# NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation and Matter

NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation Matter is an essential resource material that is essential for your Class 12 and entrance exam preparation. NCERT solution of Class 12 Physics Chapter 11 eases your work of notes preparation and revision. It consists of dual nature of radiation and matter important questions with answers, important tables, exemplary questions and worksheets.

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## NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation Matter

Maxwell’s equations of electromagnetism and Hertz experiments on the generation and detection of electromagnetic waves in 1887 strongly established the wave nature of light. The NCERT solutions are one of the best tools to prepare for Class 12 physics exam. Students must prepare this chapter well to score well in their examination. The Class 12 physics NCERT solutions are given below so that students can understand the concepts of this chapter in-depth.

#### Concepts involved in Chapter 11 Dual Nature of Radiation Matter

1. Introduction
2. Electron Emission
3. Photoelectric Effect
1. Hertz’s observations
2. Hallwachs’ and Lenard’s observations
4. Experimental Study Of Photoelectric Effect
1. Effect of intensity of light on photocurrent
2. Effect of potential on photoelectric current
3. Effect of frequency of incident radiation on stopping potential
5. Photoelectric Effect And Wave Theory Of Light Ex
6. Einstein’s Photoelectric Equation: Energy Quantum Of Radiation
7. Particle Nature Of Light: The Photon
8. Wave Nature Of Matter

Some of the topics covered in this chapter are as follows. The minimum energy needed by an electron to come out from a metal surface is called the work function of the metal. Energy (greater than the work function (φο ) required for electron emission from the metal surface can be supplied by suitably heating or applying a strong electric field or irradiating it by the light of suitable frequency. Some key points on the Dual Nature of Radiation Matter are given below.

Free electrons in a metal are free in the sense that they move inside the metal in a constant potential (This is only an approximation). They are not free to move out of the metal. They need additional energy to get out of the metal.

Observations on photoelectric effect imply that in the event of matter-light interaction, absorption of energy takes place in discrete units of hν. This is not quite the same as saying that light consists of particles, each of energy hv.

### Class 12 Physics NCERT Solutions Dual Nature of Radiation Matter Important Questions

Q.11.1: Find the:

(a) Maximum frequency, and

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

Ans:

Electron potential, V = 30 kV = 3 × 104 V

Hence, electron energy, E = 3 × 104 eV

Where, e = Charge on one electron = 1.6 × 10-19 C

(a) Maximum frequency by the X-rays = ν

The energy of the electrons:

E = hν

Where,

h = Planck’s constant = 6.626 × 10-34 Js

Therefore, $v = \frac{E}{h}$

= $\frac{1.6\times 10^{-19}\times 3 \times 10^{4}}{6.626\times 10^{-34}}$ = 7.24 x 1018 Hz

Hence,  7.24 x 1018 Hz is the maximum frequency of the X-rays.

(b) The minimum wavelength produced:

$\lambda =\frac{c}{v}$

= $\frac{3\times 10^{8}}{7.24\times 10^{18}}$ = 4.14 x 10-11 m = 0.0414 nm

Q.11.2: The work function of caesium metal is 2.14 eV. When light of frequency 6 × 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,

(c) maximum speed of the emitted photoelectrons?

Ans:

Work function of cesium, $\Phi _{o}$ = 2.14eV

Frequency of light, v = 6.0 x 1014 Hz

(a) The max energy (kinetic) by the photoelectric effect:

K = hν – $\Phi _{o}$

Where,

h = Planck’s constant = 6.626 x 10-34 Js

Therefore,

K = $\frac{6.626\times 10^{34}\times 6\times 10^{14}}{1.6\times 10^{-19}}$ – 2.14

= 2.485 – 2.140 = 0.345 eV

Hence, 0.345 eV is the maximum kinetic energy of the emitted electrons.

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

K = eVo

Therefore, Vo = $\frac{K}{e}$

= $\frac{0.345\times 1.6\times 10^{-19}}{1.6\times 10^{-19}}$

= 0.345 V

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

(c) Maximum speed of photoelectrons emitted = ν

Following is the kinetic energy relation:

K = $\frac{1}{2}mv^{2}$

Where,

m = mass of electron = 9.1 x 10-31 Kg

$v^{2}=\frac{2K}{m}$

= $\frac{2\times 0.345\times 1.6\times 10^{-19}}{9.1\times 10^{-31}}$

=0.1104 x 1012

Therefore, ν = 3.323 x 105 m/s = 332.3 km/s

Hence332.3 km/s is the 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?

