Kamal A.A. 1000 Solved Problems in Classical Physics: An Exercise Book

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760 15 Optics

15.64



nλ



50 × 1 × 5 × 10

−5

= 0.05 cm



50 × 25 × 5 × 10

−5

= 0.25 cm

15.3.6 Polarization

15.65 Path difference introduced by the plate

x = (n

− n

)t = (1.642 − 1.478) ×4 × 10

−5

= 6.56 ×10

−6

Phase difference

δ =

2πx

(2π)(6.56 × 10

−6

)

6 × 10

−7

= 68.66 rad

15.66 I

= I

max

cos

θ (Malus’ law)

cos θ =



max



√

∴ θ =±45, ±135

◦

15.67 tan θ

= μ = 1.33 (Brewster’s law)

∴ θ

= 53

◦

Therefore the elevation of the sun is 90 −53

◦

= 37

◦

15.68 tan Q

= μ

1.5

1.33

= 1.1278

∴ Q

= 48.4

◦

15.69 t =

4(μ

− μ

)

6 × 10

−5

4(1.553 − 1.544)

= 1.67 ×10

−3

15.70 θ = αLD

where θ is the angle of r otation of plane of polarization, α is the speciﬁc

rotation, L is the length of the tube in decimetres and D is the amount of

solvent in grams per 100 c.c:

L = 20 cm = 2dm

15.3 Solutions 761

D =

αL

13.2

◦

× 2

= 0.1g/100 c.c. = 1.0g/L

15.71 As the light passing through the ﬁrst sheet is unpolarized, the intensity I

the light transmitted by the ﬁrst sheet is given by the one-half rule:

(1)

Because the polarizing direction of the ﬁrst sheet is parallel to the y-axis,

the polarization of the light transmitted by it is also along y-axis.

Because the light reaching the second sheet is polarized, the intensity I

of the light transmitted by that sheet is given by the cosine-squared rule.

The angle θ in the rule is the angle between the polarization direction of

the incoming light, parallel to the y-axis, and the polarizing direction of the

second sheet, θ being counterclockwise from the y-axis. Thus

= I

cos

θ (Malus’ law) (2)

Because the light entering the third sheet is polarized and the polarizing

between the second and the third sheets is 90 − θ , the transmitted intensity

is again given by Malus’ law. Thus

= I

Cos

(90 − θ) = I

sin

θ (3)

From (3), (2) and (1)

= I

sin

θ = I

cos

θ sin

θ =

cos

θ(1 − cos

θ) (4)

= 0.1152 =

cos

θ(1 − cos

θ)

or cos

θ −cos

θ +0.2304 = 0

∴ cos

θ = 0.64 or 0.36

or cos θ =±0.8or ±0.6

Taking only the positive value

θ = 36.87

◦

or 53.13

◦

The polarization is along the x-axis.

15.72 Let an unpolarized light beam be incident from air on a dielectric such as

glass. The E vector for the wave can be resolved into two components – one

parallel to the plane of incidence, that is, the plane of paper and the second

762 15 Optics

one perpendicular to the plane of incidence. The former one is represented

by double arrow and is called π-component while the latter shown by dots is

known as σ -component; Fig. 15.36.

Fig. 15.36 Polarization by reﬂection • σ -component ↔ π -component

For a given dielectric, there is a particular angle of incidence i

, called the

polarizing angle or Brewster angle, at which the reﬂected beam is completely

polarized with its plane of vibration perpendicular to the plane of incidence,

that is, the reﬂected beam contains σ -component alone. Experiments show

that at the polarizing angle, the reﬂected and refracted beams are at right

angles. This leads to Brewster’s law

tan i

= μ

where μ is the refractive index. Note that at the polarizing angle the

π-component is entirely refracted containing some amount of σ-component

as well. Thus the transmitted light is only partially polarized. With the use

of a stack of glass plates the proportion of σ-component can be increased

in the reﬂected beam from multiple reﬂections and at the same time

π-component can be made richer in the transmitted beam. Thus the trans-

mitted beam is rendered plane polarized with the plane of vibration in the

plane of incidence.

