UPKAR PRAKASHAN, AGRA-2

ByDr. R.V.S. Chauhan

&Dr. H.P. Sharma

(Medical & Engineering Entrance Exams.)

© Publishers

Publishers

UPKAR PRAKASHAN(An ISO 9001 : 2000 Company)

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ISBN : 978-81-7482-504-9

Price : 315·00(Rs. Three Hundred Fifteen Only)

Code No. 345

Printed at : UPKAR PRAKASHAN (Printing Unit) Bye-pass, AGRA

Preface

The present book ‘Numerical Physics’ has been written on a well-conceived, broad based, scientific plan specially for the benefit ofstudents preparing for entrance examinations of Medical andEngineering faculties. It is equally useful for CBSE, ICSE, inter-mediate and all other 10 + 2 level examinations. The book is uniquelyuseful in systematic study and last minute revision of entire subjectmatter for these examinations and is of great help to them incommitting to memory the entire range of formulae and their short-cut and to-the-point applications.

To test your understanding and level of preparation, a number ofhighly selective numerical examples with explanatory hints and/orshort-cut solutions are given at the end of each chapter. These areimmensely useful from examination point of view.

Well-prepared students will find the book still more usefulbecause it will fill the gap, if any, in their preparation. Written ineverything at a glance style it will sharpen their intelligence andmake them quicker in solving problems, thus saving their mostvaluable time in the examination. We are satisfied that the book willbe a boon to those to whom it is meant.

Any suggestion for improvement of book or for any addition to itwill be highly welcome. These will be incorporated as far as possiblein the next edition of the book.

— Authors

CONTENTS

Chapters Pages

1. Units and Dimensions ………………………………………. 3–23

2. Kinematics ………………………………………………….. 24–38

3. Vector Analysis ……………………………………………... 39–67

4. Circular Motion ……………………………………………... 68–82

5. Newton’s Laws of Motion ………………………………….. 83–94

6. Work, Power and Energy …………………………………… 95–122

7. Universal Gravitation ……………………………………….. 123–139

8. Simple Harmonic Motion …………………………………… 140–159

9. General Properties of Matter ………………………………... 160–176

10. Elasticity ……………………………………………………. 177–185

11. Surface Tension …………………………………………….. 186–198

12. Flow of Liquids ……………………………………………... 199–209

13. Heat …………………………………………………………. 210–236

14. Sound ……………………………………………………….. 237–271

15. Light : Reflection of Light From Plane Surface …………….. 272–293

16. Refraction of Light Through Spherical Surface …………….. 294–340

17. Electricity …………………………………………………… 341–374

18. Capacity …………………………………………………….. 375–395

19. Electric Conduction ………………………………………… 396–417

20. Simple Circuit ………………………………………………. 418–439

Electromagnetism

21. Moving Charges and Magnetic Field ……………………….. 440–465

22. Galvanometers, Ammeter and Voltmeter …………………… 466–480

23. Magnetism …………………………………………………... 481–523

24. Electromagnetic Induction ………………………………….. 524–563

25. Alternating Current (A.C.) ………………………………….. 564–591

( viii )

Atomic Physics

26. Diode Valve and Triode Valve ……………………………… 592–645

27. Cathode Rays and Positive Rays ……………………………. 646–659

28. Photoelectric Effect ………………………………………… 660–668

29. Radiation ……………………………………………………. 669–676

30. Structure of Atom …………………………………………… 677–704

31. X-Rays ……………………………………………………… 705–716

32. Radioactivity ………………………………………………… 717–734

33. Nuclear Energy ……………………………………………… 735–747

34. Matter Waves and Special Theory of Relativity ……………. 748–755

PHYSICS

1 Units and DimensionsUnits—Units are of two type :(1) Fundamental units—Those units which do not depend upon other

units are called ‘fundamental units’.(2) Derived units—Those units which depend upon fundamental units

are called ‘derived units’.

