42.1 Consider a parallel beam of light of wavelength 600 nm and intensity 100 W/m2. (a) Find the energy and linear momentum of each photon.
(b) How many photons cross 1 cm2 area perpendicular to the beam in one second ?
42.2 Find the maximum wavelength of light that can cause photoelectric effect in lithium.
Sol. From table (42.1), the work function of lithium is 2.5eV. The threshold wavelength is
This is the required maximum wavelength.
42.3 A point source of monochromatic light of 1.0 mW is placed at a distance of 5.0 m from a metal surface. Light falls perpendicularly on the surface. Assume wave theory of light to hold and also that all the light falling on the circular area with radius = 1.0 × 10-9 m (which is few times the diameter of an atom) is absorbed by a single electron on the surface. Calculate the time required by the electron to receive sufficient energy to come out of the metal if the work function of the metal is 2.0 eV.
Sol. The energy radiated by the light source per second is 1.0 mJ. This energy is spread over the total solid angle 4p. The solid angle subtended at the source by the circular area mentioned is
Worked out Solved Examples
1. How many photons are emitted per second by a 5 mW laser source operating at 632.8 nm ?
Sol. The energy of each photon is
The energy of the laser emitted per second is 5 × 10-3 J. Thus the number of photons emitted per second
2. A monochromatic source of light operating at 200 W emits 4 × 1020 photons per second. Find the wavelength of light.
3. A hydrogen atom moving at a speed v absorbs a photon of wavelength 122 nm and stops. Find the value of u. Mass of a hydrogen atom = 1.67 × 10-27 kg.
Sol. The linear momentum of the photon
As the photon is absorbed and the atom stops, the total final momentum is zero. From conservation of linear momentum, the initial
momentum must be zero. The atom should move opposite to the direction of motion of the photon and they should have the same
magnitudes of linear momentum. Thus,
4. A parallel beam of monochromatic light of wavelength 500 nm is incident normally on a perfectly absorbing surface. The power through any
cross-section of the beam is 10 W. Find (a) the number of photons absorbed per second by surface and (b) the force exerted by the light beam on the surface.
Sol. (a) The energy of each of photon is
In one second, 10 J of energy passes through any cross-section of the beam. Thus, the number of photons crossing a cross-section is
This is also the number of photons falling on the surface per second and being absorbed.
(b) the linear momentum of each photon is
The total momentum of all the photons falling per second on the surface is
As the photons are completely absorbed by the surface, this much momentum is transferred to the surface per second.
The rate of change of the momentum of the surface, i.e., the force on it is
5. Figure shows a small, plane strip suspended from a fixed support through a string of length l. A continuous beam of monochromatic
light is incident horizontally on the strip and is completely absorbed. The energy falling on the strip per unit time is W.
(a) Find the deflection of the string from the vertical if the mirror stays in equilibrium.
(b) If the strip is deflected slightly from its equilibrium position in the plane of the figure, what will be the time period of the resulting oscillations ?
Sol. (a) The linear momentum of the light falling per unit time on the strip is W/c. As the light in incident on the strip, its momentum imparted to
the strip per unit time is thus W/c. This is equal to the force on the strip by the light beam. In equilibrium, the force by the light beam, the weight
of the strip and the force due to tension add to zero. If the string makes an angle q with the vertical,
This plays the role of effective g. The time period of small oscillations is
6. A point source of light is placed at the centre of curvature of a hemispherical surface. The radius of curvature is r and the inner surface is completely
reflecting. Find the force on the hemisphere due to the light falling on it if the source emits a power W.
Sol. The energy emitted by the source per unit time, i.e., W falls on an area 4תr2 at a distance r in unit time. Thus, the energy falling per unit area per
unit time is (4תr2 ). Consider a small area dA at the point P of the hemisphere (figure). The energy falling per unit time on it isWda/4תr2 .
The corresponding momentum incident
.
Suppose the radius OP through the area dA makes an angle q with the symmetry axis OX. The force on dA is along this radius.