Dielectrics such as glass, water can partially or fully polarize light by reﬂec-

tion. If the surface is horizontal, the light is partially or fully polarized hori-

zontally resulting in a bright spot (the glare) on the surface where reﬂection

takes place. Such a glare from horizontal surfaces is eliminated by mounting

the lenses in polarizing glasses with their polarizing direction vertical.

Fig. 15.37 Brewster

windows mounted on a laser

tube

15.3 Solutions 763

Brewster windows are used in laser technology (Fig. 15.37). The tubes of gas

lasers are sealed by mounting them in such a way that laser light is incident

at Brewster’s angle. Mirrors mounted outside the tube reﬂect the light back

and forth through the tube. A small hole in one mirror permits the laser light

to leave. Because the windows are tilted to the axis such that light is incident

at Brewster’s angle, the parallel component can traverse back and forth in the

tube with minimum attenuation. The perpendicular component after several

traversals gets substantially attenuated. Thus the light leaving the laser is

polarized in the parallel direction

tan θ

= μ = 1.5

∴ θ

= 56.3

◦

, which is also the inclination of the Brewster window.

Subject Index

Bernoulli’s equation, 394, 398–403, 405

Capacitors, 358, 459, 461, 463–464, 476–481,

517–533, 586, 634, 637, 639–641, 643,

653–654, 657, 660, 662

Centre of mass, 3, 10–12, 36–46, 48–49, 51,

55–56, 58, 61, 63, 67, 74, 83, 87, 136,

138–139, 142–143, 148, 151–152,

154,

163, 166, 168, 171, 174, 181, 183, 197,

238, 248, 268–269, 271, 273–274, 323

Collisions, 5, 20, 34, 50, 58–63, 77–95, 146,

148, 247, 409, 414, 420, 425–426, 429,

440, 661

Coriolis acceleration, 149–150, 184–188

Coupled systems of masses and springs,

251–253, 273–279

Damped vibrations, 241–243, 253–254,

279–286

Diffraction,

721–723, 751–760

Doppler effect, 343, 354–355, 384–385

Elasticity, 341–342, 410, 413, 423–424,

452–455

Electric circuit, 535–578

Electric ﬁeld, 459, 462, 464–477, 481–506,

508–509, 515, 517, 521, 523, 531, 533,

584–586, 595, 602–605, 626, 629, 634,

642–643, 645, 647–648, 657, 680–683,

689,

711

Electric potential, 460–461, 465–466,

469–473, 483, 485, 585, 695

Electromagnetic waves, 598, 635–636,

642–649, 665–692, 696, 712, 717

Electromagnetism, 579–630, 631–702

Electrostatics, 459–533, 634–635

EMF, 538–544, 552–561, 599, 633

Energy

elastic, 454

electrostatic, 464, 475, 477, 514, 516

gravitational, 194, 205, 209, 336, 392

heat, 440–441, 456, 679

internal,

412, 422–423, 442, 448, 450, 452

kinetic, 6, 26, 32, 50, 54, 59, 61–63, 69, 76,

79–80, 82, 84–86, 88, 90, 92, 95, 98,

106, 124–125, 128–131, 136, 139–140,

146, 157–158, 161, 163–167, 170, 174,

177, 182–183, 213, 217, 221, 223–224

232–233, 241, 246–247, 270, 277–278,

287–288, 297, 299–300, 303, 307, 312,

319, 324, 326–327, 330, 336, 365, 370,

392, 415, 440, 454, 502, 579, 633,

643, 671

magnetic, 581–582, 592, 595, 622–625,

643, 670, 675, 679

mechanical, 98, 106, 125, 127, 217,

440–441

potential,

48, 54, 57, 69, 76, 88, 98, 106,

124–126, 128–131, 157, 161, 163, 165,

167, 170, 174, 177, 181, 189–190, 194,

