My Physics Tutor

My Physics Tutor (MCQ in Physics with Detailed Solutions)

B. Chatterjee • S. Gayen First Web-draft

To our Parents

About the authors

Dr. Bhramar Chatterjee is a researcher in theoretical physics. Her research interest includes cosmology and quantum gravity. She has substantial contribution in calculation of tunneling probability through black-hole horizon. Dr. Chatterjee obtained her B.Sc. & M.Sc. Degree from VisvaBharati and Ph.D. degree from Homi Bhaba National Institute (Saha Institute of Nuclear Physics).

Dr. Sirshendu Gayen is an experimental physicist working on spectroscopic studies of matter in confined geometry under extreme conditions like ultra-low temperature, high magnetic field etc. The domain of his research includes magnetism, magneto-transport, superconductivity and electronic properties of topological materials. Dr. Gayen is an alumni of R. K. M. Residential College, Narendrapur and University of Kalyani. He received his Ph.D. degree from University of Calcutta (Saha Institute of Nuclear Physics).

Preface This book is intended for the college/university level students in physics. Major part of the physics syllabus in Indian Universities is covered through multiple-choice questions (MCQ) with detailed solutions and in-depth discussions. The problems are collected from different national level examinations like National Eligibility Test (NET), Graduate Aptitude Test in Engineering (GATE), Joint Entrance Screening Test (JEST), Joint Admission Test for M.Sc.(JAM), entrance test of Tata Institute of Fundamental Research (TIFR) etc. as well as various popular and renowned books in physical and mathematical sciences (some of them are listed in the bibliography). In MCQs, apparently, it seems easy to pick the correct answer out of four given options. However, intuition or guess mostly do not work. Sometimes simple checks are not sufficient and may tempt to select a provocative answer. The known tricks of dimension analysis, checking the asymptotic behavior or singularities, consideration of symmetries are useful, yet may not be sufficient when some involved concepts are associated. Here, we have discussed the alternatives in addition to straight-forward calculations to obtain the correct answer in simpler ways with better understanding of the key concepts. We begin with a warm up test paper in the first chapter with fifty hand-picked questions from different areas of physics to revise the basics. Rest part is arranged topic wise in ten chapters, namely, mathematical methods, classical mechanics, fluid dynamics & general properties of matter, electromagnetism, basic circuit theory & electronics, waves & optics, quantum mechanics, thermodynamics & statistical mechanics, condensed matter physics, nuclear & particle physics followed by chapter-wise solutions and detailed discussions. We assure that a thorough study of this book will certainly be a boost to the students in understanding the basics of the subject. We hope that our sincere effort will be of great help to students, in particular, the competitive examination aspirants. This primary draft with more than 700 problems is a flyer of the enlarged and complete version which is planned to be self-contained with introductory notes to all the chapters. Any response/suggestions/feedback for improvement of the present volume will be greatly appreciated and can be sent directly to [email protected], [email protected] Mohali, January, 2017

Bhramar Chatterjee Sirshendu Gayen

Physical Constants & Useful Relations Rest-mass energy of electron, mc2 = 0.511 MeV Rest-mass energy of proton, mp c2 ' 938 MeV 1 Room temp. thermal energy, kB T ' eV = 250 meV 40 ≡ 5 micron ≡ 60 THz ≡ 16 ps 1.24 nm Useful energy conversion factor, hc = 1.24 keV.nm ⇒ λ = E (keV) Useful energy conversion factor, ~c ' 197 MeV.fm = 197 eV.nm ~ = 2.426 pm Compton wavelength of electron, − λC = mc e2 Classical radius of electron, r0 = = 2.8179 fm 4π0 mc2 ~ 4π0 ~2 = ' 0.53 ˚ A (in semi-classical picture) First Bohr radius of H-atom, a0 = 2 me mαc − 1 r0 λC e2 ' = − = Fine structure constant, α = 4π0 ~c λC a0 137 electrostatic energy between two e− at a separation − λC = =

rest-mass energy of electron velocity of in first Bohr orbit of H-atom c, velocity of light in vacuum e−

1 e2 1 = m(αc)2 = 13.6 eV 2 4π0 a0 2 = value of ground-state energy IH α 1 e2 13.6 eV Rydberg’s constant, R = P = = /hc = hc 4πa0 2 4π0 a0 1.24 keV.nm 2 e = ~αc = 1.44 eV.nm Useful electrostatic factor, 4π0 e~ Bohr magneton, µB = = 9.274 × 10−24 JT−1 2m Wien’s displacement law: λmax T = 2.898 × 10−3 m.K h von Klitzing constant, RK = 2 = 25 kΩ (related to quantization of conductance) e

Ionization potential of H-atom, IPH =

[

Useful dimensions:

r

{

[h] [∆E∆t] [eV ][t] [V ] [V ] ∼ ∼ ∼ ∼ ∼ [R]} [e2 ] [e2 ] [e2 ] [e/t] [I]

µ h ] = [ 2 ] = resistance e

s r r e 2 1 µ 1 1 V {µ 0 0 = 2 ⇒ [ ]∼[ ] ∼ [ ] ∼ [ ] ≡ [R]} 2 2 2 2 c c 4πx e /t I

[~] = [action] =

M L2 T

~ [ mc ]=L

[ mc~ 2 ] = T

x The following relation can be used to find r0 and − λC : −

e2 4π0 r0

= mc2 =

~c − λC

α = aλ0C i.e. a0 r0 = − λ2C hence, a0 : − λC : r0 = α1 : 1 : α a0 (∼ 10−11 m) > − λC (∼ 10−13 m) > r0 ' 2R0 (∼ 10−15 m) where R0 is typical range of r0 = − λC

nuclear force

For H-like atoms: (Recall the classical motion of a particle in a circle in an inverse square force field: E = −T = 12 V where E ≡ total energy, T ≡ kinetic energy, V ≡ potential energy) 2

Ze 1 Zαc 2 1 Energy eigen-value, En = − 21 4π 2 = − 2 µ( n ) ∝ 0 aµ n

Z 2µ n2

2

where aµ = 4π0 Ze~2 µ =

~ Zµαc

(Centripetal force is provided by the electrostatic force, hence, 2 µvn L2 n 2 ~2 n 2 ~2 Ze2 = µr 2 = r 3 = µr 3 ⇒ rn = 4π0 Ze2 µ 4π0 rn n n n Inverse proportionality of rn on Z is expected as more Z, (i.e. stronger attraction), causes the orbit  to shrink) in semi-classical picture 2 2

2

2

n ~ n n ~ th 2  Radius of n Bohr orbit, rn = n aµ = 4π0 Ze2 µ = Zµαc ∝ Zµ ∝ Zn Velocity of electron in nth Bohr orbit, vn = Zαc n   3/2 Time period, Tn ∝ rn (Kepler’s third law) ∝ n3 Expectation value of < r b > (b is a ±integer) in an energy eigen-state ∝ Z −b

mc2 : IPH : ∆E (fine structure correction to energy of H-atom) = 1 : α2 : α4

Instead of an introduction ....

we would like to start with a sample paper which includes 10 multiple-choice questions. Ten minutes may be sufficient to answer these questions because no calculation is required at all! (The solutions are discussed in chapter 13, pages 115-119.) This sample paper may help the readers to judge whether they will be interested in the rest or not.



 1 2 3 Q0.1: If A =  0 2 5 , then the trace of of A−1 is 0 0 3 (a) 1/6 (b) 11/6

(c) 6

(d) 1

Q0.2: The Fourier transform of a Hermite-Gauss function is a (a) Lorentzian (b) Bessel function (c) Legendre polynomial (d) Hermite-Gauss function . Q0.3: Consider the elastic collision in one dimension between a heavy point-mass M moving with speed V and a light point-mass m( M) at rest. The speed of the light mass after collision is √ (a) V (b) 2V (c) 3V (d) 3V / 2 Q0.4: Consider the motion of a non-expandable straight rod on smooth horizontal xyplane. One end of the rod is pulled away with a constant speed along x-axis while the other end is constrained to move along the y-axis. If the rod is inclined at an angle 45◦ to the x-axis at t = 0 instant, the trajectory of the midpoint of the rod will follow (a) an ellipse, (b) a parabola, (c) the arc of a circle, (d) a rectangular hyperbola. 1

2

My Physics Tutor

Q0.5: A tiny ball of negligible mass is connected by four identical massless springs to the corners of a cube those are vertices of a regular tetrahedron as shown in the figure. It is in equilibrium at the centre of the cube. The angle between any two springs is √ (a) π2 (b) cos−1 (− √12 ) (c) cos−1 (− 13 ) (d) cos−1 (− 23 )

Q0.6: Consider the electromagnetic wave, propagating in vacuum, with the electric field ~ = 4E0 cos(3x + 4y − 500t)bi + 3E0 cos(3x + 4y − 500t + π)b vector given by E j. The direction of the Poynting vector is along j) (b) 15 (3bi + 4b (a) √12 (bi + b j) (c) 51 (4bi + 3b j) (d) b k

Q0.7: Two finite, identical, solid bodies of constant heat capacity are used as heat reservers of an engine. Their initial temperature are T1 and T2 (T1 > T2 ), respectively. If both reservers attain the same temperature, Tf . What will be Tf for the entropy to be maximum ? √ √ √ 2 T1 T2 2 T T T2 ) (a) T1 +T (b) (d) ( T (c) − 1 2 1 2 T1 +T2

Q0.8: The excited state of a system is g-fold degenerate whereas the ground state is non-degenerate. What will be the entropy of such a system having N numbers of distinguishable particles in the temperature limit T → +∞ (a) NkB ln2 (b) NkB ln(1 + g) (c) kB ln(g − 1)/N! (d) 0 Q0.9: An electron initially found to have the z component of spin + ~2 . A measurement of its spin component along x-axis is carried out, but the result was not recorded. In final measurement, its spin component along z-axis is carried out. What is the probability of getting the value + ~2 ? (a) 0 (b) 1/4 (c) 1/2 (d) 1

Q0.10: A monochromatic beam of X-rays with wavelength λ is incident at an angle θ on a crystal with lattice spacings a and b as sketched in the figure. A condition for there to be a maximum in the diffracted X-ray intensity is (a) 2acosθ =λ (b) 2asinθ = λ √ 2 2 (d) 2bcosθ = λ (c) 2 a + b sinθ = λ

Bhramar Chatterjee, Sirshendu Gayen

a b

Contents

1 Warm up

5

2 Mathematical methods

13

3 Classical mechanics

27

4 General properties of matter etc.

41

5 Electromagnetism

45

6 Basic circuit theory & elctronics

57

7 Waves & optics

67

8 Quantum mechanics

75

9 Atomic & molecular physics

85

10 Thermodynamics & statistical mechanics

89

11 Condensed matter physics

101

12 Nuclear & particle physics

111

13 Solution to introductory problems

115

Recommended books

119

Index

128

3

4


My Physics Tutor

CHAPTER

1

Warm up

p Q1.1: Which one is true for the function f (x) = ln( xa + 1 + ( xa )2 ) with real a ? f (x) is (a) an odd function (b) an even function (c) neither odd nor even (d) can be odd or even depending upon the sign of a. Q1.2: The number of point of intersection between x2 + y 2 + z 2 = 2 & x + y = 2 is (a) 0 (b) 1 (c) 2 (d) infinite. Q1.3: A straight line makes angles θx , θy , θz with the coordinate axes. The value of sin2 θx√ + sin2 θy + sin2 θz is √ (a) 1/ 3 (b) 1 (c) 3 (d) 2 

 0 0 1 Q1.4: If  1 0 0  represents a rotation matrix, then the angle of rotation and the 0 1 0 axis of rotation are, respectively, (a) 90◦ , x = y = z (b) 120◦ , x√= y = z (c) 90◦ , x/ 2 = y = √ z ◦ (d) 120 , x = y = z/ 2 Q1.5: The day, after 331 days from a Monday, will be a (a) Sunday (b) Monday (c) Wednesday

(d) Thursday

Q1.6: The total number of rectangles (including squares) in a 20 × 20 chessboard is 20 20 21 P P P (a) n2 (b) n3 (c) ( n)2 (d) (20 C2 )2 n=1

n=1

n=1

5

6

My Physics Tutor

Q1.7: A particle performs a random walk along a straight line. If it starts at the origin, the probability that after taking six equal steps it will be back at the origin is (a) 20/64 (b) 5/64 (c) 1/2 (d) 0

(1 + 31 − 21 + 51 +

1 7

1 2

−

1 4

− 14 + 19 +

1 11

Q1.8: If the sequence (1 −

+

(a) S/ln2

1 3

+

1 5

−

1 6

+ 71 · · · ) converges to S, then the value of

− 61 + · · · ) is

(b) 23 ln2

(c) S

(d) ∞

Q1.9: The function f (x) is single valued and infinitely differentiable in the range (−∞, ∞). d The value of ea dx f (x) where a is a constant, is (a) af (x) (b) f (x)/a (c) f (ea x) (d) f (x + a) R dx n −1 x Q1.10: √2ax−x [ a − 1]. The value of n is 2 = a sin (a) -1/2 (b) -1

(c) 0

Q1.11: If ~r = rb r is the position vector in three-dimension, then ∇. (a) 1/r (b) 2/r (c) 3/r

(d) 1 rb r2

is (d) 4πδ(~r)

Q1.12:A spool with wire wound on it is placed on a rough table. The free end of the wire is pulled to the right with a force F~ at an angle θ (0 < θ < 90◦ ) as shown in the figure. Which of the following statement is true ? (a) The spool will roll on clockwise. F (b) The spool will roll on anti-clockwise. (c) The spool will skid without rolling. θ (d) The spool will roll on clockwise/anti-clockwise or skid depending upon the value of θ.