By symmetry, the resultant force on the hemisphere is along OX. The component of dF along OX is
If we project the area dA on the plane containing the rim, the projection is dA cosq. Thus, the component of dF along OX is,
The net force along OX is
7. A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carries an intensity
I, find the force exerted by the beam on the sphere.
Sol. 
Let O be the centre of the sphere and OZ be the line opposite to the incident beam (figure). Consider a radius OP of the sphere making an angle
q with OZ. Rotate this radius about OZ to get a circle on the sphere. Change q to q + dq and rotate the radius about OZ to get another circle on
the sphere. The part of the sphere between these circles is a ring of area 2pr2 sinq dq. Consider a small part DA of this ring at P. Energy of the
light falling on this part in time Dt is
The momentum of this light falling on DA is DU/c along QP. The light is reflected by the sphere along PR. The change in momentum is
along the inward normal. The force on DA due to the light falling on it, is
This force is along PO. The resultant force on the ring as well as on the sphere is along ZO by symmetry. The component of the force on DA along ZO
The force acting on the ring is
The force on the entire sphere is
Note that integration is done only for the hemisphere that faces the incident beam.
8. Find the threshold wavelengths for photoelectric effect from a copper surface, a sodium surface and a cesium surface.
The work functions of these metals are 4.5 eV, 2.3eV and 1.9eV respectively.
Sol. If l0 be the threshold wavelength and be the work function,
9. Ultraviolet light of wavelength 280 nm is used in an experiment on photoelectric effect with lithium (j = 2.5 eV) cathode. Find
(a) the maximum kinetic energy of the photoelectrons and (b) the stopping potential.
Sol. (a) The maximum kinetic energy is
10. In a photoelectric experiment, it was found that the stopping potential decreases from 1.85 V to 0.82 V as the wavelength of the incident light is
varied from 300 nm to 400 nm. Calculate the value of the planck constant from these data.
Sol. The maximum kinetic energy of a photoelectron is
If V1, V2 are the stopping potential at wavelengths l1 and l2 respectively.
11. A beam of 450 nm light is incident on a metal having work function 2.0 eV and placed in a magnetic field B. The most energetic electrons
emitted perpendicular to the field are bent in circular arcs of radius 20 cm. Find the value of B.
Sol. The kinetic energy of the most energetic electrons is
The linear momentum =
When a charged particle is sent perpendicular to a magnetic field, it goes along a circle of radius
12. A monochromatic light of wavelength l is incident on an isolated metallic sphere of radius a. The threshold wavelength is l0 which is larger than l.
Find the number of photoelectrons emitted before the emission of photoelectrons will stop.
Sol. As the metallic sphere is isolated, it becomes positively charged when electrons are ejected from it. There is an extra attractive force on the photoelectrons.
If the potential of the sphere is raised to V, the electron should have a minimum energy j + eV to be able to come out. Thus,
emission of photoelectrons will stop when
The charge on the sphere needed to take its potential to V is
The number of electrons emitted is, therefore,
l
13. Light described at a place by the equation E = (100 V/m) [sin (5 × 1015 s–1) t + sin (8 × 1015 s–1)t]
falls on a metal surface having work function 2.0 eV. Calculate the maximum kinetic energy of the photoelectrons.
Sol. The light contains two different frequencies. The one with larger frequency will cause photoelectrons with largest kinetic energy.
This larger frequency is
The maximum kinetic energy of the photoelectrons is
Question for short answer
1. Can we find the mass of a photon by the definition p = mv ?
2. Is it always true that for two sources of equal intensity, the number of photons emitted in a given time are equal ?
3. What is the speed of a photon with respect to another photon if (a) the two photons are going in the same direction and
(b) they are going in opposite directions ?
4. Can a photon be deflected by an electric field ? By a magnetic field ?
5. A hot body is placed in a closed room maintained at a lower temperature. Is the number of photons in the room increasing ?