Ans:

Photoelectric cut-off voltage, Vo = 1.5 V

For emitted photoelectrons, the maximum kinetic energy is:

Ke = eVo

Where,

e = charge on an electron = 1.6 x 10-19 C

Therefore, Ke = 1.6 x 10-19 x 1.5 = 2.4 x 10-19 J

Therefore, 2.4 x 10-19 J is the maximum kinetic energy emitted by the photoelectrons.

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 a 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?

Ans:

Monochromatic light having a wavelength, λ = 632.8 nm = 632.8 × 10-9 m

Given that the laser emits the power of, P = 9.42 mW = 9.42 × 10-3 W

Planck’s constant, h = 6.626 × 10-34 Js

Speed of light, c = 3 × 108 m/s

Mass of a hydrogen atom, m = 1.66 × 10-27 kg

(a) The photons having the energy as:

E = $\frac{hc}{\lambda }$

= $\frac{6.626\times 10^{-34}\times 3\times 10^{8}}{632.8\times 10^{-9}}$

= 3.141 x 10-19 J

Therefore, each photon has a momentum of :

P = $\frac{h}{\lambda }$

= $\frac{6.626\times 10^{-34}\times 3\times 10^{8}}{632.8\times 10^{-9}}$

= 1.047 x 10-27 kg m/s

(b) Number of photons/second arriving at the target illuminated by the beam  = n

Assuming the uniform cross-sectional area of the beam is less than the target area.

Hence, equation for power is written as:

P = nE

Therefore, n= $\frac{P}{E}$

= $\frac{9.42\times 10^{-3}}{3.141\times 10^{-19}}$

= 3 x 1016 photons/s

(c) Given that, momentum of the hydrogen atom is equal to the momentum of the photon,

P= 1.047 x 10-27 kg m/s

Momentum is given as:

P=mv

Where,

ν = speed of hydrogen atom

Therefore, ν = $\frac{p}{m}$

= $\frac{1.047\times 10^{-27}}{1.66\times 10^{-27}}$ = 0.621 m/s

Question 11.5: The energy flux of sunlight reaching the surface of the earth is 1.388 × 103 W/m2. How many photons are incident on the Earth per second/square meter? Assume an average wavelength of 550 nm.

Ans:

Sunlight reaching the surface of the earth has an energy flux of

$\phi$ = 1.388 × 103 W/m2

Hence, power of sunlight per square metre, P = 1.388 × 103 W

Speed of light, c = 3 × 108 m/s

Planck’s constant, h = 6.626 × 10-34 Js

$\lambda$ = 550nm = 550 x 10-9is the average wavelength of the photons from the sunlight

Number of photons per square metre incident on earth per second = n

Hence, the equation for power be written as:

P = nE

Therefore, n=$\frac{P}{E}$

= $\frac{P\lambda }{hc}$

= $\frac{1.388\times 10^{3}\times 550\times 10^{-9}}{6.626\times 10^{-34}\times 3\times 10^{8}}$ = 3.84 x 1021 photons/m2/s

Therefore, 3.84 x 1021 are the photons that are incident on the earth per square meters.

11.6: In an experiment on the 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.

Ans:

Given that the slope of cut-off voltage (V) versus frequency (v) being:

$\frac{V}{v}$ = 4.12 x 10-15 Vs

V and frequency being related by the equation as:

Hν = eV

Where,

e = Charge on an electron = 1.6 x 10-19 C

h = Planck’s constant

Therefore, h = e x $\frac{V}{v}$

= 1.6 x 10-19 x 4.12 x 10-15 = 6.592 x 10-34 Js

Therefore, 6.592 x 10-34 Js is the Planck’s constant that is determined from the above equation.

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?

Ans:

Power of the sodium lamp P = 100W

Wavelength of the emitted sodium light, $\lambda$ = 589nm

= 589 x 10-9 m

Planck’s constant, h = 6.626 x 10-34 Js

Speed of light, c = 3 x 108

(a)

The energy per photon associated with the sodium light is given as:

E = $\frac{hc}{\lambda }$

E = $\frac{6.626\times 10^{-34}\times 3\times 10^{8}}{589\times 10^{-9}}$

= 3.37 x 10-19 J = $\frac{3.37\times 10^{-19}}{1.6\times 10^{-19}}$ = 2.11 eV

(b)

Number of photons delivered to the sphere = n

The equation for power can be written as:

P = nE

Therefore, n=$\frac{P}{E}$

= $\frac{100}{3.37\times 10^{-19}}$ = 2.96 x 1020 photons/s

Therefore, 2.96 x 1020 photons are delivered every second to the sphere.