Q1.13: A string of negligible thickness is wrapped several times around a cylinder on a rough horizontal surface. A man standing at a distance l from the cylinder holds one end of the string and pulls the cylinder without slipping towards him as shown in the figure. The length of the string passed through the hand of the man while the cylinder reaches his hands is (a) l/2 (b) l (c) 2l (d) πl

Q1.14: The work done by a force F~ = radius, centered at origin, is (a) 0 (b) 2πA Bhramar Chatterjee, Sirshendu Gayen

Ab θ r

2`

to move a unit mass along the circle of unit (c) A

(d)

A 4π

Chapter 1. Warm up

7

Q1.15: The Lagrangian for a three particle system is given by L = 21 (η˙1 2 + η˙2 2 + η˙3 2 ) − a2 (η1 2 + η2 2 + η3 2 − η1 η3 ) where a is a real constant, then one of the normal frequency, ω is given √ √ by 2 2 2 2 (d) ω 2 = 2a2 (a) ω = a /2 (b) ω = a / 2 (c) ω 2 = 2a2 Q1.16: A particle of charge q and mass m is placed in a constant electric field in presence of uniform gravitational field. The most general trajectory of the particle is a (a) straight line (b) circle (c) parabola (d) conic section. Q1.17: What is the flux through one face of a cube of edge a when a charge q is located at one of the distant corners ? 2 (a) 0 (b) 6q0 (c) 24q 0 (d) qa 30 Q1.18: A point charge +q is located at a distance d from two grounded large conducting plates intersecting at right angles as shown in the figure. Approximate potential at a distance r d from the point of intersection of plates is (a) 1/r (b) 1/r 2 (c) 1/r 3 (d) 1/r 4

Q1.19: The potential of the surfaces of a hollow metallic cube is zero except one surface which is insulated from the others and held at a constant potential V0 . The potential at the centre of the cube is (a) 0 (b) V0 /6 (c) V0 /2 (d) 2V0 /3

d

+q d

V0

Q1.20: Consider the surface charge distribution: ρ(~r) = σ0 δ(r − R)cos2 θ (where θ is the polar R angle)3 over a hollow sphere of radius R. Component of the dipole moment (≡ ~rρ(~r)d r) along z-axis is 3 3 (a) 0 (b) −σ0 R3 (c) σ04πR (d) 4πσ30 R Q1.21: Two coherent sources of different intensities send waves which interfere. The ratio of maximum intensity to the minimum intensity is 25. The intensities of the sources are in the ratio (a) 25 : 1 (b) 5 : 1 (c) 9 : 4 (d) 625: 1 Q1.22: A polarized light beam passes normally through a rotating polarizer. If ω be the angular speed of rotation of the polarizer, then the frequency of intensity variation of the emergent beam is (a) 0 (b) ω/2 (c) ω (d) 2ω Bhramar Chatterjee, Sirshendu Gayen

8

My Physics Tutor

Q1.23: The ratio of equivalent resistance between A and B for the open and closed condition of the switch, S is (a) 15:14 (b) 15:2 (c) 5:3 (d) 1:1

Q1.24: A 20 V − 50 Hz ac source is connected across a series RC circuit. If the voltage across R is 12 V , then the voltage across C will be (a) 8 V (b) 10 V (c) 12 V (d) 16 V Q1.25: A classical gas of non-interacting molecules of mass m confined within certain volume is in thermal equilibrium at temperature T . If vx , vy , vz are the cartesian velocity components then the value of (vx + bvy )2 is 2) 2 3 kB T (c) 2m kB T (d) bm kB T (a) 0 (b) (1+b m Q1.26: For a thermodynamic system, the internal energy U is related to entropy S, volume V and the number of particles N, as follows: U(S, V, N) = aS 4/3 V b where a, b are constants. What is the value of b ? (Hint: the variables mentioned here are all extensive.) (a) 1/2 (b) -1/3 (c) 2/3 (d) 1 Q1.27: A particle moves in a one dimensional potential: V (x) = a|x|α where a is a positive constant and α is a non-zero, positive integer. The heat capacity of a canonical ensemble with N such particles is α (a) 12 NkB (b) 2+α NkB (c) 1+α NkB (d) α1 NkB 2α Q1.28: For an ensemble of N non-interacting particles only three states of energies , 2 and 3 are allowed. It is in contact with a heat bath at temperature T . If C(T ) is R∞ ) dT the specific heat of the system then, what is the value of C(T T 0

(a) 0

(b) NkB ln2

(c) NkB ln3

(d) 23 NkB

Q1.29: Which one of the following curves correctly express the temperature variation of specific heat for a system having only three non-degenerate energy eigen-states (a) (b) (c) (d)

Q1.30: The energy eigen-function of a particle in a 1D box is real whereas it is complex (plane wave) for a free particle . The energy eigen-function of an electron in hydroBhramar Chatterjee, Sirshendu Gayen

Chapter 1. Warm up

9

gen atom is real for s-states whereas it is complex for other states. The criterion for energy eigen-function to be real or complex is (in addition to degeneracy) a consequence of the invariance of hamiltonian under (a) space translation (b) space rotation (c) time reversal (d) parity. Q1.31: Consider ( the one dimensional half-harmonic potential, 1 mω 2 x2 , for x > 0 V (x) = 2 ∞ for x < 0 The energy eigenvalue of the first excited state in units of ~ω is (a) 21 (b) 32 (c) 52

(d)

7 2

Q1.32: Consider the function φ(x) = hx|Tcn |0i where |0i represents the ground state of a one dimensional simple harmonic oscillator, n is a non-negative integer and x, T are space coordinate and kinetic energy, respectively. The number of nodes of φ(x) is (a) n (b) 2n (c) 2n − 1 (d) 2n + 1 Q1.33: The degeneracy of the n-th energy eigen-state of an isotropic D-dimensional simple harmonic oscillator is ? n!(D−1)! (b) (n+D+1)! (c) (n+D+1) (d) (n+D−1)! (a) (n+D−1! n!(D−1)! n!(D+1)! n!(D−1)! Q1.34: Find the expectation value hL2x i in the basis |l, mi which are simultaneous eigenstates of Lˆ2 and Lˆz i.e. Lˆ2 |l, mi = l(l + 1)~2 |l, mi and Lˆz |l, mi = m~|l, mi. (a) 0 (b) m2 ~2 (c) 21 [l(l + 1) − m2 ]~2 (d) 13 l(l + 1)~2 2 Q1.35: The eigen-value of the Hamiltonian H = aJz + bJx + cJ √ is 2 2 (a) 0 (b) cj(j + 1)~ (c) am~ + cj(j + 1)~ (d) a2 + b2 m~ + cj(j + 1)~2

ˆ Q1.36: For a two-level system H|ii = i |ii where 1 = 0 and 2 = ε. The V11 subjected to a perturbation characterized by the potential V 0 = V 21 1 1 ε . Now the energy eigen-values are 2 1 −1 (c) 0, 2ε (a) 0, ε (b) − 2ε , 2ε

system is V12 = V22 (d) 2ε , 3 2ε

Q1.37: A particle is free to move in between two impenetrable walls in one dimension. One of the walls is suddenly moved (the transition time, ∆t ~/∆E where ∆E is the typical energy difference between two states) so that the box length gets doubled. What is the probability that a particle initially in the ground state will be found in the first excited state of the new box, immediately after the change ? (a) 0 (b) 1/2 (c) 2/π (d) 32/9π 2 Q1.38: For a system consisting of two identical particles with spin I each, the ratio of the number of states symmetric in the two spins to the number of states antisymmetric Bhramar Chatterjee, Sirshendu Gayen

10

My Physics Tutor in the two spins is (a) 2I + 1

(b)

I+1 I−1

(c)

I+1 I

(d)

2I I+1

Q1.39: The ratio of the de Broglie wavelength of an electron in the first Bohr orbit to the radius of the first Bohr orbit of hydrogen atom is (a) 1836 (b) 137 (c) 2π (d) 1 Q1.40: The ground state wave-function of electron in hydrogen atom is given as: h~r|ψ1s i = Ne−r/r0 where N and r0 are constants. What is the value of hxi for an electron in hri the ground state ? (a) 0 (b) 1 (c) N2 (d) 1e

| 4! |n=2!

| 2!, | 3! | 1!

E= 0

E" 0

~ of a charged particle of charge q, mass m, spin Q1.42: The mechanical momentum , Π ~ is given by Π ~ = ~p − q A ~ where p~ is the canonical ~σ moving in a magnetic field, B ~ is the magnetic vector potential which satisfies the relation B ~ = momentum and A ~ ~ ∇ × A. Which of the following is not correct ? ~ depends upon the choice of gauge . (a) Π ~ is not the generator of space translation. (b) Π ~ does not commute with each other. (c) The components of Π 1 ~ σ .Π) ~ (~σ .Π)(~ (d) The hamiltonian for the particle can be written as 2m Q1.43: The ionization potential of hydrogen atom is 13.6 eV . The average potential energy of the electron in the first Bohr orbit is (a) 13.6 eV (b) −13.6 eV (c) 27.2 eV (d) −27.2 eV Q1.44: The binding energy of an electron in ground state of helium is 24.6 eV. The energy required to remove both electrons of helium atom is (a) 24.6 eV (b) 49.2 eV (c) 54.4 eV (d) 79.0 eV Q1.45: Through several alpha and beta decays, the radioactive nucleus 232 X90 transforms into the stable final nucleus 208 Y82 . In the process how many beta particles are emmited ? (a) 2 (b) 4 (c) 5 (d) 6 Bhramar Chatterjee, Sirshendu Gayen

E Energy

Q1.41: Consider the linear Stark effect , caused in presence → − of electric field E for n = 2 energy eigen-state of hydrogen atom (shown schematically in the figure). Which one of the followings is true for |ψ1 i and |ψ4 i ? (a) both have no permanent dipole moment (b) both are eigen-state of parity (c) both are eigen-state of Lˆ2 (d) both are eigen-state of Lˆz .

Chapter 1. Warm up

11

Q1.46: If photon were to have a finite mass, then the Coulomb potential between two stationary charges separated by a distance r would (a) be strictly zero beyond some distance (b) fall off exponentially for large r (c) fall off as r12 for large r (d) fall off as 1r for large r Q1.47: Which one is true for the potential energies, V (R) of two diatomic molecules of the same reduced mass shown in the figure. V (R) (a) moment of inertia is smaller for molecule 1 (b) inter-nuclear distance is larger for molecule 2 (c) separation between low-lying vibrational states is R 0 larger for molecule 2 1 (d) the zero-point vibration energy (with respect to po2 tential minima) is larger for molecule 1.

Q1.48: There are only two spins s~1 and s~2 , placed at the diagonal positions of a unit cell on a planar square lattice of side a. The spin s~1 is kept fixed perpendicular to the diagonal. The interaction energy between these two spins will be proportional to (a) as~1 .s~2 (b) s~1a.s~2 (c) s~1a.2s~2 (d) s~1a.3s~2 Q1.49: For a D-dimensional simple cubic lattice of lattice constant a, the tight binding dispersion relation is given as = A − B[cos(k1 a) + cos(k2 a) + · · · + cos(kD a)] where A and B > 0 are constants. What is the Fermi energy for one electron per unit cell ? (a) A (b) A − B (c) A − DB (d) A − DB 2 4 Q1.50: In Landau theory of phase transition , the free energy for some system can be written as F (T, m) = F0 (T ) + a(T )m2 + b(T )m4 Under what of the following conditions there will be a stable equilibrium with broken symmetry ? (a) When a = 0 6= b. (b) When a 6= 0 = b. (c) When a > 0 > b. (d) When a < 0 < b.


