12. Modern Physics

ISBN : 9789386320124

Product Name : Modern Physics for JEE Main & Advanced

(Study Package for Physics)

Product Description : Disha's Physics series by North India's

popu-lar faculty for IIT-JEE, Er. D. C. Gupta, have achieved a lot of

ac-claim by the IIT-JEE teachers and students for its quality and

in-depth coverage. To make it more accessible for the students Disha

now re-launches its complete series in 12 books based on chapters/

units/ themes. These books would provide opportunity to students to

pick a particular book in a particular topic.

Modern Physics for JEE Main & Advanced (Study Package for

Phys-ics) is the 12th book of the 12 book set.

• The chapters provide detailed theory which is followed by

Impor-tant Formulae, Strategy to solve problems and Solved Examples.

Th

ese topics are taken from our Book:

MODERN PHYSICS

• Each chapter covers 5 categories of New Pattern practice exercises for JEE - MCQ 1 correct, MCQ more than

1 correct, Assertion & Reason, Passage and Matching based Questions.

• The book provides Previous years’ questions of JEE (Main and Advanced). Past years KVPY questions are

also incorporated at their appropriate places.

• The present format of the book would be useful for the students preparing for Boards and various competitive

exams.

5. Electron, Photon, Atoms,

Photoelectric Effect and X-rays 245-300

5.1 Discharge through gases at low pressure 246

5.2 Cathode rays 246

5.3 Discovery of electron 247

5.4 Photon 248

5.5 Photo-electric effect 250 5.6 Characteristics of PEE 252 5.7 Einstein explanation of PEE 252

5.8 Compton’s effect 253

5.9 Matter waves 254

5.10 The atom 257

5.11 Bohr’s atomic model 258 5.12 Spectral lines of hydrogen atom 261 5.13 Shortcomings of Bohr’s atomic model 263 5.14 Correction for the finite mass of the nucleus 264 5.15 X-rays 268 5.16 Origin of x-rays 269 5.17 Moseley’s law 270 5.18 Properties of x-rays 271 5.19 Bragg’s law 271

Review of formulae & important points 273 Exercise 5.1 - Exercise 5.6

Hints & solutions (Ex. 5.1 - Ex. 5.6)

6. Nuclear Physics 301-336

6.1 The nucleus 302

6.2 Properties of nucleus 302 6.3 The atomic mass unit 303 6.4 Mass defect and binding energy 304 6.5 Nuclear stability 305

6.6 Radioactivity 305

6.7 Laws of radioactive disintegration 306

6.8 Half life 307

6.9 Radioactive equilibrium 309

6.10 Nuclear fission 310

6.11 Nuclear fusion 312

6.12 Pair production and pair annihilation 314 Review of formulae & important points 319 Exercise 6.1 - Exercise 6.6

Hints & solutions (Ex. 6.1 - Ex. 6.6)

7. Electronics and Communication 337-382

7.1 Energy levels and energy bands 338 7.2 Valence band and conduction band 338 7.3 Intrinsic and extrinsic

Semiconductors 339

7.4 Electron current and

Hole current 340

7.5 P-N junction diode 342

7.6 Biasing of junction diode 342

7.7 Zener diode 344

7.8 Light emitting diode 345

7.9 Photodiode 345 7.10 Rectification by junction diode 345 7.11 Transistor 347 7.12 Transistor connections or configurations 348 7.13 Transistor as an amplifier 349 7.14 Transistor as an oscillator 351 7.15 Logic gates 353

7.16 OR, AND and NOT gates 354 7.17 Combination of gates 356

7.18 Communication 359

7.19 Band width of transmission medium 361

7.20 Antenna 361

7.21 Modulation and its types 363 7.22 Various modes of propagation

of em-waves 364

Review of formulae & important points 365 Exercise 7.1 - Exercise 7.4

Hints & solutions (Ex. 7.1 - Ex. 7.4)

250

5.5 P

-

When light of certain frequency is incident on a metal surface, electrons are ejected from the metal. This phenomenon is called photoelectric effect (PEE). Electrons ejected from the metal are called photoelectrons. The photoelectric effect was first observed by Heinrich Hertz in 1887. The effect was investigated in details by Hallwachs and Lenard.

Work function

We know that metals have large number of free electrons. These electrons move freely inside the metal but can not come out from it due to attraction of the positive ions. Some energy is needed to liberate the electrons from the bondage of the attraction of the ions. The minimum energy required to liberate the electrons from the metal surface, is called work function, and is represented by W₀.