Question 11.8: The threshold frequency for a certain metal is 3.3 x 1014 Hz. If the light of frequency 8.2 x 1014 is incident on the metal, predict the cut-off voltage for the photoelectric emission.

Ans:

Threshold frequency of the metal, vo= 3.3 × 1014 Hz.

Frequency of light incident on the metal, v= 8.2 × 1014 Hz

Charge on an electron, e = 1.6 × 10-19 C

Planck’s constant, h = 6.626 × 10-34 Js

Cut-off voltage for the photoelectric emission from the metal = Vo

The equation for the cut –off energy is given as:

eVo = h(ν – νo)

Vo = $\frac{h(v-v_{o})}{e}$

= $\frac{6.626\times 10^{-34}\times(8.2\times 10^{14}-3.3\times 10^{14})}{1.6\times 10^{-19}}$ = 2.0291 V

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?

Ans:

Work function of the metal, $\Phi _{o}$ =4.2eV

Charge on an electron, e = 1.6 x 10-19 C

Planck’s constant, h = 6.626 × 10-34 Js

Wavelength of the incident radiation, $\lambda$ = 330nm = 330  × 10-9 m

Speed of light, c = 3 × 108 m/s

The energy of the incident photon is given as:

E = $\frac{hc}{\lambda }$

= $\frac{6.626 \times 10^{-34}\times 3\times 10^{8}}{330\times 10^{-9}}$

= 6.0 x 10-19 J = $\frac{6.0\times 10^{-19}}{1.6\times 10^{-19}}$ = 3.76 eV

The energy of the incident radiation is less than the work function of the metal. Hence, there is no photoelectric emission taking place.

Question 11.10:  Light of frequency 7.21 x 1014 Hz is incident in a metal surface. Electrons with a maximum speed of 6.0 x 105 m/s are ejected from the surface. What is the threshold frequency for photoemission of electrons?

Ans:

Frequency of the incident photon, ν = 488nm = 488 x 10-9m

Maximum speed of the electrons, v = 6.0 x 105 m/s

Planck’s constant, h= 6.626 x 10-34 Js

Mass of an electron, m = 9.1 x 10-31 Kg

For threshold frequency vo, the relation for kinetic energy is written as:

$\frac{1}{2}mv^{2}$ = h(ν – νo)

νo = ν – $\frac{mv^{2}}{2h}$

= 7.21 x 1014$\frac{(9.1 \times 10^{-31})\times (6\times 10^{5})^{2}}{2 \times (6.626 \times 10^{-34})}$

= 7.21 x 1014 – 2.472 x 1014 = 4.738 x 1014 Hz

Therefore, 4.738 x 1014 Hz is the threshold frequency for the photoemission of the 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.

Ans:

Wavelength of light produced by the argon laser,

$\lambda$ = 488nm = 488 x 10-9 m

Stopping potential of the photoelectrons, Vo = 0.38 V

1eV = 1.6 x 10-19 J

Therefore, Vo = $\frac{0.38}{1.6\times 10^{-19}}$ eV

Planck’s constant, h = 6.6 x 10-34 Js

Charge on an electron, e = 1.6 x 10-19 C

Speed of light, c = 3 x 108 m/s

Using Einstein’s photoelectric effect, following is the relation for the work function:

$\Phi _{o}$ of the material of the emitter as:

eVo = $\frac{hc}{\lambda } – \phi _{o}$ $\Phi _{o}$  = $\frac{hc}{\lambda }$ eVo

= $\frac{6.6\times 10^{-34}\times 3\times 10^{8}}{1.6\times 10^{-19}\times 488\times 10^{-9}} – \frac{1.6\times 10^{-19}\times 0.38}{1.6\times 10^{-19}}$

= 2.54 – 0.38 = 2.16 eV

Therefore, 2.16eV is the work function of the material with which the emitter is made.

Question 11.12: Calculate the

(a) momentum, and

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

Ans:

Potential difference, V = 56V

Planck’s constant, h = 6.6 x 10-34 Js

Mass of an electron, m = 9.1x 10-31 Kg

Charge on an electron, e = 1.6 x 10-19C

(a) At equilibrium, the kinetic energy of each electron is equal to the accelerating potential i.e., we can write the relation of velocity (v) of each electron as:

$\frac{1}{2}mv^{2}$ = eV

$v^{2}=\frac{2eV}{m}$

Therefore, v = $\sqrt{\frac{2\times 1.6\times 10^{-19}\times 56}{9.1\times 10^{-31}}}$

= $\sqrt{19.69\times 10^{12}}$

= 4.44 x 106 m/s

The momentum of each accelerated electron is given as:

p = mv

= 9.1 x 10-31 x 4.44 x 106 = 4.04 x 10-24 Kg m/s

Therefore, 4.04 x 10-24 Kg m/s is the momentum of each electron.