12


My Physics Tutor

CHAPTER

2

Mathematical methods

p Q2.1: Which one is true for the function f (x) = ln( xa + 1 + ( xa )2 ) with real a ? f (x) is (a) an odd function (b) an even function (c) neither odd nor even (d) can be odd or even depending upon the sign of a. Q2.2: The given curves in the figure are represented, respectively, by the equations 1 y = a−x ; : 2 y = b−x ; : 3 y = logc x; : 4 y = logd x

: where a, b, c, d all are real constants. Inspecting the curves select the correct option. (a) a > b; c > d (b) a < b; c > d (c) a > b; c < d (d) a < b; c < d

Y

1

2

3 4

Q2.3: Which of the following is/are true ? (a) log3 108 > log5 375 (b) log 1 0.3 > log 1 0.3 3

(c) log5 9.8 =

4

a+2b−1 1−a

where a = log10 2 and b = log10 7

(d) log(tan1◦) · log(tan2◦ ) · · · log(tan89◦ ) = log(tan1◦)+log(tan2◦ )+ · · · + log(tan89◦)

Q2.4: The roots of the equation x2 − kx + k = 0 are a and b. The real value of k for which a2 + b2 is minimum is (a) 0 (b) 1 (c) 2 (d) 3 13

X

14

My Physics Tutor

Q2.5: For what values of the constant a, the curves y = (a − 6)x2 − 2 and y = 2ax + 1, do not intersect ? (a) −6 < a < 3 (b) −1/2 < a < 6 (c) 0 ≤ a < ∞ (d) −∞ < a < −3 Q2.6: If for the equation x3 − 3x2 + kx + 3 = 0, one root is negative of another, then the value of k is (a) -1 (b) 1 (c) -3 (d) 3 Q2.7: For what values of real k, the equation x4 = x + k will have at least two real roots? √ 3 π e (a) π > k ≥ e (b) e > k ≥ π (c) k > −3/ 256 (d) k ≥ 4−4/3 − 4−1/3 Q2.8: The closest distance from the√origin to the curve 3x2 + 10xy + 3y 2 = 9 is √ √ (a) 1/ 3 (b) 3 (c) 3 (d) 3/2 2 Q2.9: The equation of the plane that is tangent to the surface xyz = 8 at the point (1, 2, 4) is (a) x + 2y + 4z = 12 (b) 4x + 2y + z = 12 (c) x + 4y + 2 = 0 (d) x + y + z = 7 Q2.10: The distance of the point√ (1, 2, 3) from the plane x √+ 2y = 1 is (a) 0 (b) 1 2 (c) 3/2

√ (d) 4 5

Q2.11: The shortest distance between two straight lines 3x = −2y = 6z and 10(x − 2) = 6(1 − y) = 15(z + 2) is √ √ √ (a) 0 (b) 2 (c) 1 3 (d) 4 5 q √ p √ Q2.12: The value of |40 2 − 57| − 40 2 + 57 is √ √ √ (c) - 3 19/2 (a) -19/2 (b) -10 (c) -6 2 Q2.13: The number 3n + 1 is a multiple of 10. Then the positive integer n can be written in terms of m (= 0, 1, 2, 3, · · · ) as (a) 4m2 + 2 (b) 2(2m + 1) (c) 2(m2 + m + 1) (d) (m + 1)(m + 2) Q2.14: For what minimum positive integral value of n, the integer 82n2 + 1 will be a perfect square ? (a) 16 (b) 17 (c) 18 (d) 19 Q2.15: Which of the following is/are factor(s) of 737 + 1337 + 1937 ? (a) 3 (b) 5 (c) 7 Q2.16: The remainder obtained, dividing the number (100110011001)2 by 91, is (a) 0 (b) 1 (c) 8

(d) 13 (d) 9

Q2.17: When the cube root of 13683∗0393208 is divided by 3, the remainder is 2. The digit in place of ∗ is (a) 5 (b) 6 (c) 7 (d) 9 Bhramar Chatterjee, Sirshendu Gayen

Chapter 2. Mathematical methods

15

Q2.18: The last three digits of the decimal representation of 111100 − 5 is (a) 016 (b) 376 (c) 666 19

Q2.19: The last two digits of the decimal representation of 1919 (a) 29 (b) 37 (c) 79 Q2.20: How many digits does 21000 contain ? (a) 252 (b) 302

(d) 996

is

(c) 336

(d) 99 (d) 479

Q2.21: The number of positive integers up to 999 which are not divisible by 2, nor by 3, nor by 5 is (a) 193 (b) 266 (c) 307 (d) 367 Q2.22: The remainder, when 220 + 330 + 440 + 550 + 660 is divided by 7, is (a) 0 (b) 1 (c) 6

(d) 5

Q2.23: The remainder when (1032+n +3×434+n +5) is divided by 9 is same for all positive integral values of n. What is the value of the remainder ? (a) 0 (b) 3 (c) 6 (d) 7 Q2.24: Which of the following is/are true ? (a) 3960 is a factor of 20172017 + 19432017 (b) 5353 − 3333 is a multiple of 10. (c) The digit at the unit place of the number 9

9·

··

9

is 9.

(d) The digit at the unit place of the number 272017 − 212017 + 232017 is 9. Q2.25: Which of the following is/are true ? (a) 1 + 3 + 5 + · · · + 99 = 502 . (b) The number 270 + 370 is divisible by 13. (c) 151 + 251 + 351 + 451 + 551 + 651 is a multiple of 7. (d) The remainder of the division of the number 1! + 2! + 3! + · · · + 50! by 7 is 5. Q2.26: The remainder of the division of the number

144 P

n! by 12 is

n=1

(a) 1

(b) 5

(c) 9

(d) 11

Q2.27: Consider two infinite series a = 1 + sin2 θ + sin4 θ + sin6 θ + sin8 θ + · · · and b = 1 + cos2θ + cos4 θ + cos6 θ + cos8 θ + · · · , where 0 < θ < π/2. The value of (a + b) is same as √ (a) a2 + b2

(b) πab

(c) ab

(d) π


16

My Physics Tutor

Q2.28:

∞ P

m=1

(−1)m+1 = m2

(a) 3/4 Q2.29:

N P

(b) e/2

(d) π 2 /12

(c) π/4

1 = n(n+1)

n=1 N N +1

(b)

(a)

Q2.30: The value of (a) 2N N Q2.31: The value of (a) 2N N

N P

n=0

N P

n=0

N 2 n n

N n

N 2N −1

(c)

2N 2N +1

(d)

N −1 N +2

is (b)

2N N −1

(c)

2N N +1

(c)

2N N +1

(d)

2N −1 N +1

is

(b)

2N N −1

Q2.32: The series 1 − 31 + 51 − 17 + 91 − (a) 3ln2/2

1 11

(d) N2N −1

+ · · · converges to

(b) πln2

(c) π/2

(d) π/4

Q2.33: A complex function of one real variable t is represented in parameter form as: z(t) = x(t) + iy(t) = eict where c is a complex number. As t changes, the curve traces by z in the complex plane is (a) a straight line (b) a circle (c) a cycloid (d) a spiral √ Q2.34: Given i = −1, then ii is (a) real (b) purely imaginary defined

(c) of the form x + iy with x 6= 0, y 6= 0

Q2.35: The principal value of ln(−1) is [i = (a) i (b) π Q2.36: The value of cos(π − i) where i = (a) i (b) iπ

√

√

−1]

−1 is

(c) iπ

(c) −lnπ

(d) not

(d) 2iπ

(d) −cosh(1)



 a b c Q2.37: If D = det  p q r , then the value of x y z  −x a −p q  is det  y −b z −c r 

(a) D

(b) −D


(c) 2D

(d) D/2


17



 3 −3 3 3 −3  is Q2.38: The rank of the matrix, A =  −3 3 −3 3 (a) 0

(b) 1

(c) 2 

(d) 3 

0 0 B Q2.39: What is the rank of the matrix A3n×3n =  0 B 0  where Bn×n is a matrix B 0 0 with all entries 1 ? (a) 1

(b) 3 

1 0  1 2 Q2.40: Consider the matrix, M = 3   5 3 0 1 (a) 6

0 0 1 2

(c) 3n 

(d) 3n − 3

0 0  . The value of detM is 0  3

(b) 7

(c) 18

(d) 486



 1 1 1 Q2.41: Consider the matrix, A =  1 1 1 . The value of T rA10 is 1 1 1 (b) 39

(a) 3

(c) 310

(d) 311

 1 1 1 Q2.42: Consider the matrix, A =  1 1 1 . Then A10 is equal with 1 1 1 

(a) A

(b) 3A

(c) 9A

(d) 10A



 3 1 0 −2 X Q2.43: For the matrix, M =  −3 4 1 , what is the value of λ2i ? i=1 2 1 3 (a) 23 (b) 20 (c) 14 Q2.44: Consider the matrix, A =

(a)

0

√1 2

3 2

1

√

!

(b)

π 6

Q2.45: Consider the matrix, A =

1 2 2 1

1 0 0 −1

(d) 8

) is . Then cos( πA 6

1 2 3 −1

(c)

√

3 2

0

0 0

(d)

√

3 4

1 −1 −1 1

. Then det eA is Bhramar Chatterjee, Sirshendu Gayen

18

My Physics Tutor (a) 0

(b) 1

(c) e

(d) e + 1/e

Q2.46: Consider the unitary matrix Un×n = eiKn×n where Kn×n is a Hermitian matrix. If det Un×n = −1, then the principal value of Tr Kn×n is (a) 1 (b) e (c) π/2 (d) π 0 1 1 0 . Which one of the and B = Q2.47: Consider two matrices, A = 0 0 0 0 followings is the expression for eA+B 1 1 (a) eA eB (b) eA eB e(− 2 [A, B]) (c) eA eB e( 2 [A, B]) (d) none of these. Q2.48: The diagonal elements of a square matrix are 0, 1, 2, 3 and all the off-diagonal elements are 0. The rank of the matrix is (a) 1 (b) 2 (c) 3 (d) 4 Q2.49: Which of the following cannot be the eigenvalues of a real 3 × 3 matrix ? (a) -1, 0, 1 (b) 0, i, 1 (c) 1, 2i, -2i (d) 1, eiθ , e−iθ Q2.50: A real matrix A satisfies the equation A2 − 3A + 2 = 0, then the trace of A is (a) 1 (b) 2 (c) 3 (d) 4 Q2.51: A real matrix M satisfies the equation M 3 + pM 2 + qM + r = 0 where p, q, r are constants. The trace of M is (a) -p (b) -p/r (c) r (d) p+q+r Q2.52: If ~σ ≡ (σ1 , σ2 , σ3 ) represents the Pauli matrices for spin- 12 particles, then the √ correct expression for ~σ × ~σ is (here i = −1) (a) 0 (b) I (c) iI (d) 2i~σ Q2.53:√For the Pauli matrices, σi (i = 1, 2, 3), what is the value of σ1 σ2 σ3 ? (here i = −1 and I is the identity matrix) (a) 0 (b) I (c) iI (d)−iI Q2.54: If ρ = 12 [I + √13 (σx + σy + σz )] where σ’s are Pauli matrices and I is the identity matrix, then the trace of ρ2017 is (a) 1 (b) 1/2 (c) 22017 (d)2−2017 Q2.55: Which an acceptable form of U which makes U † σy U a diagonal one where one is 0 −i σy = i 0 i 1 1 1 0 1 + i 1 1 1 1 (a) (b) √2 (c) √2 (d) −1 i i −i 1−i 0 1 1 Q2.56: Select the correct option for the matrices which project onto the eigenvectors of 0 −i σy = i 0 Bhramar Chatterjee, Sirshendu Gayen

Chapter 2. Mathematical methods [ given that σx = (a) ±iσz σx

1 0 0 1 ] and σz = 0 −1 1 0 (b) 12 (1 ± σy ) (c) 21 (1 ± σz σx )

19

(d) 12 (1 ± iσz σx )

Q2.57: The matrix Pn×n is diagonalizable with the property P 2 = P and T r[P ] = n − 1. The value of det[P ] is (a) 0 (b) 1 (c) n − 1 (d) n Q2.58: The number of independent components in a tensor Tijkl... of rank r with no symmetries in d-dimensional space is (a) dr (b) r d (c) r+d−1 Cr (d) r+d−1 Cd Q2.59: The number of independent components in a totally symmetric tensor Sijkl... of rank r in d-dimensional space is (a) dr (b) r d (c) r+d−1 Cr (d) r+d−1 Cd Q2.60: The number of independent components in a totally anti-symmetric tensor Aijkl... (i.e. changes sign under exchange of any two indices) of rank r in d (≥ r)-dimensional space is (a) dr (b) r d (c) d Cr (d) r Cd Q2.61: Consider a fourth rank Cartesian tensor Tijkl in three-dimension. Tijkl satisfies following identities: (i) Tijkl = Tjikl ; (ii) Tijkl = Tijlk ; (iii) Tijkl = Tklij , then the number of independent components in a tensor Tijkl is (a) 15 (b) 21 (c) 24 (d) 27 Q2.62: Consider the totally anti-symmetric tensor abc of rank 3 in 3-dimension where exchange of any two indices causes change in sign. If a repeated index indicates summation on that index, then the value of cab dab is (a) 0 (b) δcd (c) 2δcd (d) 6δcd Q2.63: The number of diagonals of a uniform polygon of 24 sides is (a) 196 (b) 252 (c) 276

(d) 300

Q2.64: There are 25 married couples at a party — each of them shakes hand with every persons other than his/her spouse. The number of total handshakes is (a) 256 (b) 448 (c) 774 (d) 1200 Q2.65: The total number of squares in a chessboard is (a) 64 (b) 204 (c) 1092

(d) 1296

Q2.66: The number of rectangles with unequal arms (i.e. excluding squares) in an unconventional chessboard of size 6×8 is (a) 756 (b) 623 (c) 512 (d) 133 Q2.67: How many 5-digit natural numbers are there with no repeated digits ? (a) 452 (b) 4536 (c) 27216 (d) 33696 Q2.68: How many 5-digit natural numbers are there in which every digit is smaller than the digit to its left ? (a) 400 (b) 308 (c) 252 (d) 126 Bhramar Chatterjee, Sirshendu Gayen

20

My Physics Tutor

Q2.69: How many 5-digit natural numbers are there in which every digit is greater than the digit to its left ? (a) 400 (b) 308 (c) 252 (d) 126 Q2.70: A dice is thrown five times. What is the probability of 6 coming up exactly three times ? (a) 1/10 (b) 5/196 (c) 125/3888 (d) 5/7776 Q2.71: If two ideal dice are rolled once, what is the probability of getting at least one ‘6’ ? (a) 11/36 (b) 1/36 (c) 10/36 (d) 5/36 Q2.72: What is the probability of obtaining at least one six in four throws of a die ? (a) 11/36 (b) 25/36 (c) 125/1296 (d) 671/1296 Q2.73: The standard deviation for the value of a single throw of a die is close to (a) 0.4 (b) 1.7 (c) 6.28

(d) 6

Q2.74: A button when pressed generates +1 or -1 randomly with equal probability. The button is pressed six times. What is the probability that the sum will be zero, assuming that individual presses are independent to each other ? (a) 5/16 (b) 5/64 (c) 9/16 (d) 2/3 Q2.75: In how many ways can the letters of the word PHYSICS be arranged ? 7! (a) 2!5! (b) 7! (c) 7! 5! 2!