Work functions of some photometals

Metal Work function (ev) Metal Work function (eV) Cesium 1.9 Calcium 3.2 Potassium 2.2 Copper 4.5 Sodium 2.3 Silver 4.7 Lithium 2.5 Platinum 5.6 Threshold frequency

The minimum frequency of the radiation which is just enough to liberate the electrons from the metal surface is called threshold frequency. Thus is f₀ is the threshold frequency, then work function

W₀ = hf₀, ...(1) where h is called Plank¢s constant, whose value is 6.626 × 10–34 _J-s.

Threshold wavelength

We know that the product of frequency and corresponding wavelength of any radiation is a constant and equal to c. Thus if l₀ is the threshold wavelength, then

l₀ = 0

c

f . ...(2)

Hence threshold wavelength is the longest wavelength which can cause photoelectric effect. Thus photoelectric effect takes place only if l £ l or ₀, f ³ f₀.

Kinetic energy of the photoelectrons

When light is incident on a metal surface, photons collide with the free electrons at or just within the surface of a metal, they transfer their energy to the electrons. Only some electrons will get this energy. In a particular collision, the photon may give all of its energy to the free electrons. If the energy of the photon is more than the work function W₀, the electron may come out of the metal. It is not necessary that if energy transfer to an electron is more than W₀, then the electron will come out from the metal surface. The electron after getting the energy, may lose its energy due to collisions with the atoms of the metal. Only an electron near the surface gets the extra energy and comes out from the metal surface.

Thus the kinetic energy of the photoelectrons emitted from the metal surface ranges from zero to a maximum value (K_max). If E is the energy of incident photons, then the maximum kinetic energy of any electron can be;

K_max = E – W₀. ...(3) Equation (3) is called Einstein¢s photoelectric equation.

Experimental set-up of PEE

The experimental set-up to study the photoelectric effect is shown in fig. 5.10.

When monochromatic light of frequency greater than f₀ is incident on the cathode,

Fig. 5.9

Chapter 5

251

E

, P

,

A

, P

X-

photoelectrons are emitted from it and they move towards anode A. Initially, the space between the cathode and the anode contains a number of electrons making up electron cloud. This negative charge repels the fresh electrons coming from the cathode. The electrons of maximum kinetic energy are able to reach the anode and constitutes a photocurrent. If anode is made positive with respect to the cathode, the emitted electrons are attracted by the anode and the photoelectric current increases. With the increase in anode potential, the photoelectric current increases and becomes maximum. Thereafter current will not increase with the increase in anode potential. This maximum value of current is called the saturation current (i_s). This will happen when all the emitted electrons by the cathode in any time interval are attracted by the anode . Fig. 5.11 shows the photoelectric current i with anode potential V.

Stopping potential

When anode is given negative potential with respect to the cathode, the photoelectric current decreases. For a particular value of anode potential, the photoelectric current becomes zero. The minimum negative anode potential at which photoelectric current becomes zero is called stopping or cut off potential V₀. To stop the photoelectric current, we must ensure that even the fastest electron will not reach the anode. Thus stopping potential is related to the maximum kinetic energy of the ejected electrons. If V₀ is the stopping potential, then

K_max = eV₀. ...(4)

5.6 C

1. Effect of intensity of incident light

When the intensity of the light increases, more number of photons strike with the photometal and thereby liberate more number of electrons. Because of this, photo current increases. As the frequency of the incident light is same, so maximum kinetic energy and hence slopping potential remains same.

2. Effect of frequency of incident light

When the intensity of incident light is kept constant and its frequency increases, the number of photons remains same but their kinetic energy increases. Therefore the emitted electrons are same in number but of greater kinetic energy and hence stopping potential also increases. Fig. 5.12 shows the variation of photocurrent with frequency of light f.

3. Effect of photometal

When intensity and frequency of incident light are kept constant and photo-metal is changed, the stopping potential V₀ versus frequency f are parallel straight lines. This shows that the slope V₀/f is same for all metals and is equal to universal constant (h/e). If the graph is plotted between K_max and f, then there is straight line. Slope of which gives the value of h (Fig. 5.13 b).

4. Effect of time

Metal starts emitting electrons as soon as light is incident on it and so there is no time lag between incident light and emitted electrons.

5.7 E

Einstein forwarded the Plank¢s quantum theory to explain photoelectric effect. According to him light is made of small energy bundles, called photons. The energy of photon is proportional to the frequency f. That is

E = hf,

where h is a universal constant, called Plank¢s constant. He made the following Fig. 5.10

Fig. 5.11

252

O

M

P

1. The photoelectric effect is the result of collisions between photons of incident light and free electrons of the metal.