(b) de Broglie wavelength of an electron accelerating through a potential V, is given by the relation:

$\lambda$ = $\frac{12.27}{\sqrt{V}}$ Ao

= $\frac{12.27}{\sqrt{56}}$ x 10-19 m = 0.1639 nm

Therefore, 0.1639 nm is the de Broglie wavelength of each electron.

Question 11.13: What is the:

(a) Momentum,

(b) Speed, and

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

Ans:

Kinetic energy of the electron, EK = 120 eV

Planck’s constant, h = 6.6 × 10-34 Js

Mass of an electron, m = 9.1 × 10-31 Kg

Charge on an electron, e = 1.6 × 10-19 C

(a) For an electron, we can write the relation for kinetic energy as:

Ek$\frac{1}{2}mv^{2}$

Where, v = speed of the electron

Therefore, $v^{2}=\sqrt{\frac{2eE_{k}}{m}}$

= $\sqrt{\frac{2\times 1.6\times 10^{-19}\times 120}{9.1\times 10^{-31}}}$

= $\sqrt{42.198\times 10^{12}}$

= 6.496 × 106 m/s

Momentum of the electron, p = mv = 9.1 × 10-31 × 6.496 × 106

= 5.91 × 10-24 kg m/s

Therefore, 5.91 × 10-24 Kg m/s is the momentum of the electron.

(b) speed of the electron, v = 6.496 × 106 m/s

(c) de Broglie wavelength of an electron having a momentum p, is given as:

$\lambda =\frac{h}{p}$

= $\frac{6.6\times 10^{-34}}{5.91\times 10^{-24}}$  = 1.116 x 10-10 m = 0.112 nm

Therefore, 0.112 nm the de Broglie wavelength of the electron.

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.

Ans:

Wavelength of light of a sodium line, $\lambda$ = 589nm = 589 x 10-9 m

Mass of an electron, me = 9.1 x 10-31 Kg

Mass of a neutron, mn = 1.66 x 10-27 Kg

Planck’s constant, h = 6.6 x 10-34 Js

(a) For the kinetic energy K, of an electron accelerating with the velocity v, we have the relation:

K = $\frac{1}{2}mv^{2}$ . . . . . . . . . . . . . . . (1)

We have the relation for de Broglie wavelength as:

$\lambda = \frac{h}{m_{e}v}$

Therefore, $v^{2} = \frac{h^{2}}{\lambda ^{2}m_{e}^{2}}$ . . . . . . . . . . . (2)

Substituting equation (2) in equation (1), we get the relation:

K = $\frac{1}{2}$ $\frac{m_{e}h^{2}}{\lambda ^{2}m_{e}^{2}}$

= $\frac{h^{2}}{2\lambda^{2}m_{e} }$ . . . . . . . . . . . . (3)

= $\frac{(6.6\times 10^{-34})^{2}}{2\times (589\times 10^{-9})^{2}\times 9.1\times 10^{-31}}$

= 6.9 x 10-25 J = $\frac{6.9\times 10^{-25}}{1.6\times 10^{-19}}$ = 4.31 x 10-6 eV

Hence, the kinetic energy of the electron is 6.9 x 10-25 J

(b) Using equation (3), we can write the relation for the kinetic energy of the neutron as:

= $\frac{h^{2}}{2\lambda ^{2}m_{n}}$

= $\frac{(6.6\times 10^{-34})}{2\times (589\times 10^{-9})^{2}\times 1.66\times 10^{-27}}$

= 3.78x 10 -28 J

= $\frac{3.78\times 10^{-28}}{1.6\times 10^{-19}}$

= 2.36 x 10-9 eV = 2.36 neV

The neutron has the kinetic energy of 3.78 x 10-28 J or 2.36 neV.

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?