6. Should the energy of a photon be called its kinetic energy or its internal energy ?
7. In an experiment on photoelectric effect, a photon is incident on an electron from one direction and the photoelectron is emitted
almost in the opposite direction. Does this violate conservation of momentum ?
8. It is found that yellow light does not eject photoelectrons from a metal. Is it advisable to try with orange light ? With green light ?
9. It is found that photosynthesis starts in certain plants when exposed to the sunlight but it does not start if the plant is exposed only to
infrared light. Explain
10. The threshold wavelength of a metal is l0. Light of wavelength slightly less than l0 is incident on an insulated plate made of this metal.
It is found that photoelectrons are emitted for sometime and after that the emission stops. Explain.
11. Is p = E/c valid for electrons ?
12. Consider the de Broglie wavelength of an electron and a proton. Which wavelength is smaller if the two particles have
(a) the same speed (b) the same momentum (c) the same energy ?
13. If an electron has a wavelength, does it also have a colour ?
OBJECTIVE - I
1. Planck constant has the same dimension as
(A) force x time (B) force x distance (C) force x speed (D*) force x distance x time
2. Two photons having
(A) equal wavelengths have equal linear momenta
(B) equal energines have equal linear momenta
(B) equal frequencies have equal linear momenta
(D*) equal linear momenta have equal wavelength
3. Let p and E denote the linear momentum and energy of a photon. If the wavelength is decreased
(A*) both p and E increase (B) p increases and E decreases
(C) p decreases and E increases (D) both p and E decrease
4. Let nr and nb be reprectively the number of photons emitted by a red bulb and a blue bulb of equal power in a given time.
(A) nr = nb (B) nr < nb (C*) nr > nb
(D) the information is insufficient to get a relation between nr and nb
5. The equation E = pc is valid
(A) for an electron as well as for a photon (B) for an electron but not for a photon
(C*) for a photon but nor for an electron (D) neither for an electron nor for a photon
6. The work function of a metal is hv0. Light of frequency v falls on this metal. The photoelectric effect will take place only if
(A*) v > v0 (B) v > 2v0 (C) v >v0 (D) v < v0 /2
7. Leigth of wavelength l falls on a metal having work function hc/l0. Photoelectric effect will take place only if
(A) l ³ l0 (B) l ³ 2l0 (C*) l £ l0 (D) l < l0 /2.
8. When stopping potential is applied in an experiment on photoelectric effect, no photocurrent is observed. This means that
(A) the emission of photoelectrons is stopped
(B*) the photoelectrons are emitted but are reabsorbed by the emitter metal
(C) the photoelectrons are accumulated near the collector plate
(D) the photoelectrons are dispersed from the sides of the apparatus.
9. If the frequency of light in a photoelectric experiment is double, the stopping potential will
(A) be doubled (B) be halved
(C*) become more than double (D) become less than double
10. The frequency and intensity of a light source are both doubled, the stopping potential statements.
(a) The saturation photocurrent remains almost the same.
(b) The maximum kinetic energy of the photoelectrons is doubled
(A) Both (a) and (b) are true. (B*) (a) is true but (b) is false
(C) (a) is false but (b) is true (D) Both (a) and (b) are false
11. A point source of ligth is used in a photoelectric effect. If the source is removed farther from the emitting metal, the stopping potential
(A) will increase (B) will decrease
(C*) will remain constant (D) will either increase or decrease
12. A point source causes photoelectric effect from a small metal plate. Which of the following curves may represent the
saturation photocurrent as a function of the distance between the source and the metal ?