(d)

5! 2!

Q2.76: How many 4-digit numbers can be formed which are multiple of 10 with no repetitions ? (a) 256 (b) 335 (c) 472 (d) 504 Q2.77: In how many ways a non-negative integer n can be taken as the sum of three non-negative integers nx , ny and nz ? 2 2 (n+2) (a) 2n + 1 (b) n(n+1) (c) (n 1!+1) (n 2!+2) (d) (n+1) 2 1 2 Q2.78: In a 1D random walk problem, the probability of taking a forward step is double of the probability of taking it backward. If each step is completely uncorrelated to previous step, then what is the probability that the final position will be one step away in the forward direction after taking 3 steps ? (a) 1/2 (b) 2/3 (c) 4/9 (d) 8/27 Q2.79: A random walker takes a step of unit length in the positive direction with probability 2/3 and a step of unit length in the negative direction with probability 1/3. The mean displacement of the walker after n steps is (a) n/3 (b) n/8 (c) 2n/3 (d) 0 Q2.80: Two drunks start out together at the origin, each having equal probability of making a step simultaneously to the left or right along the x axis. The probability that they meet after n steps is 2n! (b) 21n 2n! (c) 21n 2n! (d) 41n n! (a) 41n n! 2 n!2 Bhramar Chatterjee, Sirshendu Gayen


21

Q2.81: Consider 1D random walk problem with three equally likely outcomes in time interval ∆t — a step taken to the left or to the right (bypa distance ∆x) or no move. The standard deviation of net displacement X (i.e. x2 − x2 ) after N time intervals p p √ each of duration ∆tpis (a) ∆x N (b) ∆x N/2 (c) ∆x N/3 (d) ∆x 2N/3 Q2.82: On a planar grid, only upward steps and steps to the right are allowed. If each step-length is unity, then in how many ways one can reach the point (5, 5), starting from (1, 1) ? (a) 45 (b) 70 (c) 72 (d) 120

Q2.83: On a planar grid, only upward steps and steps to the right are allowed. If each step-length is unity, then in how many ways one can reach the point (5, 5), starting from (1, 1) without avoiding the points (2, 2), (3, 3), (4, 4) ? (a) 8 (b) 12 (c) 15 (d) 16

Q2.84: In a square lattice a random walker moves from a lattice point to one of its four neighbouring points with equal probability. The walker starts at the origin and takes 3 steps. The probability that during this walk no site is visited more than once is (a) 12/27 (b) 27/64 (c) 3/8 (d) 9/16 Q2.85: A random walker starts out from a lamp-post. Each step he takes is of fixed length of L and can be taken along either of the four directions — north, south, east or west with equal probability. What is the probability that he will be within a circle of radius 2L around the lamp-post after 3 steps ? (a) 5/16 (b) 5/64 (c) 9/16 (d) 2/3 Q2.86: In a triangular lattice a particle moves from a lattice point to one of its six neighbouring points with equal probability, as shown in the figure. The probability that the particle is back at its starting point after three steps is (a) 5/18 (b) 1/6 (c) 1/18 (d) 1/36


22

My Physics Tutor

Q2.87: In a system, signal flows through the components A, B, C, D connected as shown in the figure. Assume A, B, C and D function independently. If the probabilities of their failure are pA , pB , pC and pD , respectively, what is the probability that the signal will pass successfully ? (a) 1 − [1 − (1 − pA )(1 − pB )]pC pD (b) 1 − (pA + pB )(1 − pC )(1 − pD ) (c) (1 − pA )(1 − pB )(1 − pC pD ) (d) 1 − (pA + pB )pC pD

B

A C D

Q2.88: There are on average 20 buses per hour at a point, but at random times. The probability that there are no buses in five minutes is closest to (a) 0.07 (b) 0.60 (c) 0.36 (d) 0.19 (x−µ)2

Q2.89: Consider the 1D Gaussian function f (x) = σ√12π e− 2σ2 . The full width at half maxima (FWHM) is √ (a) 2σ (b) 2µ (c) µ + 2σln2 (d) 2σ 2ln2 Q2.90: What is the average value of x2 for one dimensional normal distribution in the 2

given form p(x) = (a)

µ2 2σ2

(x−µ) √1 e− 2σ 2 σ 2π 2 (b) µ2

? (c) µ2 + σ 2

(d) σ 2

x

Q2.91: If the distribution function of x is f (x) = xe− λ over the interval 0 < x < ∞, the mean value of x is (a) λ (b) 2λ (c) λ/2 (d) 0 2

Q2.92: The mean value of random variable x with probability density p(x) = is (a) 0 (b) µ2 (c) − µ2

x +µx √1 e− 2σ 2 , σ 2π

(d) σ

Q2.93: A particle is confined to move within 0 ≤ x ≤ a and it is equally probable to be found √ anywhere within the allowed region. The position √ uncertainty is (a) a/ 2 (b) 2a/3 (c) a/ 12 (d) a/4π Q2.94: Which one of the following functions with same periodicity has minimum number of harmonics: (a) cosine wave (b) square wave (c) triangular wave (d) half-rectified sine wave Q2.95: Which one of the following is not correct (a is a real constant) ? 1 x x+a x (a) lim x−a = e2a (b) lim x1/x = 1 (c) lim tanx = 0 (d) lim cos x = 1 2 x x→∞

x→∞

x→0

x→0

Q2.96: Which one of the following is not correct (n is a positive constant) ? x =1 (c) lim | xen |= ∞ (d) lim | (a) lim xx = 1 (b) lim cosx x x→0+

x→∞


x→∞

x→∞

lnx xn

|= 0


23

Q2.97: The function f (x) is single valued and infinitely differentiable in the range (−∞, ∞). d The value of eax dx f (x) where a is a constant, is (a) af (x) (b) f (ea x) (c) ea f (x) (d) f (x + a) Q2.98: Fourier transform of which of the following functions does not exist ? 2 2 (a) e−|x| (b) xe−x (c) e−x

(d) ex

2

Q2.99: Which one of the following functions can not be an eigen-function of parity ? (a) Hermite polynomial (b) Legendre polynomial (c) Laguerre polynomial (d) Hermite-Gauss function Q2.100: If bxc denotes the greatest integer not exceeding x, then (a)

1 e−1

(b) 1

e e−1

(c)

Q2.101: If bxc denotes the greatest integer not exceeding x, then (a)

1 e−1

(b) 1

(c)

e (e−1)2

R∞ bxce−x is 0

R∞ bxce−bxc is

(d) 1 −

1 e

0

(d) 1 −

1 e

Q2.102: Consider the function bxc = the nearest integer less than x. For example, d b3.14c = 3. Then for arbitrary x, the value of dx bxc = will be given in terms of the integers n as P P P (a) f (x − n) (b) |x − n| (c) δ(x − n) (d) 0 n

n

n

R∞ ax2 +bx+c Q2.103: The value of the integral e dx where a(> 0), b, c are constants is −∞ p p p p b2 (a) b c/a (b) π/a (c) b π/a (d) e 4a +c π/a

Q2.104: The value of (a) 0

R∞

−∞

(x2 + 1)δ(x2 − 3x + 2)dx is (b) -2

Q2.105: The value of the integral (a) mπ

−∞

R∞

−∞

xsinx dx x2 +a2

R1 0

(b) 2π

√ dx −lnx

(d) 7

(c) em /2π

(d) π/em

(m > 0) is

(a > 0) is (c) ea /2π

(b) π/a

Q2.107: The value of the integral (a) π

cosmx dx 1+x2

(b) π/m

Q2.106: The value of the integral (a) aπ

R∞

(c) 5

(d) π/ea

is (c) π/2

Q2.108: The coefficient of x9 in the polynomial expansion of the function |x| < 1, is (a) 1 (b) 3 (c) 9

(d) x(1+x) , (1−x)3

√

π

for all (d) 81


24

My Physics Tutor

x Q2.109: The coefficient of xn in expansion of the function (1−x) 3 , for all |x| < 1, is Tn (called the triangular number). Then which of the following is not true ? (a) Tn + Tn−1 = n2 (b) 1 + 2 + 3 + · · · + n = Tn (c) 13 + 23 + 33 + · · · + n3 = Tn2 (d) 1 + 3 + 5 + · · · + (2n + 1) = Tn

Q2.110: For a given infinite series f (x) = x + 3x2 + 6x3 + 10x4 + · · · + what is the value of f (x = 4/5) ? (a) 1 (b) 64 (c) 100

n(n+1) n x 2

+ ···,

(d) 125

Q2.111: The generating function for a set of polynomials in x is given by f (x, t) = (1 − 2xt + t2 )−1 The third polynomial (order t2 ) in this set is (a) 2x2 + 1 (b) 2x2 − x

(c) 4x2 − 1

(d) 4x2 + 1

Q2.112: Which one of the followings is not true for Legendre polynomial of order l, Pl (x) R1 2 (a) Pl (1) = 1 (b) Pl (−1) = (−1)l (c) Pl=1 (x) = 1 (d) [Pl (x)]2 dx = 2l+1 −1

−

Q2.113: What is the maximum number of extrema of the function f (x) = Pk (x)e where Pk (x) is an arbitrary polynomial of degree k (a) k + 1 (b) k + 2 (c) k + 3

2 x4 + x2 4

(d) k

Q2.114: Which of the following is a solution of the differential equation: (x2 −y)y 0+2xy 2 = dy and c is a constant 0 where y 0 = dx 2 2 xy (a) y = ce (b) y = cex y (c) y = cexy (d) y = cxey Q2.115: Select the correct differential equation for which the graph shown here is a solution. (a) y 00 + 4y 0 + 4y = 0 (b) y 00 + 2y 0 + 2y = 0 (c) y 00 − 3y 0 + 2y = 0 (d) y 00 + 4y = 0

2

y

x

4

Q2.116: The partial differential equation ∂∂ 2yt + b2 ∂∂4 xy = 0, where b > 0 is a constant, represents (a) diffusive conduction, (b) transverse propagating wave, (c) transverse vibration of a beam, (d) transverse stationary wave in a string. Q2.117: If rb is the unit vector in the radial direction in three-dimension, then ∇.b r is (a) 1/r (b) 2/r (c) 3/r (d) 4πδ(~r) Bhramar Chatterjee, Sirshendu Gayen


25

Q2.118: In spherical polar coordinates (r, θ, φ), which one of the following expressions is the correct (a) ∇.ˆ r=0 (b) ∇.θˆ = 0 (c) ∇.φˆ = 0 (d) ∇ × θˆ = 0 Q2.119: For 3D position vector ~r, the value of ∇.(r~r) = ? (a) 3 (b) 1.5r (c) 3r Q2.120: The value of the integral