Ans:

(a) Mass of the bullet, m = 0.040 Kg

Speed of the bullet, v = 1.0 km/s = 1000 m/s

Planck’s constant, h = 6.6 x 10-34 Js

de Broglie wavelength of the bullet is given by the relation :

$\lambda = \frac{h}{mv}$

= $\frac{6.6\times 10^{-34}}{0.040\times 1000}$

= 1.65 x 10-35 m

(b) Mass of the ball, m = 0.060 Kg

Speed of the ball, v = 1.0 m/s

de Broglie wavelength of the ball is given by the relation:

= $\lambda = \frac{h}{mv}$

= $\frac{6.6\times 10^{-34}}{0.060\times 1}$

= 1.1 x 10-32 m

(c) Mass of the dust particle, m = 1 x 10-9 Kg

speed of the dust particle, v = 2.2 m/s

de Broglie wavelength of the dust particle is given by the relation:

= $\lambda = \frac{h}{mv}$

= $\frac{6.6\times 10^{-34}}{2.2\times 1\times 10^{-9}}$

= 3.0 x 10-25 m

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 the electron.

Ans:

Wavelength of an electron $\lambda _{e}$ and a photon $\lambda _{p}$ ,

$\lambda _{e}$ = $\lambda _{p}$ = $\lambda$ = 1 nm = 1 x 10-9 m

Planck’s constant, h = 6.63 x 10-34 Js

(a) The momentum of an elementary particle is given by de Broglie relation:

$\lambda = \frac{h}{p}$ $p=\frac{h}{\lambda }$

It is clear that momentum depends only on the wavelength of the particle. Since the wavelengths of an electron and a photon are equal, both have an equal momentum.

Therefore, p = $\frac{6.63\times 10^{-34}}{1\times 10^{-9}}$

= 6.63 x 10-25 Kg m/s

(b) The energy of a photon is given by the relation:

E = $\frac{hc}{\lambda }$

Where,

Speed of light, c = 3 x 108 m/s

Therefore, E = $\frac{6.63\times 10^{-34}\times 3\times 10^{8}}{1\times 10^{-9}\times 1.6\times 10^{-19}}$

= 1243.1 eV = 1.243 keV

Therefore, the energy of the photon is 1.243 keV.

(c) The kinetic energy (K) of an electron having momentum p, is given by the relation:

K = $\frac{1}{2}\frac{p^{2}}{m}$

Where, m = Mass of the electron = 9.1 x 10-31 Kg

p = 6.63 x 10-25 Kg m/s

Therefore, K = $\frac{1}{2}\times \frac{(6.63\times 10^{-25})^{2}}{9.1\times 10^{-31}}$

= 2.415 x 10-19 J

= $\frac{2.415\times 10^{-19}}{1.6\times 10^{-19}}$ = 1.51 eV

1.51eV is the kinetic energy of the electron.

Question 11.17:

(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 300 K.

Ans:

(a) de Broglie wavelength of the neutron, $\lambda$ = 1.40 x 10-10 m

Mass of a neutron, mn = 1.66 x 10-27 Kg

Planck’s constant, h = 6.6 x 10-34 Js

Kinetic energy (K) and velocity (v) are related as:

K = $\frac{1}{2}m_{n}v^{2}$ ……… (1)

de Broglie wavelength ($\lambda$) and velocity (v) are related as:

$\lambda = \frac{h}{m_{n}v}$ ………. (2)

Using equation (2) and equation (1), we get:

K = $\frac{1}{2}\frac{m_{n}h^{2}}{\lambda ^{2}m_{n}^{2}}$

= $\frac{h^{2}}{2\lambda ^{2}m_{n}}$

= $\frac{(6.63 \times 10^{-34})^{2}}{2\times (1.40\times 10^{-10})^{2} \times 1.66\times 10^{-27} }$

= 6.75 x 10-21 J

= $\frac{6.75\times 10^{-21}}{1.6\times 10^{-19}}$

= 4.219 x10-2 eV

Hence, the kinetic energy of the neutron is 6.75 x 10-21 J or 4.219 x 10-2 eV.

(b) Temperature of the neutron, T = 300K

Boltzmann constant, k = 1.38 x 10-23 Kg m2 s-2 K-1

Average kinetic energy of the neutron:

K’ =  $\frac{3}{2}$ kT

= $\frac{3}{2}$ x 1.38 x 10-23 x 300

= 6.21 x 10-21 J

The relation for the de Broglie wavelength is given as:

$\lambda ‘=\frac{h}{\sqrt{2K’m_{n}}}$

Where,

mn = 1.66 x 10-27 Kg

h = 6.6 x 10-34 Js

K’ = 6.75 x 10-21 J

Therefore, $\lambda ‘=\frac{6.63\times 10^{-34}}{\sqrt{2\times 6.21\times 10^{-21}\times 1.66\times 10^{-27}}}$

= 1.46 × 10-10 m = 0.146 nm

Therefore, 0.146nm is the de Broglie wavelength of the neutron.