205, 212–214, 217, 221, 224, 232–233,

240–241, 246–247, 260, 271, 277–278,

287–288, 294–295, 297, 299–300, 303,

307, 312, 319

, 324, 326–327, 330, 336,

370, 392, 440, 461–462, 466, 471, 473,

478, 483, 488, 500, 502, 505–506, 532,

595, 633

spring, 79, 336

total, 48, 84, 182, 191–192, 198–199, 213,

217, 219, 221, 223–224, 233, 236, 245,

271, 351, 370–371, 513, 656

Equilibrium, 3, 9, 12–

13, 44–46, 57, 67, 69,

76, 106, 109, 112, 117–119, 125–127,

132, 136, 200, 238, 246–248, 250, 252,

765

766 Subject Index

255–256, 260, 269, 272, 274, 288, 290,

293, 295, 312, 332, 346, 384, 402, 425,

440, 442, 454, 461–462, 474, 476, 484,

487, 489, 496, 507, 526, 620, 629

Faraday’s law, 582–583, 596–599, 621,

625–629, 634–635, 647, 660,

683, 685

Field

combined, 505

electric, 459, 462, 464–477

, 481–506,

508–509, 515, 517, 521, 523, 531, 533,

583–585, 595, 602–605, 626, 629, 634,

642–643, 647–648, 680–683, 689, 711

external, 621

force, 48

gravitational, 189, 195–196, 210–211, 482

magnetic, 579–580, 582–585, 590–593,

595–607, 612, 616, 619, 622, 627–629,

642–644,

647, 654, 657, 674–675,

681–684

potential, 250

Fluid dynamics, 391–408

Force

average, 59, 62, 628

centrifugal, 106, 119, 125, 130, 137, 166,

173, 400, 402

centripetal, 104–106, 116–118, 120–122,

133, 160, 204, 219, 225, 504, 516, 602

conservative, 48, 191, 292

coriolis, 137, 150, 188

damping, 253, 282

external, 40, 47–51, 59, 71, 122

frictional,

35, 67–68, 75, 104, 116,

118–122, 126, 133, 142, 145, 169, 242

gravitational, 47, 67–68, 70, 173, 189,

193–195, 202–204, 219, 225, 461, 468,

484, 489–490, 492, 496, 603, 620, 626

horizontal, 10, 52–53, 56

internal, 47–48

lateral, 150, 188

magnetic, 581, 593–595, 601–602, 604,

619

–621, 626–627, 682

resultant, 49, 95, 113, 130, 204, 397, 594

stretching, 105, 454–455

vector, 9, 59, 406

Gauss’ law, 462–463, 473–476, 478, 481–482,

506–517, 524, 532–533, 634–635,

665–666, 668

Geometrical optics, 713–715, 725–728

Gravitation, 47, 67–68, 70, 88, 173, 189–233,

260, 296, 310, 336

, 392, 461, 467–469,

482, 484, 489–490, 492, 496, 533, 603,

620, 626

Group velocity, 351–352, 375, 636, 649–650,

692–696, 698–699, 702

Hamiltonian mechanics, 287–338

Heat, 409–457

Heat transfer, 410–411, 418–420, 433–439

Instruments, 537–538, 544–547, 564–569

Interference, 339, 365, 636, 705–710, 717–722,

740–

755

Kinematics, 1–46, 103–133

Kinetic theory of gases, 414–416, 425–429

Kirchhoff’s law, 538, 547–552, 569–578

Lagrangian mechanics, 287, 289

Latent heat, 419–420, 439–441

Loop-the-loop, 107, 112–114, 129–133, 144

Magnetic dipole moment, 581–582, 595,

622–625

Magnetic energy, 581–582, 592, 595, 622–625,

643, 670, 675, 679

Magnetic force, 581, 593–595, 619–622

Magnetic induction, 580–581, 587–

593, 597,

606–619, 621, 634, 668, 684

Matter, 409–457

Maxwell’s equations, 634, 642–648,

665–696, 700

Moment of inertia, 9, 54–56, 135–136,

138–141, 146, 151–157, 161–163, 167,

169, 177, 180, 183, 248, 250, 268, 270,

274–275, 296

Motion in a horizontal plane, 104–105,

107–111, 114–123

Motion in a vertical plane, 106–107, 112,

123–128

