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**Exercise : ** Solutions of Questions on Page Number : **407**

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Q1 :
**
**

Find the

**(a)** maximum frequency, and

**(b)** minimum wavelength of X-rays
produced by 30 kV electrons.

**Answer :**

Potential of the electrons,
*V*= 30 kV = 3
×10^{4}
V

Hence, energy of the electrons,
*E* = 3
×10^{4}
eV

Where,

*e*= Charge on an electron = 1.6
×10^{ - 19}C

**(a)**Maximum frequency produced by the
X-rays = *ÃŽÂ½*

The energy of the electrons is given by the relation:

*E* =
*h**ÃŽÂ½*

Where,

*h*=
Planck's
constant = 6.626
×10^{ - 34}Js

Hence, the maximum frequency of X-rays produced is

**(b)**The minimum wavelength produced by
the X-rays is given as:

Hence, the minimum wavelength of X-rays produced is 0.0414 nm.

Answer needs Correction? Click Here

Q2 :
**
**

The work function of caesium metal is 2.14 eV. When light
of frequency 6
x 10^{14}
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?

**Answer :**

Work function of caesium metal,

Frequency of light,

**(a)**The maximum kinetic energy is given
by the photoelectric effect as:

Where,

*h*=
Planck's
constant = 6.626
×10^{ - 34}Js

Hence, the maximum kinetic energy of the emitted electrons
is

0.345 eV.

**(b)**For stopping potential, we can write the
equation for kinetic energy as:

Hence, the stopping potential of the material is 0.345 V.

**(c)**Maximum speed of the emitted
photoelectrons = *v*

Hence,the relation for kinetic energy can be written as:

Where,

*m*= Mass of an electron = 9.1
×10^{ - 31}kg

Hence, the maximum speed of the emitted photoelectrons
is

332.3 km/s.

Answer needs Correction? Click Here

Q3 :
**
**

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

**Answer :**

Q4 :
**
**

Monochromatic light of wavelength 632.8 nm is produced by ahelium-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?

**Answer :**

Q5 :
**
**

The energy flux of sunlight reaching the surface of the
earth is 1.388 x
10^{3}W/m^{2}.
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.

**Answer :**

Q6 :
**
**

In an experiment on photoelectric effect, the slope of the
cut-offvoltage versus frequency of incident light is
found to be 4.12 x
10^{-15}V
s. Calculate the value of
Planck's
constant.

**Answer :**

Q7 :
**
**

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?

**Answer :**

Q8 :
**
**

The threshold frequency for a certain metal is 3.3
x
10^{14} Hz. If light of
frequency 8.2 x
10^{14}Hz is incident on the
metal, predict the cutoff voltage for the photoelectric
emission.

**Answer :**

Q9 :
**
**

The work function for a certain metal is 4.2 eV. Will this metal givephotoelectric emission for incident radiation of wavelength 330 nm?

**Answer :**

Q10 :
**
**

Light of frequency 7.21 x
10^{14}Hz is incident on a
metal surface. Electrons with a maximum speed of 6.0
x
10^{5}m/s are ejected from the
surface. What is the threshold frequency for photoemission of
electrons?

**Answer :**

Q11 :
**
**

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.

**Answer :**

Q12 :
**
**

Calculate the

**(a)** momentum, and

**(b)** de Broglie wavelength of the
electrons accelerated through a potential difference of 56
V.

**Answer :**

Q13 :
**
**

What is the

**(a)** momentum,

**(b)** speed, and

**(c)** de Broglie wavelength of an
electron with kinetic energy of 120 eV.

**Answer :**

Q14 :
**
**

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.

**Answer :**

Q15 :
**
**

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
x
10^{-9}kg
drifting with a speed of 2.2 m/s?

**Answer :**

Q16 :
**
**

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.

**Answer :**

Q17 :
**
**

**(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 300 K.

**Answer :**

Q18 :
**
**

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

**Answer :**

Q19 :
**
**

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)

**Answer :**

Q20 :
**
**

**(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
x
10^{11}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?

**Answer :**

Q21 :
**
**

**(a)** A monoenergetic electron beam with
electron speed of 5.20 x
10^{6}m
s^{-1}is
subject to a magnetic field of 1.30
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
10^{11}C
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.]

**Answer :**

Q22 :
**
**

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
(∝¼10^{-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.
Determine
*e*/*m*from
the data.

**Answer :**

Q23 :
**
**

**(a)** An X-ray tube produces a continuous
spectrum of radiation with its short wavelength end at 0.45
Ãƒ”¦. 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?

**Answer :**

Q24 :
**
**

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
ÃŽÂ³-rays of equal
energy. What is the wavelength associated with each
ÃŽÂ³-ray? (1BeV =
10^{9}eV)

**Answer :**

Q25 :
**
**

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
(∝¼10^{-10}W
m^{-2}).
Take the area of the pupil to be about 0.4
cm^{2}, and the average
frequency of white light to be about 6
x
10^{14}Hz.

**Answer :**

Q26 :
**
**

Ultraviolet light of wavelength 2271
Ãƒ”¦ 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
(∝¼10^{5}W
m^{-2})
red light of wavelength 6328 Ãƒ”¦
produced by a He-Ne laser?

**Answer :**

Q27 :
**
**

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.

**Answer :**

Q28 :
**
**

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:

*ÃŽÂ»*_{1}=
3650 Ãƒ”¦,
*ÃŽÂ»*_{2}=
4047 Ãƒ”¦,
*ÃŽÂ»*_{3}=
4358 Ãƒ”¦,
*ÃŽÂ»*_{4}=
5461 Ãƒ”¦,
*ÃŽÂ»*_{5}=
6907 Ãƒ”¦,

The stopping voltages, respectively, were measured to be:

*V*_{01}= 1.28
V, *V*_{02}=
0.95 V,
*V*_{03}= 0.74
V, *V*_{04}=
0.16 V,
*V*_{05}= 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 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*.]

**Answer :**

Q29 :
**
**

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 Ãƒ”¦ 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?

**Answer :**

Q30 :
**
**

Light of intensity
10^{-5}W
m^{-2}falls
on a sodium photo-cell of surface area 2
cm^{2}. 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?

**Answer :**

Q31 :
**
**

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
Ãƒ”¦, which is of the order of
inter-atomic spacing in the lattice)
(*m*_{e}=
9.11 x
10^{-31}kg).

**Answer :**

Q32 :
**
**

**(a)** Obtain the de Broglie wavelength of
a neutron of kinetic energy 150 eV. As you have seen in Exercise
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.
(*m*_{n}=
1.675 x
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.

**Answer :**

Q33 :
**
**

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?

**Answer :**

Q34 :
**
**

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.)

**Answer :**

Q35 :
**
**

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.

**Answer :**

Q36 :
**
**

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 x
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
distinguished apart from one another. This indistinguishibility
has many fundamental implications which you will explore in more
advanced Physics courses.]

**Answer :**

Q37 :
**
**

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**ÃŽÂ½*,
*p* =

But while the value of
*ÃŽÂ»*is physically
significant, the value of
*ÃŽÂ½*(and therefore,
the value of the phase speed
*ÃŽÂ½**ÃŽÂ»*)
has no physical significance. Why?

**Answer :**

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