2. The electrons of metal are bound with the nucleus by attractive forces. The minimum energy required to liberate an electron from this binding is called work function W₀.

3. The incident photon interacts with a single electron and spend energy in two parts :

(i) in liberating the electron from the metal surface,

(ii) and imparting kinetic energy to emitted electrons. Thus if hf is the energy of incident photons, then

h f = W₀ + K_max ...(i) As f = c_l and W₀ = 0 hc , l \ hc_l = max2 0 1 2 hc mv + l . ...(ii)

Above equation is known as Einstein photo-electric equation. It should be remembered that photoelectric effect will occur only if l £ l_o.

4. The efficiency of photoelectric effect is less than 1%, i.e., only less than 1% of photons are capable of ejecting electrons from the metal surface. The rest 99% of the photon energy will convert into thermal energy.

5. If V₀ is the stopping potential, then K_max = eV₀ and so hf = W₀ + eV₀ or hf e = 0 W e + V0 or V₀ = W0 h f. e e æ ö - _{+ ç ÷} è ø ...(iii)

The equation (iii) is a straight line between V₀ and f, whose slope is æ öç ÷_{è ø}h_e , which is a universal constant.

5.8 C

¢

When radiation of short wavelength (like X-rays) is incident on target of electron, the wavelength of scattered X-rays becomes longer than the wavelength of incident X-rays. This was first studied by Compton and therefore is known as Compton¢s effect.

According to him if l and l¢ are the wavelengths of incident and scattered X-rays, then shift in wavelength Dl is given by

Dl = l¢ – l = h

(

)

mc - f

where f is the scattering angle.

Fig. 5.13

253

E

, P

,

A

, P

X-

mc is a constant, called the Compton wavelength.

Compton shift depends only on the scattering angle. This phenomenon supported the view that both momentum and energy are transferred via photons.

305

6.6 R

The spontaneous disintegration of nucleus, is called radioactivity. This phenomenon was discovered by Henry Becquerel in 1896. Some radioactive elements are ; uranium, radium, thorium, polonium etc. Madam Curie discovered radium which is 106 _{times more} radioactive than uranium. Radioactivity is the nuclear phenomenon which does not depend on any physical or chemical changes.

Nuclear radiations

Rutherford discovered that radioactive elements emit two types of particles : a and b-particles. Each parent nucleus changes into daughter nucleus after emitting a and b particles until it changes into stable nucleus. Later, Villard discovered g-radiations along with a and b radiations. When radiations from any radioactive element enter into electric or magnetic field, they split into three parts : a-rays, b-rays and g-rays.

Fig. 6.3. Deflection of a, b and g-rays in electric and magnetic fields. Properties of a, b and g- rays

Feature a- rays b -rays g - rays 1 Identity Helium nucleus ₂He 4 Fast moving electron _-1b 0 Photon

2 Charge +2e –e Zero

3 Mass 4mp (mp = 1.87 × 10–27 kg) me = 9.1 × 10–31 kg rest mass zero

4

. Speed ;10 m/s7 1 to 99% of speed of light speed of light 3 × 108 _m/s

5

Effect of electric and

magnetic field

Deflected Deflected Not deflected

6 Ionisation power in comparison to g -rays 10000 100 1

7 Penetration _power less (can be stopped by _paper) 100 times a 10000 times a 8 Equation of _decay zXA¾¾®-a z 2- YA+zHeA zXA¾¾®z 1+YA+-1e0+ n zXA¾¾®zXA+ g

Chapter 6

Nuclear Physics

306

O

M

P

6.7 L

Rutherford and Soddy discovered the following laws :

(i) Radioactive disintegration is random. It is the matter of chance to disintegration any of the atom first.

(ii) During disintegration, either a-particle or b- particle is emitted at a time. Both the particles are never emitted simultaneously.

(iii) When an a-particle is emitted, a new atom is formed whose atomic number is decreased by two and mass number by four. Thus for an atom X of atomic number Z and mass number A,

A A 4 4

zX ¾¾®z 2- Y - +2He + g -rays.

(iv) Similarly, when a b-particle is emitted, a new atom is formed whose atomic number is increased by one but mass number remains the same. Thus for an atom X,

A A 0

zX ¾¾®z 1+Y + b + g --1 rays.