(A) a (B) b (C) c (D*) d
13. A nonmonochromatid light is used in an experiment on photoelectric effect. The stopping potential
(A) is related to the mean wavelength (B) is related to the longest wavelength
(C*) is related to the shortest wavelenth (D) is not related to the wavelength
14. A proton and an electron are acceleration by the same potential diffenrence. Let le and lp denote the de
Broglie wavelength of the electron and the proton respectively
OBJECTIVE - II
1. When the intensity of a light source is increased,
(A*) the number of photons emitted by the source in unit time increases
(B*) the total energy of the photons emitted per unit time increases
(C) more energetic photons are emitted
(D) faster photons are emitted
2. Photoelectric effect supports quantum nature of light because
(A*) there is a minimum frequency below which no photoelectrons are emitted
(B*) the maximum kinetic energy of photoelectrons depends only on the frequency of light and not on its intensity
(C*) even when the metal surface is faintly illuminated the photoelectrons leave the surface imme diately
(D) electric charge of the photoelectrons is quantized
3. A photon of energy hv is absorbed by a free electron of a metal having work function j < hv
(A) The electron is sure to come out
(B) The electron is sure to come out with a kineitc energy hv - j
(C) Either the electrons does not come out or it comes out with a kinetic energy hv - j
(D*) It may come out wiht a kinetic energy less than hv - j
4. If the wavelength of light in an experiment on photoelectric effect is doubled,
(A) the photoelectric emission will not take place (B*) the exposure time is increased
(C) the intensity of the source is decreased (D*) the exposure time is decreased
5. The photocurrent in an experiment on photoelectric effect increases if
(A*) the intensity of the source is increased (B) the exposure time is increased
(C) the intensity of the source is decreased (D) the exposure time is decreased
6. The collector plate in an experiment on photoelectric effect is kept vertically above the emitter plate. Light source is put on and a
saturation photocurrent is recorded. An electric field is switched on which has vertically downward direction
(A) The photocurrent will increase (B*) The kinetic energy of the electrons will increase
(C) The stopping potential will decrease (D) The threshold wavelength will increase
7. In which of the following situations the heavier of the two particles has smaller be Broglie wavelength? The two particles
(A*) move with the same speed (B) move with the same linear momentum
(C*) move with the same kinetic energy (D*) have fallen through the same height
Exercise
1. Visible light has wavelengths inthe range of 400 nm to 780 nm. Calculate the range of energy of the photons of visible light (in joules).
Ans. 2.56 × 10–19 J to 5.00 × 10–19 J
2. Calculate the momentum of a photon of light of wavelength 500 nm.
Ans. 1.33 × 10–27 kg-m/s
3. An atom absorbs a photon of wavelength 500 nm and emits another photon of wavelength 700 nm.
Find the net energy absorbed by the atom in the process.
Ans. 1.1 × 10–19 J
4. Calculate the number of photons emitted per second by a 10 W sodium vapour lamp. Assume that 60% of the consumed
energy is converted into light. Wavelength of sodium light = 590 nm. HCV_Ch.42_4
Ans. 1.77 × 1019
5. When the sun is directly overhead, the surface of the earth receives 1.4 × 103 W/m2 of sunlight. Assume that the light is monochromatic
with average wavelength 500 nm and that no light is absorbed in between the sun and the earth’s surface. The distance between the sun
and the earth is 1.5 × 1011 m.
(a) Calculate the number of photons falling per second on each square metre of earth’s surface directly below the sun.
(b) How many photons are there in each cubic metre near the earth’s surface at any instant ?
(c) How many photons does the sun emit per second ?
Ans. (a) 3.35 × 1021 (b) 1.2 × 1013 (c) 9.9 × 1044
6. A parallel beam of monochromatic light of wavelength 663 nm is incident on a totally reflecting plane mirror. The angle of incidence is 60°
and the number of photons striking the mirror per second is 1.0 × 1019. Calculate the force exerted by the light beam on the mirror.
Ans. 1.0 × 10–8 N
7. A beam of white light is incident normally on a plane surface absorbing 70% of the light and reflecting the rest. If the incident
beam carries 10 W of power, find the force exerted by it on the surface.
Ans. 7.43 × 10–8 N
8. A totally reflecting, small plane mirror placed horizontally faces a parallel beam oflight as shown in figure. The mass ofthe mirror is 20g.