R

allspace

(a) 0

(b) 4π

(d) 4r

e−r/r0 ∇. rrb2 d3 r, for r0 to be a constant, is (c) −16π 2

(d) 4πr03

~ = ~r3 over the surfaces of a cube of side a with Q2.121: The surface integral of a vector A r edges parallel to the axes, centered at the origin is (a) 0 (b) 2π (c) 4π (d) 4πa3 Q2.122: The curl of velocity known as the vorticity vector for the velocity, ~v = ω ~ × ~r with constant ~ω is (a) 0 (b) 2~ω (c) 2~r (d) 2~v √ Q2.123: The value of the area under the curve r = r0 θ; (0 ≤ θ ≤ 2π) where r0 is a positive constant is (a) r02 (b) πr02 (c) 2πr02 (d) π 2 r02 Q2.124: For a cylinder of radius r and height h, what is the value of h/r which maximizes its total surface area subject to the constraint that its volume is constant ? (a) 1/3 (b) 2 (c) π (d) 4π/3 Q2.125: For a given total surface area of a cylindrical can (curved wall + top & bottom disks), the volume will be maximum when the ratio of height/radius is (a) 0.5 (b) 1 (c) 2 (d) π Q2.126: What is the maximum volume of a right circular cylinder inscribed in a right circular cone of height H and base radius R ? 7 4 (a) 5πR2 H/16 (b) 20 πR2 H (c) 43 πR2 H (d) 27 πR2 H Q2.127: If the surface area, Sd of a sphere in d-dimension is related to the volume, Vd−1 √ Γ( d+1 ) of a sphere of same radius in (d − 1)-dimension as: Sd = 2 π Γ( d2 ) Vd−1 , then the 2 volume of a sphere of unit radius in 4-dimension is (a) π 2 /2 (b) π 2 /3 (c) 2π 2 /3 (d) 4π 2 /3 Q2.128: The volume to surface ratio of a ring torus with major radius R and minor radius r < R is (a) r/2 (b) R (c) 2πR (d) πR2 /r

r R


26

My Physics Tutor

Q2.129:

n=0

n=1

n=2

n=3

Koch snowflake , a fractal, is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process. If the length of initial n = 0 side is 1, then the perimeter length after n iteration will be (a) 31−n 4n (b) 3 4n−1 (c) 3n+1 (d) 22n+1 3n+1 Q2.130: If the initial (n = 0) area of the Koch snowflake in the previous problem is 1, then the area after n → ∞ iteration will be (a) 3/2 (b) 5/4 (c) 7/4 (d) 8/5


CHAPTER

3

Classical mechanics

~ = 0 is a Q3.1: The most general trajectory of a particle with angular momentum, L (a) straight line (b) circle (c) parabola (d) conic section. Q3.2: The position vector of a particle moving in xy-plane is given by ~r(t) = a2 eαt xˆ + b2 e−αt yˆ where a, b and α (> 0) are real constants. The trajectory of the particle is (a) a circle (b) an ellipse (c) a cycloid (d) a rectangular hyperbola Q3.3: A particle confined to move in xy-plane moves away from the origin following a spiral curve rθ = 1 (0 < θ ≤ 2π). The maximum limit along y-axis that it can travel is (a) 1 (b) e (c) π (d) ∞ Q3.4: The speed of a car, starting from rest, increases uniformly to v at a rate f , then remains constant for an interval, and finally decreases to zero at the same constant rate f . If d is the total distance traveled, then total time elapsed is (a) (v/f + d/v) (b) (v/f + d/v)/2 (c) (2v/f + d/v) (d) (v/f + 2d/v) Q3.5: The instantaneous position of a particle moving in a plane is given as ~r(t) = î sin2t + ˆj cos3t The trajectory of the particle correctly represented by

(a)

(b)

(c)

27

(d)

28

My Physics Tutor

Q3.6: Three particles X, Y and Z starts moving with constant speed v from the vertices of an equilateral triangle XYZ of unit side-length. The instantaneous direction of velocity of each is cyclic towards the next i.e. X always has its velocity along XY, Y along meet each other ? √ what time will the particles √ √ YZ and Z along ZX. At (a) 2/v (b) 3/ 2v (c) 2/ 3v (d) 2/3v Q3.7:

v/m-s-1

The temporal variation of speed of a particle moving in a straight line, as shown in the figure, is part of a circle. The average √ √ over time 2t0 is √ speed (a) πt0 / 2 (b) 3t0 /2 (c) (π − 1/2)t0 (d) ( 2π − 1)t0

Q3.8: A particle slides down in a vertical plane under gravity (θ + sinθ); y = − 5a (1 − along a smooth cycloid: x = 5a π π cosθ) where a is a constant. If ti is the time taken by the particle to slide down from rest at a point i = (1, 2, 3, 4) on the cycloid to the bottom-most point 5, then which one is correct ? (a) t1 = 2t3 (b) t2 = 3t4 (c) t2 = t3 + t4 (d) t1 t4 = t2 t3

t0

2t0

450 450

Y 0 a 2a 3a 4a 5a

10a

1 2 3

4

5

Q3.9: A particle is projected at 60◦ with the horizon with a kinetic energy K. The kinetic energy at the highest point is (a) 0 (b) K (c) K/2 (d) K/4 Q3.10: A ball is projected upward at an angle θ (with respect to the horizontal line) from ground. For what value of θ the area under the trajectory will be maximum ? (a) 30◦ (b) 45◦ (c) 60◦ (d) sin−1 (2/3) Q3.11: A ball is projected upward at an angle θ (with respect to the horizontal line) from ground. For what value of θ the maximum height will be equal to the horizontal distance traveled ? (a) 45◦ (b) 60◦ (c) tan−1 4 (d) tan−1 (3/2) Q3.12: A stone is dropped vertically from the top of a tower of height 40 m at the same time a gun is aimed directly at the stone from the ground at a horizontal distance 30 m from the base of the tower and fired. If the bullet from the gun is to hit the stone before it reaches the ground, the minimum velocity of the bullet must be, approximately, (a) 57.4 m/s (b) 27.7 m/s (c) 17.5 m/s (d) 7.4 m/s Q3.13: Consider the elastic collision in one dimension between a heavy point-mass M and a light point-mass m( M) moving towards each other with the same speed V in lab-frame. The light mass, after collision, will bounce back with a speed of (in lab-frame) √ (a) V (b) 2V (c) 2V (d) 3V Bhramar Chatterjee, Sirshendu Gayen

t/s

X

Chapter 3. Classical mechanics

29

Q3.14: Consider elastic collision between two masses m and M(> m) where the lighter mass m was initially at rest. The heavy mass M can be deflected by the maximum angle of m m M (a) 180◦ (b) sin−1 ( M ) (c) sin−1 ( m+M ) (d) sin−1 ( m+M ) Q3.15: A ping-pong ball of mass m rests on top of a heavy tennis ball of mass M( m). The balls are dropped from a large height h. At what speed the ping-pong ball will bounce back immediately after the tennis ball hits the ground ? Consider the motion is confined in one dimension and the balls bounce elastically. Also assume that√there is a small separation √ between the balls. √ √ (a) 2gh (b) 2 2gh (c) 3 2gh (d) 4 gh Q3.16: A ball bounces back to half of the height from where it is dropped on a floor. The coefficient of restitution is √ √ (d) 2 (a) 1/4 (b) 1/2 (c) 1/ 2 Q3.17: Consider a stack of n identical books (length = L) protruding over the edge of a table without the stack falling over. The possible maximum overhang is n n n n P P P P 1 1 1 2 L (a) L2 (b) (c) L (d) L n 2 n2 n(n+1) n+2 i=1

i=1

i=1

i=1

Q3.18: A weighing balance having unequal arms is used with equal alternation. Which of the following is true ? (a) the seller will gain always. (b) the buyer will gain always. (c) no one will gain. (d) the seller/buyer may gain depending upon the ratio of the arms.

Q3.19: A uniform bar is leaning against a wall. The coefficient of static friction between the bar-vertical wall and bar-floor are µv and µh , respectively. The minimum angle that the bar can form with the floor so that not to slip down is µv −1 −1 1−µv µh −1 µv +µh −1 µv µh (b) tan (c) tan (d) tan (a) tan 2 2µh 2µh µv µh

Q3.20: A uniform solid wheel of mass m and radius r is halted at a step of height h as shown in the figure. What is the minimum force to be applied horizontally at the centre of√ the wheel to raise √ the wheel over the step ? h(2r−h) (a) mg√ r+h h(2r−h) (c)mg r−h

(b)

(d)

F

h(r+h) mg √ r−h h(r+2h) mg r+h .

Q3.21: A block is at rest on a wedge which makes an angle θ with the horizontal line as shown in the figure. What should be the minimum acceleration to be imparted to the wedge in a horizontal direction so that the block falls freely in the vertical direction ? (a) g/sinθ (b) gcosθ (c) gtanθ (d) gcotθ

r

h

M


30

My Physics Tutor

Q3.22:The center of a wheel, A rolling on a plane surface moves with a speed v0 as shown in the figure. The instantaneous speed of the point B, on the rim at the same horizontal √ level, is √ (a) 0 (b) v0 (c) 2v0 (d) v0 / 2

.B

.A

Q3.23: A wheel of radius a rolls without slipping on floor with a constant forward speed v0 . The acceleration of a point fixed on the rim of the wheel (with respect to an observer at rest on the same floor) is (a) v02 /a in the forward direction (b) v02 /a towards center of the wheel (c) v02 /2a towards the point of contact (d) v02 /a along the tangent to the wheel. Q3.24: A particle moves along a spiral in horizontal plane such that the normal component of acceleration is constant (= k). The speed of the particle at a point where the radius of curvature is ρ0 is √ √ (a) k/ρ0 (b) k/ ρ0 (c) k ρ0 (d) kρ0

Q3.25: A man holds one end of a ladder of length l keeping it horizontal, while the other end rests on a cylinder as shown in the figure. The man starts moving forward so that the cylinder rolls without slipping. There is no slipping between the ladder and cylinder as well. How far the man has to move to reach the cylinder ? (a) l/2 (b) l (c) 2l (d) πl

Q3.26:One end of a massless inextensible string of length L is attached to a fixed disc of radius R. The other end of the string is connected to a point mass. The string is kept straight and tangent to the disc. The mass is kicked with a speed V in the direction perpendicular to the string as shown in the figure. Consider the motion confined within a frictionless horizontal plane. How long will it take for the string to be wrapped completely around the disc ? √ (a) πR2 /LV (b) 2RL/V (c) L2 /2RV (d) L2 /2πRV


V

R L

Chapter 3. Classical mechanics Q3.27:A massless inextensible string with a small ball tied at one end is wrapped around a fixed disc of radius R so that the ball touches the disc. If, at t = 0 instant, the ball is given a velocity V in the radial direction, what will be the length of unwound segment of the string at the instant of time t > 0? √ √ (a) πR2 /tV (b) V 2 t2 /2R (c) 2RV t (d) 2πRV t

Q3.28: A block of mass m rests on another block of mass M as shown in the figure. The friction coefficient between the two blocks is µ whereas the horizontal plane is frictionless. What is the maximum amplitude of oscillation before the mass m slips? ? )g g µmM g (a) µmg (b) µM (c) µ(m+M (d) k(m+M k k k )

31

V

R ?

k

m M

Q3.29:A chain of uniform mass density ρ per unit length is suspended vertically with its lower end touching a weighing balance. The chain is released and falls freely onto the balance. Neglecting the size of the individual links, what is the reading of the balance when a length x of the chain has fallen ? (a) ρgx (b) 21 ρgx (c) 2ρgx (d) 3ρgx

Q3.30: A uniform inextensible rope of length L and mass M is held at one end on a smooth horizontal table with the other end hanging over the edge. The length of the hanging segment is l0 . The rope is released from rest. What is the speed of the ropepwhen all of the rope p slide off the tablep ? p 2 2 (a) (L − l0 )g/L (b) g(L − l0 ) (c) 2(L2 − l02 )g/3l0 (d) (L2 − l02 )g/l0

Q3.31: A long chain of length L is held at one end on a frictionless horizontal table while the other end with a length l hanging through a smooth hole in the table. If the chain is released, the length that hangs below√the hole will change with time√ as √ −t g/L t g/l −t g/l l 2 −gt2 /L (a) l + gt (b) le (c) le (d) 2 e +e Q3.32: Two light spring balances are connected to two blocks of mass 10 kg each as 2. Two light spring balances are connected to two blocks of mass 10 kg each as shown in the figure. shown in the figure. Which one is correct ? (a) Both the scales will read 5 kg. (b) Both the scales will read 10 kg. (c) Both the scales will read 20 kg. (d) The readings may be anything 10 kg 10 kg but their sum will be 40 kg.