Fundamental units (Basic SI units)

Quantity Unit Symbol

1. Length Metre m2. Mass Kilogram kg3. Time Second s4. Temperature Kelvin K5. Amount of Substance Mole mol6. Electric Current Ampere A7. Luminous Intensity Candela cd

Supplemental Units

Quantity Unit Symbol

Plane angle Radian rad Solid angle Steradian sr

● Physical Quantity = (Number) (Unit)● Now-a-days the units chosen are atomic (not mechanical). They do not

depend upon temperature, pressure etc. They are same at all planets.Now-a-days :

1 sec = Time for 9,192631,770 Cesium (Cs133) atom vibrations1 m = 1,650,763·73 Wavelengths of Kr86 (Krypton)

1 kg = mass of 5·0188 × 1025 atoms of C12

● Before 1963, several systems of units were in use but now-a-days onlysingle system (SI system) is used.

4 | Phy.

● Before SI units, there were two systems in vogue in Mechanics, theBritish system and the metric system. In the British system the basicunits were the foot, the pound and the second. In the metric system, thebasic units were the centimeters, the gram and the second.

1 pound = 453·6 gm= 0·4536 kg

1 foot = 12 inch = 30·48 cm = 0·3048 m

● Some General Units

Unit Related Conversion/RelationQuantity

Pascal Pressure 1 Pascal = 1 Newton/m2

Torr Pressure 1 Torr = 1 mm Hg pressureAtmosphere Pressure 1 atmosphere = 1·013 × 105 N/m2

=1·013 × 106 dynes/cm2

Fermi Length 1 fermi = 10–15 mMicron(μ) Length 1 μ = 10–6 m = 104 ÅAngstrom(Å) Length 1 Å = 10–10 mLight year Length 1 light year = 9·46 × 1015 m(Distance travelledby light in one year)Shake Time 1 shake = 10–8 secBarn Area 1 barn = 10–28 m2

Bar Pressure 1 bar = 105 N/m2

= 106 dynes/cm2

X ray unit Length 1 X-ray unit = 10–13 mPar second Length 1 par sec = 3·26 light yearsSlug Mass 1 slug = 14·59 kgChandra Shekhar Limit Mass 1 Chandra Shekhar limit

= 1·4 × solar massLitre Volume 1 litre = 103 CC = 10–3 m3

Yard Length 1 yard = 3 feetMile Length 1 mile = 1760 yards = 1·6 km

● Dimensions of Physical QuantitiesThe dimensions of a physical quantity show how that quantity is

related to the fundamental (or basic) quantities through its definingequation.

e.g., density =mass

volume

Phy. | 5

∴ The dimensional formula of density

=ML3

= [ML–3]

● Some Dimensional formulae

Quantity (with relations) Dimensional formula1. Length [L]2. Area [L2]3. Volume [L3]

4. Density =Mass

Volume =

ML3

= ML–3 [ML–3]

5. Relative Density or Specific Gravity

=Density of substance

Density of water = 1 No dimension

6. Speed or Velocity =Distance

Time

=LT

= LT–1 [LT–1]

7. Acceleration (a)—

=Change of velocity (ΔV)

time (Δt)

=LT–1

T = LT–2 [LT–2]

8. Force = mass × acceleration= M × LT–2 = MLT–2 [MLT–2]

9. Weight (Force with which the earthattracts a body towards its centre)Weight = force = MLT–2 [MLT–2]

10. Tension = force = MLT–2 [MLT–2]11. Thrust = Normal force = MLT–2 [MLT–2]12. Mass [M]

13. Pressure = Thrustarea

= ML–1 T–2 [ML–1T–2]

14. Momentum (p) = mass × velocity = mv = MLT–1 [MLT–1]

15. Impulse = force × time= MLT–2 × T = MLT–1 [MLT–1]

16. Moment of force or Torque (τ )= force × perpendicular distance of direction of force from the point

= F × r = MLT–2 × L = ML2T–2 [ML2T–2]

6 | Phy.