Motion in 1-D, 1–6, 13–26

Motion in 2-D, 2, 6–8, 26–34

Motion of blocks on a plane, 52–53, 64–68

Motion of charged particles, 579–580,

583–587, 599–606

Motion on incline, 53–56, 68–75

Subject Index 767

Optics, 703–763

Oscillations, 149, 235–286, 289, 291, 294,

320, 323, 331, 350, 356, 638, 648, 655,

664, 692

Phase velocity, 344, 346, 351–352, 375, 636,

649–650, 692–696, 698–699

Physical pendulum, 238–239, 248–250,

267–272

Polarization, 481, 533, 634, 644, 647, 690–691,

711–713, 724, 760–763

Potential,

3, 12, 48, 54, 57, 69, 76, 88, 98,

106, 124–125, 128–131, 157, 161,

163, 165, 167, 170, 174, 177, 181,

189–190, 193–196, 201–214, 217, 221,

224, 232–233, 240–241, 246–247, 250,

260, 271, 277–278, 287–288, 290, 292,

294–295, 297, 299

–300, 303, 307, 319,

326–327, 330, 336, 370, 392, 440,

459–473, 475–480, 482–506, 509, 513,

515, 520–521, 524–525, 528–529, 532,

538, 540, 542, 544, 546, 552–553, 555,

563–564, 575, 582, 585–587, 595, 599,

629, 633, 637–638, 660, 668–669,

682, 685

Power, 47, 56–58, 75, 108, 114, 136, 201,

230, 244, 253, 297, 340, 353, 355, 365,

379, 381, 383, 419, 439, 469, 471, 480,

505, 536, 538–544, 548–561, 564, 570,

572, 584, 595, 622, 633, 635, 637–638,

643–646, 652–654, 656, 663, 671, 675,

679

–681, 703, 710

Poynting vector, 635, 642–648, 665–692

Resistance, 2, 6, 8, 21, 63, 149, 181, 241, 254,

280, 419, 438, 481, 535–536, 538–543,

552–569, 572, 574, 576, 583, 595–598,

627, 633, 637–639, 641, 643, 646,

657–658, 662, 675, 681–682

Reverberation, 344, 355–356, 386

RLC circuit, 637–639, 651–664

Rockets, 196–201, 213–233

Rotational dynamics, 135–188

Rotational kinematics, 103–133

Rotational motion, 55, 139–149, 157–188

Satellites, 191–192, 196–201, 213–233

Shock wave, 344, 355, 386

Simple harmonic motion, 182, 235–236,

245–248, 248, 254–267, 277, 288, 318,

325–327, 331, 473

Sound waves, 347, 352–354, 365, 378–384

Speciﬁc heat, 383, 420, 425

, 439–441, 453

Statics, 1–46

Surface tension, 341–342, 350–352, 374, 377,

409, 413–414, 425, 455–457

Thermal expansion, 410, 416–418, 430–433

Thermodynamics, 412–413, 420–423, 441–452

Torque, 9–10, 35–36, 45, 48, 54, 70, 105,

121–122, 144–147, 158, 161–162, 165,

167, 169–172, 178–181

, 239, 250, 267,

270, 275, 306, 462, 595

Torricelli’s theorem, 392, 396–397, 403–406

Variable mass, 49, 63–64, 95–102

Vibrating strings, 340, 345–350, 356–371

Viscosity, 391, 393–394, 396–398,

406–408, 468

Waves

in liquids, 341, 350–352, 373–378

in pipes, 343, 356, 389–390

in solids, 341, 350, 371–373

Work, 48, 53, 56–58, 67

–68, 75–77, 79, 86–87,

135, 142, 147–148, 163–164, 175, 177,

179, 189, 195, 197, 211, 214, 248,

265–266, 392, 396, 401, 412–413,

421–423, 425, 441–442, 445, 448,

450–452, 460, 465, 467, 470, 481, 489,

499–500, 502, 514, 532, 546, 548,

628, 634

Problem Index

Chapter 1

Kinematics and Statistics

1.1 Overtaking of a truck travelling at constant speed by a car starting from rest at

constant acceleration.

1.2 Greatest height attained by a stone projected up from an elevated point.

1.3 Meeting of two stones one projected up from the ground and the other dropped

from a height.