32

My Physics Tutor

Q3.33:Two weights 3m and m are attached to the ends of a massless string passing over a pulley which is connected to a weighing balance as shown in the figure. Assume the mass of the pulley is negligible. If the reading in the balance is W during the masses in motion, then the correct option is ? (a) W > 4mg (b) W = 4mg (c) W < 4mg (d) W = 0

Q3.34: Consider an Atwood’s machine with masses m and M (> m) connected by an inextensible string. If the mass of the pulley is m0 then the acceleration of the masses m and M is −m) −m) 0g (a) Mmmg (b) Mm+m (c) g(M (d) Mg(M M +m +m+m0 /2 0

m 3m

m0

m M

Q3.35: A man of weight W is at rest on a platform connected by ropes to the pulley system as shown in figure. With what force the man needs to pull the rope to hold the platform at rest. Assume that the weight of platform, ropes and pulleys are negligible. (a) W (b) W/2 (c) W/3 (d) W/4

Q3.36: A particle moves in a circle of radius a with speed kt where k > 0 is a constant and t represents time. The ratio of radial and cross-radial acceleration in magnitude is 2 2a (a) 0 (b) kt (c) kta (d) infinite 2 p Q3.37: A particle of mass m starts moving from x = ab (a, b are positive constants) in a potential V (x) = −ax + bx3 . What will be the maximum kinetic energy of the particle q motion ? q q q during the rectilinear (a)

4a3 27b

(b)

8a3 27b

(c)

16a3 27b

(d)

64a3 27b

Q3.38: A particle of mass m under an attractive force F~ (x) = − xk (k > 0 is a constant) starts moving towards the centre of force from a point x = a > 0. The time for it to reach the centre of force is q p pm pm 2m (a) a πm (b) a (c) k (d) a 2k k a k

Q3.39: A particle of mass m falling down vertically under gravity experiences a dragging force of kv 2 (k > 0, a constant) where v is the instantaneous speed. The terminal speed p p p in this case will be p (b) g/mk (c) k/mg (d) mk/g (a) mg/k Bhramar Chatterjee, Sirshendu Gayen


33

Q3.40: If W1 is the work done to raise a mass from the bottom of a radial shaft of depth R/2 inside the earth to the earth’s surface (assume earth as a sphere of uniform mass density) and W2 is the work done to raise the same mass to a height R/2 from earth’s surface, then the value of W1 /W2 is √ (a) 1 (b) 7/5 (c) 9/8 (d) 3/2 Q3.41: What is the gravitational potential at a point on the rim of a uniform thin circular disc of mass M and radius R ? (a) 3GM (b) 4GM (c) 3GM (d) 3GM 5R πR πR 4πR Q3.42: The gravitational self-energy of a homogeneous solid sphere of mass M and radius R is 2 2 2 (b) − 32 GM (c) − 35 GM (d) − 25 GM (a) − 43 GM R R 5R R Q3.43: For a sphere of radius R with uniform volume density of ρ, what is the gravitational pressure at the center ? (a) 43 πR3 ρG (b) 4πRρ2 G (c) 52 ρ2 G/R (d) 23 πR2 ρ2 G Q3.44:A particle of unit mass is placed on the top of a solid hemisphere of mass M with uniform density as shown in the figure. What is the gravitational force experienced by the particle due to the √ hemisphere? √ (a) ( 2 − 1)GM/a2 (b) 2GM/a2 (c) 4GM/a2 (d) 4GM/5a2 Q3.45: A particle of unit mass initially at the center of the base of a uniform solid hemisphere of mass M and radius a is taken to infinity. The work done against the gravitational force is (a) 12 GM (b) 35 GM (c) 32 GM (d) 34 GM a a a a Q3.46: The difference in total energy for two satellites of same mass m moving around earth (assume the mass to be M) in circular orbits of 2R and 4R is m m m m (b) GM (c) GM (d) GM (a) GM R 2R 4R 8R Q3.47: Imagine a particle is let to move under gravity along a straight smooth hypothetical tunnel made through the earth (mass M, radius R) connecting any two points on its surface. The time that the particle would take to go from one end to the other pthrough the tunnel isp(G is the gravitational p constant) p 3 3 (a) R /2GM (b) π R /GM (c) 4πR3 /3GM (d) 2R3 /3GM

Q3.48: A small mass slides along a parabolic wire described by z = 2(x2 + y 2 ). The wire rotates about z-axis. For what value of angular speed does the mass maintain a non-zero height under the action of gravity along −z √ ? √ √ √ (a) g (b) 2g (c) 3g (d) 4g

Q3.49: How many constants of motion are there for a particle moving in an attractive inverse square force field in 3D ? (a) 1 (b) 3 (c) 4 (d) 7. Bhramar Chatterjee, Sirshendu Gayen

34

My Physics Tutor

Q3.50: For the motion of a particle under the influence of an attractive inverse square force field, which of the following is not a constant of motion (a) areal velocity (b) linear momentum (c) angular momentum (d) Laplace-Runge-Lenz vector. Q3.51: In one dimension, the motion of a particle is bound (a) only when the motion is periodic. (b) only when the total energy is negative. (c) only when there are two classical turning points. (d) when there is at least one classical turning point with total energy negative. Q3.52: A particle is kept at rest at a distance R (earth’s radius) above the earth’s surface. Thep minimum speed with which so that it does not p it should be projected p preturn is (a) 2GM/R (b) GM/R (c) GM/2R (d) GM/4R

Q3.53: If Ve be the escape velocity and Vo be the orbital velocity of a satellite close to earth’s surface, then which onep is correct √ √ (a) Ve = 2Vo (b)Ve = 3/2Vo (c) Vo = 2Ve (d) Ve = 2Vo . Q3.54: Two particles of masses M and m start moving towards each other from infinity due their mutual gravitational force. Their relative speed when they are at a distance d apart is q q q q 2G(M +m) 2G(M +m) 2GM m (b) (c) (d) (a) (M2Gd +m) d (M +m)d M md

Q3.55: Two particles of masses M and m are released from rest to collide due to their mutual gravitational force. If initial separation is d, the time taken for them to collide qis q q q π d3 d3 d3 d3 (a) π 2G(M (b) 2π (c) (d) π +m) G(M +m) 2 G(M +m) 8G(M +m)

Q3.56: Two stars of masses m1 and m2 at a center-to-center distance of a rotate about their common center of mass under the influence of gravity. If the orbits are assumed to be circular, what is the time q q period of rotation q q (a) 2π

a3 G(m1 +m2 )

(b) 2π

a3 m1 m2 G(m1 +m2 )

(c) 2π

a3 (m1 +m2 ) Gm1 m2

(d) 2π

a3 Gm1 m2

Q3.57: How does the period of revolution, T depend on the radius of the circular orbit for a particle moving in a central potential V (r) = −K/r n (K > 0, is a constant) ? (a) r 2+n (b) r 1+n/2 (c) r 3n/2 (d) r 3/2n

Q3.58: The condition for the circular orbit of a particle to be stable under small radial oscillation in an attractive central potential obeying a power-law ∝ −1/r n is (a) 0 < n < 1 (b) −1 < n < 1 (c) −1 ≤ n ≤ 2 (d) n < 2 Q3.59: The trajectory of a particle moving in a spherically symmetric potential is given as r(t) ∝ θ(t)4 . The temporal dependence of angular displacement θ(t) is expected to be (a) θ(t) ∝ t4 (b) θ(t) ∝ t1/2 (c) θ(t) ∝ t1/4 (d) θ(t) ∝ t1/9 Bhramar Chatterjee, Sirshendu Gayen


35

Q3.60: A particle moves in a central force field f (r) ∝ r n following a spiral curve r(t) = r0 eαθ(t) (n, r0 , α are constants). The value of n is (a) -1 (b) -2 (c) -3 (d) -4 Q3.61: Consider the motion of a particle of mass m in an attractive central potential V (r). When the total energy of the particle is E0 , the orbit is a circle of radius r0 . However, for total energy E (0 > E > E0 ) with same angular momentum, the motion is bound but the orbit is neither circular nor closed. In such non-circular orbit, if the particle takes T time to complete an angular revolution (from θ = 0 to back to θ = 2π) then which oneq is the correct expression q for2 T q mr02 mr E (r0 ) mr0 (a) 2π | dV | | (b) 2π (c) 2π V (r00)E0 (d) V2πE E dr r0

Q3.62:Consider the transfer (called the Hohmann transfer) of a satellite of mass m around the earth (earth mass = M) from one circular orbit of radius R to another of radius 2R. At position A, an instantaneous thrust is given to increase the kinetic energy by ∆K1 . When the satellite reaches B following an elliptic orbit, another instantaneous thrust increases the kinetic energy by ∆K2 to restore it in the larger circular orbit. The value of (∆K1 + ∆K2 ) is (ignore the effects of other planetary objects) m m m m (a) GM (b) GM (c) GM (d) GM R 2R 4R 8R

R

A

2R

B

Q3.63: Consider the Hohmann transfer as described in the previous problem. The time required processqis (ignore the effects of other q planetaryobjects) q in this transfer 3/2 3/2 3/2 3 3 3 2π 1 2πM m (a) G(M +m) 2 R (b) π G(M +m) 2 R (c) G(M R (d) +m) 2 q Mm R3/2 π G(M +m) Q3.64: A point-mass is thrown horizontally with a speed v along a frictionless track consisting of a horizontal part and a vertical semicircle of radius a as shown in the figure. What is the minimum value of v so that after leaving the semicircle at the top it will be projected to fall back on the initial position from where it was thrown (the distance, L ≥ 2a) ? √ √ √ √ (a) ag (b) 2ag (c) 2 ag (d) 5ag

A small block of mass m slides down from rest along a fricQ3.65: tionless vertical rail as shown in the figure. What should be the minimum value of the initial height h so that the block will not loose contact with the rail at the top of the circular part ? (a) 2R (b) 5R/2 (c) 3R (d) 7R/3

D Y

/

m

h 2R


36

My Physics Tutor

Q3.66:A pendulum of length l is released from horizontal rest position. The string runs into a peg as shown in the figure. What is the largest value of r for which the string remains taut at all times ? √ (a) l/2 (b) l/3 (c) 2l/5 (d) l/2 2

l

r

Q3.67: For rolling down of a solid cylinder over an inclined plane without slipping, the speed of the center of mass is vroll when it has fallen a vertical height h from rest. If vslide be the speed of the center of mass at the same position for frictionless slide without p rolling, then vslide /vroll is √ (a) 3/2 (b) 1/2 (c) 1/ 2 (d) 1

Q3.68: A point particle slides without friction down a smooth fixed sphere, starting at the top (the angle with vertical line, θ = 0). At what value of θ does the particle leave the sphere ? p √ (a) cos−1 (2/3) (b) cos−1 (1/3) (c) cos−1 (1/ 2) (d) cos−1 3/2. Q3.69: A spring of force constant k is stretched by x . It takes twice as much work to stretch a second spring by x/2. The force constant of the second spring is, (a) k (b) 2k (c) 4k (d) 8k

Q3.70: Consider small amplitude oscillation in one dimension for a system consisting of two tiny identical balls connected by a spring of natural length l. When the balls are charged equally by some amount of charge, the length of the spring becomes 2l. If ωi and ωf are the frequencies before and after charging, respectively, then √ (a) 2ωi = ωf (b) ωi = ωf (c) ωi = 2ωf (d) ωi = 2ωf Q3.71: A particle of mass m executes simple harmonic motion so that its displacement is given by x = 3sinωt + 4cosωt. Its total energy in units of mω 2 is (a) 3.5 (b) 4.5 (c) 8.0 (d) 12.5 Q3.72: A particle of unit mass executes small amplitude simple harmonic oscillation about the equilibrium position under the influence of a one dimensional potential V (x) = 6x(x − 1). The time period of oscillation is √ √ (a) π/2 (b) π/ 2 (c) π/ 3 (d) π Q3.73: A particle of mass m is subjected to one-dimensional potential V (x) = − 21 kx2 + λ 4 x where k, λ are positive constants. What is the angular frequency of small 4 amplitude oscillations aboutq the minimum of the potential ? q q q (a)

k m

(b)

2k m

(c)

λ m

(d)

√

λ m

Q3.74: A solid cylinder of mass M and radius R rolls forward without slipping on a moving platform. If the platform moves with an acceleration of A, what is the acceleration of the cylinder ? (a) A (b) A/2 (c) A/3 (d) 2A



37

A ball of mass m is attached to two identical springs and Q3.75: can move along a smooth horizontal rod as shown in the figure. The springs are unstreched when they are vertical. The ball is displaced by a small horizontal distance A and released to execute small oscillation. If the time period T depends on A as T ∝ Aη , then the value of η is (ignore the effect of gravity) (a) -1 (b) 0 (c) -1/2 (d) 3/2

k

m k

Q3.76: A small ball of mass m, attached to a massless string, oscillates simple harmonic ally in the vertical plane under gravity so that the maximum tension is twice the minimum tension. The value of the maximum tension is (a) 5mg/2 (b) 3mg/4 (c) 3mg/2 (d) mg Q3.77:Two equal masses m are connected to springs having equal spring constant, k as shown in the figure. The normal are p p p p frequencies (b) pk/m, p3k/m (a) p k/m, p2k/m (c) 2k/m, 3k/m (d) k/2m, k/m Q3.78: Three particles of mass m, m and M are connected through three unstretched identical massless springs, each of spring constant k. They are constrained to move in a circular path as shown in the figure. Which of the follwing is not a normal mode frequency for small amplitude oscillation ?p p p (a) 0 (b) k/M (c) 3k/m (d) (1 + 2m/M)k/m

M

m

m

Q3.79:Two masses of m and 4m are connected to springs of spring constant, k and 3k, respectively, as shown in the p figure. The p normal frequencies are p p (b) pk/m, p3k/m (a) p k/2m,p 3k/2m (c) k/m, 4k/3m (d) k/2m, k/m Q3.80: A solid cube of edge a, suspended vertically from one of its edges, executes small oscillation under gravity. Theq length of equivalent simple pendulum is (a)

√ 2 2 3

a

(b)

2 5

a

(c)

√1 2

a

(d)

3 2

a

Q3.81: A solid sphere of radius R, suspended vertically from one point on its surface, executes qgravity. The time period q of small oscillation isq q small oscillation under 3R 7R (b) 2π 5g (c) 2π 2R (d) 2π Rg (a) 2π 2g 5g Bhramar Chatterjee, Sirshendu Gayen

38

My Physics Tutor

Q3.82: Seven identical circular planar disks, each of radius r are welded symmetrically as shown in the figure. The assembly is suspended vertically from a point on the rim and executes small oscillation under gravity. The time period of small oscillation q q q q is 55r 4πr 9r (b) 2π (c) 2π (d) 2π (a) 2π 181r 6g 6g g 2g Q3.83: A cylindrical shell of mass m has an outer radius b and an inner radius a. The moment of inertia of the shell about the axis of the cylinder is: (a) 12 m(b2 − a2 ) (b) 21 m(a2 + b2 ) (c) 14 m(b2 − a2 ) (d) m(a2 + b2 ) Q3.84:A tiny mass slides under gravity without friction inside a semicircular depression of radius a inside a fixed block placed on a horizontal surface, as shown in the figure. The equation of motion of the mass in the x-direction will be q 2 g g (b) x¨ = − a x 1 − xa2 (a) x¨ = − a x q q 2 3g x2 x 1 + (d) x ¨ = − x 1 − xa2 . (c) x¨ = − 2a g a2 a Q3.85: A particle can slide freely along a fixed, smooth wire with a cycloidal shape: x = c(θ + sinθ),

y = 0,

z = c(1 − cosθ)

where c is a constant

If the particle executes oscillation under gravity in the vertical plane (i.e. zx-plane), what q is the period of oscillation q q p (a) π gc (b) 2π gc (c) 2π gc (d) 4π gc .