17. Work (W) = force × displacement= MLT–2 × L = ML2T–2 [ML2 T–2]

18. Light year = Distance = L [L]

19. Power (P) = worktime

= ML2T–2

T = ML2T–3 [ML2T–3]

20. Kinetic Energy (EK)

EK =12 mv2

= M × [LT–1] 2 = ML2 T–2 [ML2T–2]21. Potential Energy (EP)

EP = mgh

= M × LT–2 × L = ML2 T–2 [ML2T–2 ]22. Energy = Work = ML2T–2 [ML2T–2]23. Heat or Amount of heat (Q )

Q = energy = ML2T–2 [ML2T–2]24. Calorie = Unit of heat

= ML2T–2 [ML2T–2]25. Mechanical Equivalent of heat (J)

J =Mechanical Work done (W)

Heat produced (Q)

= ML2T–2

ML2T–2 = 1 No dimension

26. Time period (T)= time taken in one vibration = T [T]

27. Frequency (n) = 1

Time period =

1T

= T–1 [T–1]

28. Wavelength (λ) λ = L [L]29. Wave number (⎯υ)

= No of waves in unit length

= 1λ

= 1L

= L–1 [L–1]

30. Angle or Angular displacement (θ)

= arc

radius =

LL

= 1 No dimension

31. Angular Velocity (ω)

= angular displacement (Δθ)

time (Δt)

= 1T

= T–1 [T–1]

Phy. | 7

32. Angular acceleration (α)

α =Change in angular velocity (Δω)

time(Δt)

=T–1

T = T–2 [T–2]

33. Angular momentum (J)J = mv r

= M × LT–1 × L = ML2T–1 [ML2T–1]34. Planck 's Constant (h)

h =Eυ

= energy

frequency

=ML2T–2

T–1 = ML2T–1 [ML2T–1]

35. Angular Impulse= Torque × time

= ML2T–2 × T = ML2T–1 [ML2T–1]36. Universal Gravitation Constant (G)

G =Fr2

m1m2 =

MLT–2 × L2

M × M= M–1L3T–2 [M–1L3T–2]

37. Intensity of Gravitational field (I)

I =forcemass

= MLT–2

M= LT–2 [LT–2]

38. Gravitational Potential (V)(Work done in moving a unit massfrom infinity to that point)

V =workmass

= ML2T–2

M = L2T–2 [L2T–2]

39. Moment of Inertia (I)I = mr2 = ML2 [ML2]

40. Surface Tension

= forcelength

= MLT–2

L = MT–2 [MT–2]

41. Force Constant or Spring Constant (k)

k =Fx

=MLT–2

L = MT–2 [MT–2]

8 | Phy.

42. Coefficient of Viscosity (η)

η =F

6πrv =

MLT–2

L × LT–1

= ML–1T–1 [ML–1T–1]

43. Velocity Gradient

= ΔvΔx

= LT–1

L = T–1 [T–1]

44. Stress = forcearea

= MLT–2

L2

= ML–1T–2 [ML–1T–2]

45. Longitudinal strain

= Change in LengthOriginal Length

= LL

= 1 No dimension

46. Volume strain

= Change in volume (v)Original volume(V)

= L3

L3 = 1 No dimension

47. Shearing Strain or Angle of shear (φ)

= arc

radius =

LL

= 1 No dimension

48. Lateral strain

= Change in diameterOriginal diameter

= LL

= 1 No dimension

49. Modulus of Elasticity (E) or Young 'smodulus (Y) or Bulk modulus (K) ormodulus of rigidity (η)

= stressstrain

= F/A

strain

= MLT–2/L2

1 = ML–1T–2 [ML–1T–2]

50. Compressibility

= 1

Bulk modulus =

1K

= 1

ML–1T–2 = M–1LT2 [M–1LT2]

Phy. | 9

51. Poisson 's ratio (σ)

σ =Lateral strain

Longitudinal strain = 1 No dimension

52. Amplitude (A)= maximum displacement = L [L]

53. Avogadro No. (N)= Pure number = no dimension No dimension

54. Intensity of Wave (I) orIlluminance (E) or Emissivepower (e) or Solar Constant (S)

= energy

area × time =

ML2T–2

L2 × T = MT–3 [MT–3]

55. Gas Constant (R)

R =PVT

= ML–1T–2 × L3

θ= ML2T–2 θ–1 [ML2T–2 θ– 1]

56. Vander Waal 's Constants (a &b)

∴ equation is ( )P+ aV2 (V–b)= RT

∴ a

V2= Pressure (dimensionally)

