CHAPTER -11 DUAL NATURE OF RADIATION AND MATTER

Consider a 20 W bulb emitting light of wavelength 5000A °
and
shining on a metal surface kept at a distance 2m. Assume that
the metal surface has work function of 2 eV and that each atom
on the metal surface can be treated as a circular disk of radius
1.5 A
°
.
(i) Estimate no. of photons emitted by the bulb per second.
[Assume no other losses]
(ii) Will there be photoelectric emission?
(iii) How much time would be required by the atomc disk to receive
energy equal to work function (2 eV)?
(iv) How many photons would atomic disk receive within time
duration calculated in (iii) above?
(v) Can you explain how photoelectric effect was observed
instantaneously?
[Hint: Time calculated in part (iii) is from classical consideration
and you may further take the target of surface area say 1cm2
and estimate what would happen?]

Consider a thin target (10–2m square, 10–3m thickness) of sodium,
which produces a photocurrent of 100μA when a light of intensity
100W/m2 (λ = 660nm) falls on it. Find the probability that a
photoelectron is produced when a photons strikes a sodium
atom.

A neutron beam of energy E scatters from atoms on a surface
with a spacing d = 0.1nm. The first maximum of intensity in the
reflected beam occurs at θ = 30°. What is the kinetic energy E of
the beam in eV?

Two particles A and B of de Broglie wavelengths λ1 and λ2 combine
to form a particle C. The process conserves momentum. Find the
de Broglie wavelength of the particle C. (The motion is one
dimensional).

Two monochromatic beams A and B of equal intensity I, hit a
screen. The number of photons hitting the screen by beam A is 

twice that by beam B. Then what inference can you make about
their frequencies?

Assuming an electron is confined to a 1nm wide region , find the
uncertainty in momentum using Heisenberg Uncertainty
principle (Ref Eq 11.12 of NCERT Textbook). You can assume
the uncertainty in position Δx as 1nm. Assuming p 􀀀 Δp , find
the energy of the electron in electron volts.

Consider a metal exposed to light of wavelength 600 nm. The
maximum energy of the electron doubles when light of
wavelength 400 snm is used. Find the work function in eV.

There are two sources of light, each emitting with a power of
100 W. One emits X-rays of wavelength 1nm and the other visible
light at 500 nm. Find the ratio of number of photons of X-rays to
the photons of visible light of the given wavelength?

Do all the electrons that absorb a photon come out as
photoelectrons?

There are materials which absorb photons of shorter wavelength
and emit photons of longer wavelength. Can there be stable
substances which absorb photons of larger wavelength and emit
light of shorter wavelength.

A proton and an α-particle are accelerated, using the same
potential difference. How are the deBroglie wavelengths λp and
λa related to each other?

Sketch a graph b/w ?of incident radiation and stopping potential for a given photosensitive material. What information can be drawn from the value of intercept on the potential axis.     

IF the frequency of incident radiation an photocell is doubled for same intensity ,what charges will you observe in.    

(i) Kinetic energy of photo electrons

(ii) Photoelectric current

(iii) Stopping potential.

State the laws of Photoelectric effect. Explain it on the basis of Einstein equation.     

Show the graphical variation of photocurrent with intensity of incident radiation at constant potential difference b/w electrons and the graphical variation of photocurrent of incident radiation.     

A proton and an alpha particle  are accelerated through the same potential .Which one  them has Higher De Broglie wave length.     

Radition of frequency 10¹⁵ Hz are in incident on two photosensitive surfaces A  and B. Following observations are  recorded:

Surface A: No photoemission takes place.
Surface B: photoemission takes place but photoelectrons hace zero energy.

Explain the above observations on the basis of Einstein’s photo electric equation.     

Calculate the number of photons emitted per second by transmitter of 10 KW power; radio waves of frequency 6×10⁵ Hz.     

Explain the effect of increase of intensity and potential difference on photoelectrons kinetic energy.

Show the graphical variation of stopping potential with the frequency of incident radiation .How do we determine the Planck’s constant using Graph.     

A Source of light is placed at a distance of 50 c.m from a photocell and cut –off potential is found to be V_o .If the distance b/w source and photocell is made 25 c.m. What will be new cut –off potential.     