(v) The rate of decay of atoms is proportional to the number of undecayed atoms present at any instant. Thus if N is the number of atoms at any instant, then rate of disintegration æç_è-dN_dt ö÷_ø is ; dN dt æ_- ö ç ÷ è ø µ N or æç_è-dN_dt ö÷_ø = lN … (i) where l is a constant and is called decay or disintegration constant. From equation (i), we have

dN

N = -ldt

If N₀ and N are the number of atoms at t = 0 and at any time t, then

0 N N dN N

ò

ò

This shows that the number of atoms of a radioactive substance decreases exponentially with time.

307

N

P

Note:

1. The life time of atoms of radioactive substances are from zero to infinity. 2. If a radioactive substance decays simultaneously a and b-particles with

disintegration constants l₁ and l₂ respectively, then effective value of l is : l = l + l_{1 2}

and N = ( 1 2)t

0

N e- l +l .

3. The b-particle is not present initially in the nucleus but it is produced by the disintegration of neutron into a proton. Thus

1

0n = 1H1+ b + n-1 0 (antineutrino). When a proton is converted into neutron, a positron is emitted. Thus

1 1 0

1H ¾¾®0n + b + n1 (neutrino)

4. When a nucleus emits a g- particle, neither the atomic number nor the mass number changes.

6.8 H

The half life period of a radioactive substance is defined as the time in which one-half of the radioactive substance is disintegrated. If N₀ is the initial number of radioactive atoms, then after a half life t_1/2, the number of atoms present are 0

2 N . Thus at t = t_1/2 N = 0 2 N = _{N e0} -lt1/ 2 or elt1/ 2 = 2 \ lt1/ 2 = ln2 or t1/ 2 = ln 2 l = 0.693 l We can form a working formula line as follows : After t t= 1/ 2, N = ₂0 N = N₀(1/ 2)1 t = 2t_{1/ 2}, N = 0 4 N ₂ 0(1/ 2) N = -1/ 2 t nt= , N = N₀

( )

Average or mean life

The life time of radioactive atoms ranges from zero to infinity. The average life of all the atoms of a radioactive substance can be defined as :

T = sum of the life times of all the atoms total number of atoms If dN are the number of atoms with life time t, then

308

O

M

P

ò

ò

_ò

T = 1 l. … (2) As 1/ 2 0.693 t l = \ T = 1/ 2 1/ 2 1 _1.44 (0.693 /t )= t

Activity of radioactivity substance

The disintegration of radioactive atoms per unit time is called activity. Thus activity A is: A = dN_dt = 0 t dN e dt -l = (-l)N e0 -lt or A = lN. … (4)

If A ₀ is the initially activity, so

A₀ = lN₀. \ 0 A A = 0 N N As N N/ ₀=e-lt, \ 0 A A = e-lt. Units of activity

The SI unit of activity is becquerel.

1 becquerel = 1 disintegration /s 1 curice = 3.7 × 1010 _{disintegration/s} 1 rutherford = 106 _{disintegration /s}

6.9 R

Suppose a parent nuclide A with disintegration constant l decay into daughter_A nuclides B with disintegration constant l which changes into nuclide as follows :_B

309

N

P

N_A atoms of A and N_B atoms of B. At any time t,

the net rate of formation of B = rate of disintegration of A – rate of disintegration of B Thus dN_dtB = lANA- lBNB As NA = N e0 -lAt \ dNB dt = lAN e0 -lAt – lB BN or dN_B+ l_{B B}N dt = ₀ At AN e-l l Multiplying the equation by _elBt, we have

Bt _B B _{B B} el dN +el l N dt = ( ) 0 B A t AN el -l l or ( Bt) B d N el = ₀ ( B A)t AN el -l dt l After integration, we get

. Bt B N el = A 0 ( B A)t B A N el -l C æ l ö + ç_{l - l} ÷ è ø … (i) At t = 0, N_B = 0, \ C = – A 0 B A N æ l ö ç_{l - l} ÷ è ø .

Substituting the value of C in equation (i), we get

N_B = A 0 ( At Bt) B A N e-l e-l æ l ö -ç_{l - l} ÷ è ø … (1)

Special cases : If parent nuclide A is of long life than B, then l << l_{A B}, and from equation (1), we have

B N = A B(1 Bt) B N e-l l -l … (2)

After time t which is longer than half-life of B but shorter than half life of A, then

Bt e-l < <1 and N_A =N₀ and so B N = A A B N l l or l_{A A}N = l_{B B}N . … (3)