⇒ a = PV2 = ML–1T–2 × (L3)2

= ML+5T–2 [ML+5T–2]and b = volume = L3 [L3]

57. Rydberg Constant [R)

R =1λ [ ] ∴ 1λ = R ( )1n12 –

1n22

=1L

= L–1 [L–1]

58. Wien 's Constant (b)b = λm T (product is constant)

= Lθ [Lθ]59. Stefan's Constant (σ)

σ =e

T4 [e = energy emitted

per unit area per unit time]

=ML2T–2/L2 × T

θ4 = MT–3θ–4 [MT–3θ–4]

10 | Phy.

60. Decay Constant or DisintegrationConstant (λ)

λ =0·693

Half Life =

1T

= T–1 [T–1]

61. Diffusion Constant (D)

∴

D = – n (X2 – X1)(n2 – n1)

Here n = no. of particles passingper unit area per unit timen1& n2 = No. of particles per unitvolume at distance X1 and X2respectively

∴ D =

No.of particlesarea × time

× distance

No. of particlesvolume

=

1L2T

× L

1L3

= L2T–1 [L2T–1]

62. Boltzmann's Constant (k)

k =R(Gas Constant)N(Avogadro no.)

=ML2T–1θ–1

1 = ML2T–1θ–1 [ML2T–1θ–1]

63. Thermal Capacity of a body

= ms = QΔT

[

∴

Q = msΔT]

=ML2T–2

θ = ML2T–2θ–1 [ML2T–2θ–1]

64. Specific heat (s )

s =Q

mΔT [

∴

Q = ms ΔT]

=ML2T–2

M × θ = L2T–2θ–1 [L2T–2θ–1]

65. Latent heat (L)

L =Qm

[

∴

Q = mL]

=ML2T–2

M = L2T–2 [L2T–2]

Phy. | 11

66. Coefficient of Thermal Conductivity (K)

K =Ql

A(θ1–θ2)t

[ ] ∴ Q = KA (θ1–θ2)tl =

ML2T–2× LL2 × θ × T

= MLT–3θ–1 [MLT–3θ–1]

67. Water Equivalent of Calorimeter (W)W= mass of water = M [M]

68. Thermal Current (H)H = Rate of flow of heat

=Qt

= ML2T–2

T= ML2T–3 [ML2T–3]

69. Temperature Gradient

=ΔθΔx

= θL

= θL–1 [L–1 θ]

70. Thermal Resistance (R)

R =θ1–θ2

H

=θ1–θ2Q/t

= (θ1–θ2)t

Q

=θ × T

ML2T–2 = M–1L–2T3θ [M–1L–2T3θ]

71. Entropy (S) =amount of heat

temperature

=ML2T–2

θ = ML2T–2θ–1 [ML2T–2θ–1]

72. Coefficient of linear expansionof solid (α)

=ΔL

L × Δθ = θ–1 [θ–1]

Similarly volume coefficient ofgas and Pressure coefficient of gashas dimensions [θ–1]

73. Temperature Coefficient of resistance

=1

temperature

=1θ = θ–1 [θ–1]

12 | Phy.

Electricity1. Charge (Q)

Q = It = A T [AT]

2. Intensity of Electric field (E)

E =Fq

[ ∴

F =qE]

=MLT–2

AT = MLT–3A–1 [MLT–3A–1]

3. Electric potential (V)

V =Work

Charge =

ML2T–2

AT= ML2T–3A–1 [ML2T–3A–1]

● Electromotive force (emf) alsohave this dimension

4. Potential Gradient

=ΔVΔX

= ML2T–3A–1

L= MLT–3A–1 [MLT–3A–1]

5. Permittivity of Vacuum (ε0)

ε0 =θ1– θ24πr2F

⎣⎢⎢⎡

⎦⎥⎥⎤

∴F =

14πε0

Q1Q2 r 2

=AT × AT

L2 × MLT–2

= M–1L–3T4A2 [M–1L–3T4A2]6. Permittivity of medium (ε)

Dimension of ε is same as that of ε0 [M–1L–3T4A2]