When a monochromatic yellow colored light beam is incident an a given photosensitive surface, photo electrons are not ejected to green colored monochromatic beam.
What will happen if the same surface is exposed to (i)violet and (ii) red colored  monochromatic beam of light.     

The wavelength λ of a photon and De Broglie wavelength is of  an electron have  the same value .Show that energy of photon is 2 λ mc/n times  the energy of the electron.     

A particle of mass M at decays into two particle of masses m₁ and m₂having velocity v₁ and v₂ respectively .Find the ratio of de Broglie wavelength of two particles.     

Derive an expression for the de Broglie Wavelength of an electron moving under potential difference of  V volt.     

Ultra violet light of wavelength 2271 A^o is incident on two photo sensitive material having work function W₁ and W₂ (W₁ >> W₂). In which case will the kinetic energy of emitted electrons be greater ?     

An electron and a proton have same De broglie wavelength associated with them. How are their Kinetic energies related to each other ?

With what purpose was famous Davisson - Germer experiment with electrons per formed ?     

How does the maximum kinetic energy of photoelectrons vary with work function of metal ?     

State Einstein’s Photoelectric equation

In an experiment on photoelectric effect , the following graph were obtained. Name the characteristics of incident radiation that was kept constant.     

The stopping potential in an experiment  in Photoelectric effect is 1.5 eV. What is the maximum kinetic energy of the photoelectron.     

Two metals P and Q have work function  2ev and 4ev  respectively. Which of the two metal have smaller threshold wavelength.      

If the intensity of the incident radiation in a photocell is increased. How does the stopping potential vary?     

The  frequency  of incident radiation is greater than threshold frequency ₀ in a photocell .How will the stopping potential  vary if frequency is increased.     

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 = hν, p = 

But while the value of ÃŽÂ»is physically significant, the value of ÃŽÂ½(and therefore, the value of the phase speed ÃŽÂ½ÃŽÂ») has no physical significance. Why?

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.]

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.

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^-15m 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.)

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?

(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^-27kg)

(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.

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 Ô¦, which is of the order of inter-atomic spacing in the lattice) (m_e= 9.11 x 10^-31kg).

Light of intensity 10^-5W m^-2falls 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?

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 Ô¦, ÃŽÂ»₂= 4047 Ô¦, ÃŽÂ»₃= 4358 Ô¦, ÃŽÂ»₄= 5461 Ô¦, ÃŽÂ»₅= 6907 Ô¦,

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 x 10^-19C). 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.]

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 Ô¦, ÃŽÂ»₂= 4047 Ô¦, ÃŽÂ»₃= 4358 Ô¦, ÃŽÂ»₄= 5461 Ô¦, ÃŽÂ»₅= 6907 Ô¦,

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 x 10^-19C). 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.]

Monochromatic radiation of wavelength 640.2 nm (1nm = 10^-9m) 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.

Ultraviolet light of wavelength 2271 Ô¦ 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 (∝¼10⁵W m^-2) red light of wavelength 6328 Ô¦ produced by a He-Ne laser?

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^-10W 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 10¹⁴Hz.

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 = 10⁹eV)

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

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^-2mm of Hg). A magnetic field of 2.83 x 10^-4T 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/mfrom the data.

(a) A monoenergetic electron beam with electron speed of 5.20 x 10⁶m s^-1is subject to a magnetic field of 1.30 x 10^-4T 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 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.]

(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 10¹¹C 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?

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)

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

(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.

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.

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^-9kg drifting with a speed of 2.2 m/s?

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.

What is the

(a) momentum,

(b) speed, and

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

Calculate the

(a) momentum, and

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

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.

Light of frequency 7.21 x 10¹⁴Hz is incident on a metal surface. Electrons with a maximum speed of 6.0 x 10⁵m/s are ejected from the surface. What is the threshold frequency for photoemission of electrons?

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

The threshold frequency for a certain metal is 3.3 x 10¹⁴ Hz. If light of frequency 8.2 x 10¹⁴Hz is incident on the metal, predict the cutoff voltage for the photoelectric emission.

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?

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^-15V s. Calculate the value of Planck's constant

The energy flux of sunlight reaching the surface of the earth is 1.388 x 10³W/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.

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?

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

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?

Find the

(a) maximum frequency, and

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