1.4 Distance travelled by a particle in 3 s when x = A sin πt.

1.5 A man of height 1.8 m moves away at 7 m/s from a lamp 6 m high. To ﬁnd the

speed at which the tip of shadow moves.

1.6 The displacement x of a particle is described by 3t =

√

3x + 6. To ﬁnd x

when v = 0.

1.7 A particle projected up passes the same height h in 2 and 10 s. To ﬁnd h.

1.8 Overtaking of cars with constant acceleration and constant deceleration.

1.9 Choice of route in the ﬁeld and on the road so that a boy may reach the desti-

nation in minimum time.

1.10 Location of water drops from the nozzle of a shower which fall at regular time

intervals.

1.11 Velocity–time graph for an object thrown upwards from the roof of a building.

1.12 A ball dropped into a lake from a diving board.

1.13 One stone is dropped from h = 44.1 m, another is thrown down 1 s later. To

ﬁnd u of the second stone if both the stones strike the ground simultaneously.

1.14 A ball is seen to move up and down before a window 2 m high for t = 1s

overall. To ﬁnd the height above the window to which the ball rises.

1.15 In the last second of a free fall a body covered three-fourth of its total path. To

ﬁnd t and h.

1.16 Relative velocity of wind.

1.17 Time for a bolt to hit the ﬂoor of an elevator which accelerates with constant a.

1.18 Deceleration of a car and truck to avoid a rear-end collision.

1.19 Relative velocity and distance of closest approach of two ships.

769

770 Problem Index

1.20 A packet is dropped from an ascending balloon at 9.8 m/s at a height of 98 m.

To ﬁnd time for the packet to reach the ground.

1.21 Time of ascent/time of descent and initial speed/ﬁnal speed when a body is

thrown up with heavy air resistance.

1.22 Motion of a body in free fall with air resistance proportional to velocity.

1.23 In prob. (1.22) air resistance is proportional to the square of velocity.

1.24 Under the assumption of velocity square air resistance to ﬁnd h to which a

body will rise when projected upwards with velocity u.

1.25 Loss in kinetic energy when a body thrown upwards returns with air resistance.

1.26 Given velocity components dx/dt = 6 + 2t and dy/dt = 4 + t, to ﬁnd x(t)

and y(t), v and a.

1.27 Two objects are projected horizontally in opposite directions with velocity u

and u

from a tower. To ﬁnd time when v

is ⊥ to v

and the distance of

separation.

1.28 An object projected up from the foot of a tower crosses the top of a tower in

time t

and recrosses it in time t

.Ift

is the time for free fall from the top of

tower then t

√

1.29 Maximum range of a shell ﬁxed at angle θ up an incline of α is obtained when

θ = α/2 + π/4.

1.30 A stone thrown from ground over horizontal ground just clears three walls

separated by r and 2r. The inner wall is 15/7 as high as the outer ones which

are equally high. To ﬁnd n if R = nr.

1.31 The velocity and angle of projection so that a ball may be thrown through two

openings in the windows of a house.

1.32 Monkey and hunter problem.

1.33 To show (a) tan α = 4h/R and (b) h = gT

/8.

1.34 T , R and time to reach y

max

and shape of ﬂight.

1.35 Explosion of a projectile at the highest point of the trajectory.

1.36 Radius of curvature of the trajectory of a projectile.

1.37 Path of a boat which is rowed with constant velocity equal to that of the river,

with the ﬂow always directed towards the opposite point to the starting point.

1.38 A ball thrown from a height against a wall bounces and hits the ground. To

locate the landing spot.

1.39 Three forces acting on a particle as in the diagram.

1.40 Torque, angular acceleration and angular velocity.

1.41 Minimum coefﬁcient of friction to prevent a container from sliding in an accel-

erated track.

1.42 Minimum force required for a wheel to climb up an obstacle.

1.43 Centre of mass of a semicircular wire.

1.44 Centre of mass of a semicircular disc.

1.45 Centre of mass of a solid hemisphere.

1.46 Centre of mass of a hollowed circular disc.

1.47 Centre of mass of the earth–moon system.

1.48 Centre of mass of CO molecule.

1.49 Centre of mass of NH

molecule.