7. Dielectric Constant (K)

K =εε0

= 1 No dimension

8. Electric Dipole moment (p)p = q × 2l = AT × L

= ATL = LTA [LTA]

9. Capacity (C)

C =QV

= Q

W/Q =

Q2

W

=(AT)2

ML2T–2

= M–1L–2T4A2 [M–1L–2T4A2]

Phy. | 13

10. Electric Resistance (R)

R =VI =

W/QI

= WQI

= ML2T–2

AT × A = ML2T–3A–2 [ML2T–3A–2]

11. Potential Energy of a capacitor (U)

U =12 CV2 =

12 QV =

Q2

2C

= ML2T–2 (Unit of energy) [ML2T–2]

12. Current density (J)

J =current (I)area (A)

= AL2

= L–2A [L–2A]

13. Specific Resistance orElectric Resistivity (ρ)∴

R = ρ lA

⇒ ρ =ARl

= L2 × ML2 T–3A–2

L

= ML3T–3A–2 [ML3T–3A–2]14. Electric Conductivity or Specific

Conductance (σ)∴

σ =1ρ =

1ML3T–3A–2

= M–1L–3T3A2 [M–1L–3T3A2]

15. Intensity of magnetic field (B)∴

F = IBl sin θ

⇒ B =F

Il sin θ =

MLT–2

AL

= MT–2A–1 [MT–2A–1]

16. Magnetic flux (φ)φ = BA cos θ

= MT–2A–1× L2

= ML2T–2A–1 [ML2T–2A–1]

17. Magnetic moment (M)

M = n I A (n → no. of turns incoil . A → area, I → current)

= 1 × L2× A = L2A [L2A]

14 | Phy.

18. Pole Strength (m)

m =M2l

= L2AL

= LA [LA]19. Self Inductance (L)

e = –L d Idt

⇒ L = – e × dt

d I

= –

WQ

× dt

d I =

ML2T–2 × TAT × A

= ML2T–2A–2 [ML2T–2A–2]

● Mutual inductance M ( )e = –M d Idthave same dimension as that of L.

20. Magnetic Energy (U)

U =12 LI2 = ML2T–2 [ML2T–2]

21. Permeability of Vacuum (μ0)

∴ F =

μ04π

× 2I1I2

r

or, μ0 =F × 4πr

2I1I2 =

MLT–2 × LA × A

= ML2T–2A–2 [ML2T–2A–2]

22.LR

(L = inductance, R = Resistance)

LR

= Time constant of L –R circuit

= dimension of time = T [T]23. RC (R = Resistance, C = Capacitance)

RC = Time Constant of R – C Circuit= dimension of time = T [T]

24. LC (L = inductance, C = Capacitance)∴

frequency of electrical oscillation

n =1

2π√⎯⎯LC

or n2 =1

4π2LC

Phy. | 15

⇒ LC =1

4π2n2 =

1( )T–1 2

=1

T–2 = T2 [T2]

25. LI2 (L = inductance, I = current)∴

Magnetic potential energy U

=12 LI2

∴ LI2 = 2U = Energy= ML2T–2 [ML2T–2]

26. Power of Lens (P)

P =1f

= 1 metref metre

=100 cmf cm

= LL

= 1 No dimension

27. Enthalpy (H)∴

H = U + PV = Total heat content

= ML2T–2 [ML2T–2]28. ε0 μ0 (ε0 = Permittivity of free space

μ0 = Permeability of free space)∴

ε0 μo =1C2

(C = Speed of light)

=1

( )LT–1 2

=1

L2T–2 = L–2T2 [L–2T2]

29. VI or I2Rt or V2

R (V = Potential difference)

∴VI = I2Rt

=V2

R = P = Power

=Wt

= ML2T–2

T= ML2T–3 [ML2T–3]

30. VIt or I2Rt or V2tR

(R = resistance)

∴ VIt = I2Rt = V2t R

= Work = ML2T–2 [ML2T–2]

Quicker Numerical Physics

Publisher : Upkar Prakashan ISBN : 9788174825049 Author : Dr. R.V.S. Chauhanand Dr. H.P. Sharma

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