Question 11.18: Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).

Ans:

The momentum of a photon having energy (hv) is given as:

p = $\frac{hv}{c}$ = $\frac{h}{\lambda }$ $\lambda = \frac{h}{p}$ ………. (i)

Where,

$\lambda$ = wavelength of the electromagnetic radiation

c = speed of light

h = Planck’s constant

De Broglie wavelength of the photon is given as:

$\lambda = \frac{h}{mv}$

But, p = mv

Therefore, $\lambda = \frac{h}{p}$  ……………(ii)

Where, m = mass of the photon

v = velocity of the photon

From equation (i) and (ii) it can be concluded that the wavelength of the electromagnetic radiation and the de Broglie wavelength of the photon are equal.

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-mean-square speed of molecules at this temperature. (Atomic mass of nitrogen = 14.0076 u)

Ans:

Temperature of the nitrogen molecule, T = 300 K

Atomic mass of nitrogen = 14.0076 u

Hence, mass of the nitrogen molecule, m = 2 × 14.0076 = 28.0152 u

But, 1 u = 1.66 × 10-27 kg

Therefore, m = 28.0152 ×1.66 × 10-27 kg

Planck’s constant, h = 6.63 × 10-34 Js

Boltzmann constant, k = 1.38 × 10-23 J/K

We have the expression that relates mean kinetic energy $(\frac{3}{2}kT)$ of the nitrogen molecule with the root mean square speed (Vrms ) as:

$\frac{1}{2}mv_{rms}^{2}$ = $(\frac{3}{2}kT)$

Vrms = $\sqrt{\frac{3kT}{m}}$

For nitrogen molecule, the de Broglie wavelength is given as:

$\lambda =\frac{h}{mv_{rms}}$ = $\frac{h}{\sqrt{3mkT}}$

= $\frac{6.63\times 10^{-14}}{\sqrt{3\times 28.0152\times 1.66\times 10^{-27}\times 1.38\times 10^{-23}\times 300}}$

= 0.028 x 10-9 m = 0.028 nm

Therefore, the nitrogen molecule is 0.028 nm is the de Broglie wavelength.

Dual Nature of Radiation Matter along with other concepts are very crucial for your CBSE Class 12 board examination. It is very important for the students to understand the topics deeply to avoid difficulty in understanding higher concepts.

BYJU’S provide you with complete NCERT Class 12 Physics solutions to make you well equipped for the all-important CBSE Class 12 Board examination. BYJU’S also provides downloadable pdf of Chapter wise NCERT Solutions for Class 12 for subjects along with exemplar problems.

## Frequently Asked Questions on NCERT Solutions for Class 12 Physics Chapter 11

### What are the topics and subtopics covered under the Chapter 11 of NCERT Solutions for Class 12 Physics?

The topics and subtopics covered under the Chapter 11 of NCERT Solutions for Class 12 Physics are –
Introduction
Electron Emission
Photoelectric Effect
Hertz’s observations
Hallwachs’ and Lenard’s observations
Experimental Study Of Photoelectric Effect
Effect of intensity of light on photocurrent
Effect of potential on photoelectric current
Effect of frequency of incident radiation on stopping potential
Photoelectric Effect And Wave Theory Of Light Ex
Einstein’s Photoelectric Equation: Energy Quantum Of Radiation
Particle Nature Of Light: The Photon
Wave Nature Of Matter

### What is photoelectric effect in the Chapter 11 of NCERT Solutions for Class 12 Physics?

The photoelectric effect occurs because the electrons at the surface of the metal tend to absorb energy from the incident light and use it to overcome the attractive forces that bind them to the metallic nuclei. This concept can be clearly understood by the students using the NCERT Solutions created by the faculty at BYJU’S. The solutions contain explanations provided in a comprehensive manner to boost the exam preparation of students.

### Can students depend on the NCERT Solutions for Class 12 Physics Chapter 11 from BYJU’S?

The NCERT Solutions for Class 12 Physics Chapter 11 from BYJU’S are designed with the main aim of helping students to focus on the important concepts. Each and every minute detail is explained with utmost care to improve the conceptual knowledge among students. The solutions also contain various shortcut techniques which can be used to remember the concepts effectively. Students can refer to the solutions while answering the textbook questions and understand the method of answering them without any difficulty.