Q3.86: One end of a uniform rod of mass m and length l is attached to a fixed smooth pivot while the other end is gently released from rest when the rod is horizontal. Select the correct equation of motion involving the angular displacement (θ) from a vertical line. Assume that the potential energy to be zero when the rod is horizontal (θ = π/2). (a) θ¨ + gl sinθ = 0 (b) θ¨ + 2gl sinθ = 0 (c) θ¨ + 3lg sinθ = 0 (d) θ¨ + 3g sinθ = 0 2l Q3.87:The support of a pendulum of mass m and length l is free to slide along a frictionless horizontal rail as shown in the figure. If the mass of the support is M, then which one is the correct equations of motion of this dumbbell oscillator ? m ¨ θ + mg sinθ = 0 (b) M θ¨ + mg sinθ = 0 (a) MM+m l l mg g ¨ ¨ (c) (M + m)θ + l sinθ = 0 (d) θ + l sinθ = 0


M

l

m


39

Q3.88: AB is a piston rod of length d. The point A is constrained to move along the x-axis. The point B moves in a circle of radius a < d with a constant angular speed of ω as shown in the figure. At t = 0, A was at the maximum distance from the centre O. Which one of the following is correct ? ω

(a) the motion is simple harmonic , but not periodic. (b) the motion is periodic, but not simple harmonic . (c) the motion is simple harmonic and periodic as well. (d) the motion is neither periodic nor simple harmonic .

Q3.89:A particle of mass m swings around in a horizontal circle hanging through a massless string of length l. The angle θ0 between the string and the vertical line is constant and finite. of this circular pendulum is q q The time period lcosθ0 l (a) 2π g (b) 2π q g q 1 2 l (d) 2π gl 1 + 16 θ0 + · · · (c) 2π gsinθ 0

a

d

B

2`

O

A

g

θ0

l

m

Q3.90: Consider the motion of a straight rod of constant length l on smooth horizontal xy-plane. One end of the rod is pulled away with a constant speed v along x-axis while the other end is constrained to move along the y-axis. If the rod is aligned along the y-axis at t = 0 instant then what will be the speed of the midpoint of the rod when the end on the x-axis is at a distance l/2 from √ √ the origin ? (a) v (b) v/2 (c) v/ 2 (d) v/ 3 Q3.91: A boy initially stands at the origin and holds a string of length l that is attached to a toy located at (l, 0). The boy then starts walking up along the y-axis at a constant speed. The string is always tangent to the path of the toy. The correct dy expression for the instantaneous slope (= dx ) of the trajectory followed by the toy is p √ √ (d) − l2 − y 2/x (a) −(l − y)/x (b) − l2 − x2 /y (c) − l2 − x2 /x [The curve traced by the toy is called tractrix after the Latin participle tractus meaning dragged. It is the path of the rear end of a trailer initially lying on x-axis as the trailer is dragged by a tractor moving along the y-axis away from the origin.]

Q3.92: A dog at the point (L, 0) gives a chase as it sees a rabbit running away from the origin in a straight line along the y-axis. The dog runs always targeting the rabbit at the same speed as the rabbit. Which one of the following differential equations correctly represents the path y(x) followed by the dog q (a)

(b)

d2 y = dx2 2 d y y dx 2 =

dy 2 1 − ( dx )

dy 2 1 + ( dx )


x

40

My Physics Tutor (c)

d2 y x dx 2 2

q dy 2 = 1 + ( dx ) q dy 3/2 = 1 + ( dx )

d y (d) y dx 2 [Such path traced by the pursuers (follower) is called the curve of pursuit. In such motion

the pursued (leader) is always on the pursuer’s tangent.]

Q3.93: The equation of motion of a particle is given as a¨ x + bx˙ + cx = 0 where a, b, c are all positive constants. Which one of the following is the most acceptable phase space trajectory: (a)

.

(b)

x

x

.

x

(c)

x

.

x

(d)

x

.

x

x

Q3.94: The transformation (p, q) → (P = 12(p2 + q 2 ), Q) will be canonical for 1 (b) Q = tan−1 pq (c) Q = tan−1 pq (d) Q = tan−1 (pq) (a) Q = tan−1 pq

Q3.95: Which one is the generating to the canonical transforma 2 function corresponding q p p tion: p, q → P = − 2 + 1 q , Q = ln q 2 q (b) 21 (Qlnq − Q) (c) Qeq (d) qQ (a) 2 + 1 eQ


CHAPTER

4

General properties of matter etc.

Q4.1: The speed of sound in a stretched string is v when the extension is ∆l. What will be the speed of sound in stretched string when the extension is doubled within the elastic√limit (assume the mass density remains the same) (a) v/ 2 (b) v/2 (c) v (d) 2v Q4.2: A uniform rod of steel (density = ρ, Young’s modulus = Y ) of length l rotates with a constant angular speed ω in horizontal plane about a vertical axis passing through one of its ends. The elongation of the rod is 2l 2 3 2 (a) ρωl (b) ρω (c) ρω3Yl (d) 2ρω Y 2Y 5l3 Y Q4.3: A thin metallic ring of radius R rotates about a vertical axis passing through its centre. The density and tensile strength of the material is ρ and σ, respectively. The maximum angular speedqabove which the ring raptures is q q p σρ ρR πR2 σ (b) (c) (d) (a) πR 2 σρ ρR2 σ

Q4.4: If shearing strength (≡ maximum tolerable stress within the elastic limit) of µmetal is W , then the force required to punch a circular hole of radius r in a µ-metal sheet of thickness t is (a) rtW (b) 2πrtW (c) πr 2 tW (d) W πr 2 Q4.5: A solid cylinder of diameter d carries an axial load W . If the Young’s modulus and Poisson’s ratio of the material are Y and ν, respectively, then the change in diameter is πdY νY (a) 4νW (b) 4νW (c) 4πdW (d) πdW πdY 4νY

Q4.6: A solid bar is subjected to a hydrostatic pressure. The relation between the Young’s modulus (Y), Poisson’s ratio (ν) and volume compressibility (κ) of the material is (a) κY = 3(1 − 2ν) (b) Y = 3κ(1 − 2ν) (c) κY = 2ν (d) Y = 2νκ Q4.7: A solid cylinder of uniform density ρ is floating in a liquid of density σ(> ρ). The height and cross-sectional area of the cylinder are H and A, respectively. It is 41

42

My Physics Tutor pushed down slightly and released. The frequency of small oscillation is q q q p gρ gσ gHσ gHρ (b) (c) (d) (a) Hρ Aρ Hσ Aσ

Q4.8: An open cylindrical vessel partially filled with water rotates with a constant angular speed ω about its axis (z-axis) along the vertical direction. The differential equation of the free surface of water in zx-plane for steady uniform rotation is (gravitational acceleration = −gˆ z) dz dz 2 (a) g dx = ω x (b) g dx = ωx2 (c) g dx = ω2x (d) g dx = ωz 2 . dz dz

Q4.9:

Water is running out of a right circular conical funnel at a constant rate. If the initial height was H, then the instantaneous height h(t) depends upon time (t) (involving a constant k) as (a) H − kt (b) (H 2 − kt)1/2 (c) (H 3/2 − kt)2/3 (d) (H 3 − kt)1/3

h

Q4.10: Water is drained out through a hole p at the bottom of a right circular conical funnel with a speed proportional to h(t) where h(t) is the instantaneous height. If the initial height was H, then h(t) is expected to vary as (involving a constant k) (a) H − kt (b) (H 3 − kt)1/3 (c) (H 3/2 − kt)2/3 (d) (H 5/2 − kt)2/5

Q4.11: The height difference of mercury columns between the vertical tubes of a manometer is h during the flow of a gas through the manometer. The cros-sectional area of the vertical tubes are different. The wider vertical tube is connected to the narrower horizontal arm of the manometer as shown in the figure. Select the correct option when gas flows through the manometer. (a) h > 0 (b) h = 0 (c) h < 0 (d) the sign of h will depend upon the direction of flow

h mercury

Q4.12: A liquid of density ρ flows through a horizontal pipe of non-uniform cross-section. The pressure at two points are pA and pB , respectively. If the speed at these points are related as vA = v = 2vB then (pB − pA ) is (a) 3ρv 2 /8 (b) ρv 2 /2 (c) 2ρv 2 /3 (d) −5ρv 2 /8 Bhramar Chatterjee, Sirshendu Gayen

Chapter 4. General properties of matter etc.

43

Q4.13:

A non-viscous gas of density ρg is flowing through a uniform cylindrical pipe of cross-section A. The hight difference of a liquid of density ρl in a pivot tube mounted inside the pipe is h as shown in the figure. The amount of gas that flows out through the pipe per unit time q q is q q ghρg 2ghρg 2ghρl l (a) A ρl ) (b) A (d) A 2ghρ (c) 2A ρg ρl ρg

h

Q4.14: A tiny spherical object of radius r and mass density ρ falls from rest under gravity in a viscous medium of viscosity η. The approximate time in which it will practically attain the terminal velocity is (a) ηr 2 /ρ (b) ηr/ρ (c) ηr/ρ2 (d) ρr 2 /η Q4.15: Consider the steady flow of a fluid of density ρ, viscosity η through a uniform 2 cylindrical pipe of radius R. The radial distribution of speed is v(r) = v0 (1 − Rr 2 ). The amount of fluid flowing across the section of the tube per unit time is (a) R2 v0 (b) πR2 v0 (c) R2 v0 /4π (d) πR2 v0 /2 Q4.16: In the previous problem if a constant pressure difference, ∆P is maintained between two ends of the pipe, the expression for ∆P is ηv0 l 0l 0l 0l (a) ηv (b) 4ηv (c) 4ηv (d) 4πR 2 R2 R2 πR2 Q4.17: The cross-sectional radius of a pipe decreases as a function of distance (x) from one end as r(x) = r0 e−ax where a > 0 is a constant. The ratio of Reynolds numbers for two cross-section separated by a distance L is (a) eaL (b) ear0 (c) eL/r0 (d) e2πr0 /L Q4.18:

A non-viscous and incompressible fluid is pumped at a constant speed of v0 (volume flow rate is πa2 v0 ) through a circular hole of radius a into a parallel plate arrangement as shown in the figure. The parallel plates are circular and concentric, each of radius b and separated by a small distance d. The velocity within the plates is purely radial and discharge to atmospheric pressure at r = b. If density of the fluid is ρ, what is the upward force experienced by the lower plate following the Bernoulli’s principle ? (assume that the region with r < a does not contribute any force) b πρv2 a4 πρv2 a3 πρv2 a4 πρv2 a4 (b) 4d02 e− a (c) 4bd0 (d) 4d02 (a) 4d02 ln ab

a

b

Q4.19: Incompressible fluid of negligible viscosity is pumped at a constant rate Q through two small holes (Q/2 from each) into the narrow gap between two parallel concentric disks as shown in the figure. The separation between the discs is d. The velocity of Bhramar Chatterjee, Sirshendu Gayen

44

My Physics Tutor fluid particles within the discs is purely radial. The magnitude of acceleration of a fluid particle along the radial direction as a function of r is (Hint: Use the continuity equation to derive the radial velocity and take the convective derivative) 2 2 Q Q (a) 16π3Q (b) 4π2Qd2 r3 (c) 2πd (d) 2πdr 2 d2 r 3 3r 3

Q/2

Q/2

Q4.20: The dispersion relation for ocean surface wave is ω 2 = gk tanh(kh) where h is the water depth and g is the acceleration due to gravity. In the deep-water approximation h λ = 2π/k whereas in the shallow-water approximation kh 1. The ratio of deep-water group velocity (vd ) to shallow-water group velocity (vs ) i.e. vd /vs is √ (d) 2 (a) 1 (b) 1/2 (c) 1/ 2 Q4.21: A spherical bubble rises up gently from the bottom of a deep lake. Assume that there is no temperature gradient along the depth of the lake. The atmospheric pressure is equivalent to a water column of height H and surface tension of water is T . If the radius of the bubble at a depth h is r, then the radius of the bubble gets doubled when the depth is hT 6T 2T 2T (a) rρgH (b) 18 (h − 7H) − rρg (c) 34 (H − h) − rρg (d) (H − 4h) − rρg Q4.22: The equilibrium radius of a charged soap bubble is R. If the surface tension of soapqsolution is T , then the surface charge density q is q q 80 T 0 T 4T (a) (b) πR (c) 4πR (d) 4πR0 T R Q4.23: Two soap bubbles of radius 3 cm and 4 cm coalesce under isothermal condition in vacuum. The radius of the new bubble will be (a) 2.3 cm (b) 4.5 cm (c) 5 cm (d) 7 cm


CHAPTER

5

Electromagnetism

Q5.1: The dimension of e2 /h is same with the dimension of (a) conductance (b) inductance (c) mobility

(d) magnetic moment.