Assume that there isno absorption in the lens and that 30% of the light emitted by the source goes through the lens. Find the power of the
source needed to support the weight of the mirror. Take g = 10 m/s2 .
9. A 100 W light bulb is placed at the centre of a spherical chamber of radius 20cm. Assume that 60% of the energy supplied to the bulb is
converted into light and that the surface of the chamber is absorbing. Find the pressure exerted by the light on the surface of the chamber.
Ans: 100 MW
10. A sphere of radius 1.00 cm is placed in the path of a parallel beam of large aperture. The intensity of the light is 0.50 W/cm If the sphere
completely absorbs the radiation falling on it Find the force exerted by the light beam on the sphere.
Ans: 4.0 × 10 – 7 Pa
11. Consider the situation described in the previous problem. Show that the force on the sphere due to the light falling on it is the same
even if the sphere is not perfectly absorbing.
Ans : 5.2 × 10 – 9 N
12. Show that it is not possible for a photon to be completely absorbed by a free electron.
Ans :
13 Two neutral particles are kept 1m apart .Suppose by some mechanism some charge is transferred from one particle to the other and the
electronic potential energy lost is completely converted into a photon .Calculate the longest and the next smaller wavelength of the photon possible .
Ans: 860 m, 215 m
14. Find the maximum kinetic energy of the photoelectrons ejected when light of wavelength 350 nm is incident on a cesium surface .
Work function of cesium = 1.9 ev
Ans: 1.6 V
15. The work function of a metal is 2.5 × 10–19 J (a) Find the threshold frequency for photoelectric emission
(b) If The metal is exposed to a light beam of frequency 6.0 × 1014 Hz what will be the stopping potential is
Ans: (a) 3.8 × 10 14 Hz (b) 0.91 V
16. The work function of a photoelectric material is 4.0 eV. (a) What is the threshold wavelength ? (b) Find the wavlength of light for which the stopping potential is 2.5 V.
Ans: (a) 310 nm (b) 190 nm.
17. Find the maximum magnitude of the linear momentum of a photoelectron emitted when light of wavelength 400 nm falls on a metal having work function 2.5 eV
Ans: 4.2 ×10 – 25 kg/ ms
18. When a metal plate is exposed to a monochromatic beam of light of wavelength 400 nm a negative potential of 1.1 is needed to stop the photocurrent .
Find the threshold wavelength for the metal.
Ans: 620 nm
19. In an experiment on potential of 1.1 V is needed to stop the photocurrent the data collected are as follows :
wavelength (nm) : 350 400 450 500 550
stopping potential (v) 1.45 1.00 0.66 0.38 0.16
Plot the stopping potential against inverse of wavelength (1/ l ) on a graph paper and find
(a) then planck constant, (b) the work function of the emitter and (c) the threshold wavelength.
Ans : (a) 4.2 × 10 – 15 eV –s (b) 2.15 eV (c) 585 nm
20. The electric field associated with a monochromatic beam becomes zero 1.2 × 1015 times per second Find the maximum kinetic energy of the
photoelectrons when this light falls on a metal surface whose work functions is 2.0 eV
Ans : 0.48 eV
21. The electric field associated with a light wave is give by E = Eo sin [(1.57 × 107 m– 1 ) (x – ct).]
Find the stopping potential when this lights is used in an experiment on photoelectric effect with the emitter having work function 1.9 eV.
Ans: 1.2 V
22. The electric field at a point associated with a light wave is E = (100 V/m ) sin [3.0 × 1015 S – 1)t]
sin [(6.0 × 10 15 s – 1)t] If this light falls on a metal surface having a work function of 2.0 eV , what will be the maximum kinetic energy of the photoelectrons ?
Ans : 3.93 eV
23. A monochromatic light source of intensity 5m W emits 8 × 1015 photons per second .This light ejects photoelectrons from a metal surface.
The stopping potential for this setup is 2.0 V. Calculate the work function of the metal.
