundamental implications which you will explore is more advanced physics curses.

**Question 37. Answer the following questions:**

**(a) Quarks inside protons and neutrons are thought to carry fractional charges [(+2/3)e; (-1/3)e]. Why do they not show up in Millikan’s oilÂ drop experiment?****(b) What is so special about the combination e/m? Why do we not simply talk of e and m separately?****(c) Why should gases be insulators at ordinary pressures and start conducting at very low pressures?****(d) Every metal has a definite work function. Why do all photoelectrons not come out with the same energy if incident radiation is monochromatic? Why is there an energy distribution of photoelectrons?****(e) The energy and momentum of an electron are related to the frequency and wavelength of the associated matter wave by the relations**

**But while the value of A is physically significant, the value of v (and therefore, the value of the phase speed vA) has no physical significance. Why?**

**Sol. (a)** Quarks have fractional charges that are confined within a p+ or nÂ°. They are bound by nuclear forces which are strong and grow stronger if quarks are tried to pull apart. Thus quarks always remain together in form of fractional changes inside proton and neutron only.

**(c)** Gases are generally insulators at atmospheric pressures because even if they are ionised, their positive and negative ions are close together and get ionised to form neutral atoms. At low pressures, the ionised gas particles are far apart and cannot recombine, hence they are free to conduct. Due to the presence of such free ions, gas can conduct.**(d)** This is because work function gives the minimum energy required for the uppermost e- of the conduction band (least ionisation potential) to come out of the metal. Not all electrons ejected are from this level but from a different levels which requires different energy to come out. Hence, for same radiation incident, electrons may be knocked off from different levels and come out with different energy.**(e)** The absolute value of energy E (only numerical part) has no significance except as an additive constant (P.S.- the sign of energy has physical meaning) put with momentum it is not so, Similarly for matter waves frequency has no direct physical meaning but wavelength is physically

significant. The phase speed vlamda = v has no significance but group speed

**Note:** In a photoelectric effect an electron absorbs a quantum of energy of radiation (hv). So, the electron which is more tightly bound will come out with less energy than the maximum value.

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