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Question 11.1 Find the
(a) maximum frequency, and
(b) minimum wavelength of X-rays produced by 30 kV electrons.
Question 11.2 The work function of caesium metal is 2.14 eV. When light of
frequency 6 ×1014Hz 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?
Question 11.3 The photoelectric cut-off voltage in a certain experiment is 1.5 V.
What is the maximum kinetic energy of photoelectrons emitted?
Question 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?
Question 11.5 The energy flux of sunlight reaching the surface of the earth is
1.388 × 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 11.6 In an experiment on photoelectric effect, the slope of the cut-off
voltage versus frequency of incident light is found to be 4.12 × 10–15 V s.
Calculate the value of Planck’s constant.
Question 11.7 A 100W 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 11.8 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 cutoff
voltage for the photoelectric emission.
Question 11.9 The work function for a certain metal is 4.2 eV. Will this metal give
photoelectric emission for incident radiation of wavelength 330 nm?
Question 11.10 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?
Question 11.11 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 11.12 Calculate the
(a) momentum, and
(b) de Broglie wavelength of the electrons accelerated through a
potential difference of 56V.
Question 11.13 What is the
(a) momentum,
(b) speed, and
(c) de Broglie wavelength of an electron with kinetic energy of
120 eV.
Question 11.14 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 11.15 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?
Question 11.16 An electron and a photon each have a wavelength of 1.00 nm. Find
(a) their momenta,
(b) the energy of the photon, and
(c) the kinetic energy of electron.
Question 11.17 (a) For what kinetic energy of a neutron will the associated de Broglie
wavelength be 1.40 × 10–10m?
(b) Also find the de Broglie wavelength of a neutron, in thermal
equilibrium with matter, having an average kinetic energy of
(3/2) k T at 300 K.
Question 11.18 Show that the wavelength of electromagnetic radiation is equal to
the de Broglie wavelength of its quantum (photon).
Question 11.19 What is the de Broglie wavelength of a nitrogen molecule in air at
300 K? Assume that the molecule is moving with the root-meansquare
speed of molecules at this temperature. (Atomic mass of
nitrogen = 14.0076 u)
Question 11.20 (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/m is given to be
1.76 × 1011 C kg–1.
(b) Use the same formula you employ in (a) to obtain electron speed
for an collector potential of 10 MV. Do you see what is wrong ? In
what way is the formula to be modified?
Question 11.21 (a) A monoenergetic electron beam with electron speed of
5.20 × 106 m s–1 is subject to a magnetic field of 1.30 × 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 × 1011C kg–1.
(b) Is the formula you employ in (a) valid for calculating radius of
the path of a 20 MeV electron beam? If not, in what way is it
modified ?
[Note: Exercises 11.20(b) and 11.21
(b) take you to relativistic
mechanics which is beyond the scope of this book. They have been
inserted here simply to emphasise the point that the formulas you
use in part (a) of the exercises are not valid at very high speeds or
energies. See answers at the end to know what ‘very high speed or
energy’ means.
Question 11.22 An electron gun with its collector at a potential of 100 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 × 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.) Determine
e/m from the data.
Question 11.23 (a) An X-ray tube produces a continuous spectrum of radiation with
its short wavelength end at 0.45 Å. 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 11.24 In an accelerator experiment on high-energy collisions of electrons
with positrons, a certain event is interpreted as annihilation of an
electron-positron pair of total energy 10.2 BeV into two γ-rays of
equal energy. What is the wavelength associated with each γ-ray?
(1BeV = 109 eV)
Question 11.25 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 radiowaves of wavelength
500 m.
(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 × 1014 Hz.
Question 11.26 Ultraviolet light of wavelength 2271 Å from a 100 W mercury source
irradiates a photo-cell made of molybdenum metal. If 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 Å produced by a He-Ne laser?
Question 11.27 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 11.28 A mercury lamp is a convenient source for studying frequency
dependence of photoelectric 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:
λ1 = 3650 Å, λ2= 4047 Å, λ3= 4358 Å, λ4= 5461 Å, λ5= 6907 Å,
The stopping voltages, respectively, were measured to be:
V01 = 1.28 V, V02 = 0.95 V, V03 = 0.74 V, V04 = 0.16 V, V05 = 0 V
Determine the value of Planck’s constant h, the threshold frequency
and work function for the material.
[Note: You will notice that to get h from the data, you will need to
know e (which you can take to be 1.6 × 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 11.29 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 Å from a He-Cd laser placed 1 m away from the
photocell? What happens if the laser is brought nearer and placed
50 cm away?
Question 11.30 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?
Question 11.31 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 the probe equal to 1 Å, which is of the order of inter-atomic spacing
in the lattice) (me=9.11 × 10–31 kg).
11.32 (a) Obtain the de Broglie wavelength of a neutron of kinetic energy
150 eV. As you have seen in Exercise
Question 11.31, an electron beam of
this energy is suitable for crystal diffraction experiments. Would
a neutron beam of the same energy be equally suitable ? Explain.
(mn = 1.675 × 10–27 kg) (b) Obtain the de Broglie wavelength associated with thermal
neutrons at room temperature (27 ºC). Hence explain why a fast
neutron beam needs to be thermalised with the environment
before it can be used for neutron diffraction experiments.
Question 11.33 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 11.34 The wavelength of a probe is roughly a measure of the size of a
structure that it can probe in some detail. The quark 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 beams produced by a linear accelerator at
Stanford, USA. Guess what might have been the order of energy of
these electron beams. (Rest mass energy of electron = 0.511 MeV.)
Question 11.35 Find the typical de Broglie wavelength associated with a He atom in
helium gas at room temperature (27 ºC) and 1 atm pressure; and
compare it with the mean separation between two atoms under these
conditions.
Question 11.36 Compute the typical de Broglie wavelength of an electron in a metal
at 27 ºC and compare it with the mean separation between two
electrons in a metal which is given to be about 2 × 10–10 m.
[Note: Exercises 11.35 and 11.36 reveal that while the 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 distintguished apart from one another. This
indistinguishibility has many fundamental implications which you
will explore in more advanced Physics courses.]
Question 11.37 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 oil-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:
E = h ν, p = λ
h
But while the value of λ is physically significant, the value of ν
(and therefore, the value of the phase speed ν λ) has no physical
significance. Why?
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