Ans : 1.9 eV
24. Show figure is the plot of the stopping potential versus the frequency of the light used in an experiment on photoelectric effect .
Find (a) ratio h/e and (b) the work function.
Ans : (a) 4.14 × 10 – 15 Vs (b) 0.414 eV
25. A photographic film is coated with a silver bromide layer. When light falls on this film silver bromide
molecules dissociate and the film records the light there. A minimum of 0.6 eV is needed to dissociate a silver bromide molecule .
Find the maximum wavelength of light that can be recorded by the film
Ans: 2070 nm
26. In an experiment on photoelectric effect, light of wavelength 400 nm is incident on a cesium plate at the rate of 5.0 W .
The potential of the collector plate is made sufficiently positive with respect to the emitter so that the current teaches its
saturation value .Assuming that on the average one of every 106 photons is able to eject a photoelectron, find the photocurrent in the circuit .
Ans : 1.6 µ A
27. A silver ball of radius 4.8 cm is suspended by a thread in a vacuum chamber. Ultraviolet lights of wavelength 200 nm is incident on the ball
for some time during which a total light energy of 1.0 × 10 – 7 J falls on the surface. Assuming that on the average one photoelectron fine the
electric potential at the surface of the ball assuming zero potential at infinity. What is the potential at the centre of the ball?
Ans : 0.3 V in each case
28. In an experiment on photoelectric effect , the emitter and the collector plates are placed at a separation of 10 cm and are connected through
an ammeter without any cell
figure (42-E3) A, Magnetic field B exists parallel to the plates. The work function of the emitter is 2.39eV and the light incident on it has
wavelengths between 400 nm and 600 nm Find minium value of B for which the current registered by the ammeter is zero Neglect any effect of space charge.
Ans: 2.85 × 10 – 5 T
29. In the arrangement shown in figure (42 E4) , y = 10 nm d = 0.24mm and D = 1.2 m. The work function of the material of the emitter of 2.2 eV stop the photocurrent.
30. In a photoelectric experiment the collector plate is at 2.0 V with respect to the emitter plate made of copper (j = 4.5 eV.). The emitter is illuminated by a
source of monochromatic light of wavelength 200 nm Find the minimum and maximum kinetic energy of the photoelectrons reaching the collector.
Ans: 2.0 eV, 3.7 eV
31. A small piece of cesium metal (j = 1.9 eV) is kept at a distance of 20cm from a large metal plate having a charge density of 1.0 10 – 9 C/m on
the surface facing the cesium piece. A monochromatic light of wavelength 400 nm is incident on the cesium piece. Find the minimum and the maximum
kinetic energy of the photoelectrons reaching the large metal plate. Neglect any change in electric field due to the small piece of cesium present.
Ans: 22.6 eV , 3.7 eV
32. Consider the situation of the previous problem. Consider the fastest electron emitted parallel to the large metal plate. Find the displacement of this
electrons parallel to its initial velocity before it strikes the large metal plate
Ans: 9.2 cm
33. A horizontal cesium plate (j = 1.9 eV) is moved vertically downward at a constant speed n in a room full of radiation of wavelength 250 nm and above.
What should be the minimum value of n so that the vertically upward component of velocity is non positive for each photoelectron?
Ans : 1.04 10 6 m/s
34. A small metal plate (work function j ) is kept at a distance d from a singly ionized. Fixed ion. A monochromatic light beam is incident on the metal
plate and photoelectrons are emitted. Find the maximum wavelength of the light beam so that some of the photoelectrons may go round the ion along a circle.
Ans: 
35. A light beam of wavelength 400 nm is incident on a metal plate of work function 2.2 eV (a) A particular electrons absorbs a photon and makes two
collisions before coming out of the metal Assuming that 10% of the extra energy is lost to the metal in each collision find the kinetic energy of this
electrons as it comes out of the metal (b) Under the same assumptions find the maximum number of collisions the electrons can suffer before it
becomes unable to come out of metal.
Ans (a) 0.31 eV (b) 4