Q5.2: A sphere of radius R with uniform volume charge density is rotating with constant angular speed about one of its diameter. The total charge in the sphere is q. The electric field for r > R is (a) 0 (b) 4πq0 r2 rˆ (c) 4πq0 rR rˆ (d) 4πqR0 r3 rˆ Q5.3: The total charge for a spherically symmetric distribution of screened charge given α2 e−αr is by ρ(~r) = Q δ(~r) − 4π r (a) 0 (b) Q (c) 4πQ (d) Q/4π Q5.4: Four equal point charges are kept fixed at the four vertices of a square. How many neutral points (i.e. points where the electric field vanishes) will be found inside the square ? (a) 0 (b) 1 (c) 4 (d) 5 Q5.5: A spherical cavity of radius r is hollowed out from the interior of a uniformly charged sphere (volume charge density = ρ) of radius R > r as shown in the figure. The separation between the centre of the spheres is d. The electric field strength in the cavity is (a) 0 (b) ρR/30 (c) ρ(R − r)/30 (d) ρd/30

R d

r

~ be the electric field due to a unit charge at the origin then the volume integral Q5.6: If E ~ over a sphere of radius a, centered at the origin is of ∇.E (a) 0 (b) 4π (c) 1/0 (d) a3 /30 45

46

My Physics Tutor

Q5.7: What is the flux through a square surface of side a, placed at a distance a/2 from the centre to a point charge q ? (a) q/4a2 (b) q/40 a2 (c) q/60 (d) q/80 Q5.8: Two point charges +q and −q are separated by a distance √ 2d. The electric flux through the circle of radius R = d/ 3 as shown in the figure is (a) 0 (b) q/20 (c) q/4π0 (d) 3q/4π0

R -q

d

d

+q

Q5.9: Consider a unit positive charge is located at the top of a hypothetical hemisphere as shown in the figure. What is the flux through the curved surface ? √ √ (1+ 2) (2− 2) π (b) 24 (c) (d) (a) 810 8π0 40 0 Q5.10: Assume that a unit positive charge is placed at the centre of the top surface of a cylinder as shown in the figure. The height of the cylinder is equal to its radius. What is the flux through the curved surface ? √ √ √ 2) (2− 2) (a) 2√12 (b) 252π0 (c) (1+ (d) 8π0 40 0

Q5.11: Two positive and two negative charges of unit magnitude are located alternatively on a circle of radius R at equal angular separation as shown in the figure. How much work does it take to assemble this charge configuration ? (a) 0 (b) π10 R (c) π10 R ( 2√1 2 − 1) (d) 4π10 R ( √12 − 1)

+q

-q q

-q q R

+q

Q5.12:

The electro-static interaction energy for a system consisting of four identical unit charges (q = 1) located at the vertices of a regular tetrahedron of side a is (b) π10 a2 (c) π02a2 (d) π03a2 (a) 2π30 a2

+q a

+q

+q +q

Q5.13: Two point charges +q and −q are located at diagonally opposite corners of a square. If the potential at the corner P is taken as 1 Volt then the potential (in √ Volt) at the center O√is (d) 1/ 2 (a) 0 (b) 1 (c) 2


+q

P O -q

Chapter 5. Electromagnetism

47

Q5.14:

A sphere made of some dielectric material is placed in an other~ as shown in the figure. Which wise homogeneous electric field E of the following is true for the strength of the net electric field at the points A, B and C ?

E

BB

A

C

(a) EA = EB = EC (b) EA > EB > EC (c) EA > EB < EC (d) EA < EB > EC Q5.15: A particle of mass √ m with a charge +q is released from rest from a distance d = 3R from the centre of a ring of radius R as shown in the figure. The ring carries a charge −q distributed uniformly over its rim. What will be the speed of the particle when qit will pass through q the centre of q the ring q (a)

q2 2π0 Rm

(b)

q2 4π0 Rm

(c)

q2 12π0 Rm

(d)

R d

+q

-q

3q 2 4π0 Rm

Q5.16: Two thin concentric metal spheres of radius R and 2R carries charges Q and 2Q, respectively. The potential difference between the spheres is V0 . If the value of the charges on the spheres is altered, the magnitude of potential difference will be (a) V0 /2 (b) 2V0 (c) 5V0 /3 (d) 7V0 Q5.17: A thin metal sphere of radius r1 is charged to a potential V1 and then placed inside a thin metal sphere (made up of two hemispheres) of radius r2 (> r1 ) so that two spheres are electrically isolated. Now if they are shorted for a while and disconnected what will be the potential of the outer spheres ? (a) 4πV10 r1 r2 (b) Vr21 r1 (c) V1 (d) 0 Q5.18:

Two concentric hollow metal spheres of radius R and 2R are at potential V0 and 2V0 , respectively. Which of the following is the correct expression for the potential in the region R < r < 2R ? (d) V0 3 − 2R (a) V0 (b) 2V0 (c) VR0 r r

2V0 V0 R 2R

Q5.19: Two point charges ±q are located at (0, ±a) as shown in the figure. The work done in bringing a unit positive point charge from (2a, 0) to (0, 2a) in a circular path is (a) 0 (b) 4πq 0 a (c) 21 4πq 0 a (d) 23 4πq 0 a

(0, 2a) (0, a) +q

(2a, 0) (0, -a) -q

Q5.20: A point charge q is located at a distance d from an infinite conducting plate. The amount of work to be done to remove the charge far away from that plate is (a) q 2 /16π0 d (b) q 2 /4π0 d (c) q 2 /4π0 (d) q 2 /2π0 d Bhramar Chatterjee, Sirshendu Gayen

48

My Physics Tutor

Q5.21: A point charge q of mass m released from rest at a distance d from an infinite conducting plate. How long does it take for the charge to to hit the plate ? (ignore gravity) √ √ √ √ (b) 2π0 md3 /q (c) 2π 3 0 md3 /q (d) 4π0 md3 /q (a) 2π0 md/q Q5.22: A point charge +q is located at a distance d from two grounded large conducting plates intersecting at right angles as shown in the figure. The magnitude of force acting on the charge is √ 2 √ 2 √ πq q2 q2 3q √ (a) 4√2π (b) (d) (c) 2 − 1) 2 2 (2 2 2 16π d 32π d d 4 2 d 0 0 0

+q

d

d

0

Q5.23: A point charge +q is located at a distance d from two grounded large conducting plates intersecting at right angles as shown in the figure. The potential at C, the mid-point of the line joining the charge and the point of intersection of the plates, is √ √ q q 3q 3 2 √1 ) ( − ( − √15 ) (c) (d) (a) 0 (b) 4√2π 4π0 d 2 π0 d 3 d 2

+q

d C

d

0

Q5.24: If the electrostatic potential V (r, θ, φ) in a charge free region has the form V (r, θ, φ) = f (r)cosθ , then the general functional form of f (r) (with two constants α and β) is β β β 2 α (b) αr + r2 (c) αr + r (d) r ln βr (a) αr + r

Q5.25: If the electrostatic potential on a spherical shell of radius R is given by, V (r = R, θ, φ) = V0 cos2 θ , then the value of V (r = 2R, θ, φ) is (a) V80 (1 + cos2 θ) (b) V80 (1 + 2cos2 θ) (c) V40 (1 − cos2 θ) (d) V80 (−2cosθ + cos3 θ).

ˆ ~ = E0 k. Q5.26: A grounded metallic hollow sphere is placed in a uniform electric field E The charge distribution on the surface of the sphere is (a) 0 (b) 30 E0 cosθ (c) 4π0 E0 cosθ (d) E0 cosθ/4π0 Q5.27: The potential at the surface of a hollow sphere of radius R is given by V (r = R, θ) = V0 cosθ, where V0 is a constant. The surface charge density on the sphere is 0 V0 (b) 23R sinθ (c) 3R0 V0 cosθ (d) 0RV0 cos2 θ (a) 4πR0 V0 sinθ Q5.28: The potential at the surface of a hollow sphere of radius R is given by V (r = R, θ) = V0 cos3θ, where V0 is a constant. The potential inside the sphere V (r ≤ R, θ) = is 0r (a) 0 (b) V0 (c) rVR0 cosθ (d) V5R cosθ[4 Rr (5cos2 θ − 3) − 3] Q5.29: Two positive and two negative charges of equal magnitude are placed alternatively on the rim of a thin insulating circle of radius R at equal angular separation as shown in the figure. How does the potential V (r) depend on the distance r from the center of the circle for r R ? (a) 1/r (b) 1/r 2 (c) 1/r 3 (d) 1/r 4 Bhramar Chatterjee, Sirshendu Gayen

+q

-q q

-q q R

+q


49

Q5.30: Three point charges each of charge q are located at (0, −a, a), (0, a, 2a) and (0, 0, −a), respectively. The net dipole moment of the charge distribution about the origin is (a) −4aqî (b) −2aqˆj (c) 2aq kˆ (d) 4aq kˆ Q5.31: The surface charge distributed over a hollow sphere of radius R given R by ρ(~r) = σ0 δ(r − R)cosθ where, θ is the polar angle. The dipole moment (≡ ~rρ(~r)d3 r) is [kˆ is the unit vector along z-axis] 3 3 (a) 0 (b) −σ0 R3 kˆ (c) σ04πR kˆ (d) 4πσ30 R kˆ ~ 0 with Q5.32: A point electric dipole of moment ~p is placed in a uniform electric field, E ~ One of the equipotential surface, in this case, is a sphere. The radius of this ~p||E. sphere q q is q q p p p 3 3 (d) 4π3p0 E0 (a) 4π0 E0 (b) 4π0 E0 (c) 0 E 2 0

Q5.33: Two hollow metal spheres of radius r1 and r2 carry charges of Q and 2Q, respectively. What will be the ratio of surface charge density, σ1 : σ2 when they are short-circuited ? (a) r1 : r2 (b) r2 : r1 (c) r1 : 2r2 (d) 1 : 2

Q5.34: Consider a spherical capacitor consisting of two concentric spherical metal shells of radius a and b (> a) with a dielectric material between them. If the permittivity of the dielectric varies radially as (r) = kr , then the capacitance is 0 k 0 k 4π0 k 0k (b) ln(b/a) (c) 4π (d) b−a (a) ln(b/a) b−a ~ to a sphere of dielectric material causes Q5.35: Application of an external electric field E a small displacement of the net positive charges of the dielectric with respect to all its negative charges resulting in the development of a uniform polarization P~ . Select ~ and P~ (= net dipole moment per unit volume). the correct relation between E ~ = − 1 P~ ~ = − 1 P~ ~ = − 1 P~ ~ = − 1 P~ (a) E (b) E (c) E (d) E 4π0 30 20 0 Q5.36: The magnitude of magnetic dipole moment associated with a charged disc of radius R carrying a uniform surface charge density σ, rotating with a constant angular speed ω is (a) σωR4 (b) σωR4/4π (c) πσωR4/4 (d) 3πσωR4/4 Q5.37: The magnitude of magnetic dipole moment associated with a charged hollow sphere of radius R carrying a uniform surface charge density σ, rotating with a constant angular speed ω is M1 whereas it is M2 for a rotating solid sphere of same radius having uniform volume-charge density ρ, rotating with same angular speed. The value of M1 /M2 is (a) σ/%R (b) σ/4π%R (c) 4πσ/3%R (d) 5σ/%R Q5.38: Consider small angle oscillation of a bar magnet in a uniform magnetic field. If the magnetic field is doubled, √ the time period of oscillation will be (a) increased by a factor of 2 Bhramar Chatterjee, Sirshendu Gayen

50

My Physics Tutor √ (b) decreased by a factor of 2 (c) increased by a factor of 2 (d) increased by a factor of 2.

Q5.39: Two concentric metal spheres carry charges ±q(t) as shown in the figure. The space between the spheres is filled with Ohmic material. Current flows radially through the material from one sphere to another. Which of the following is true for the magnetic field in three regions — inside the inner sphere (region 1), in the material between the spheres (region 2), outside the outer sphere (region 3) ? -q (a) B1 = B2 = B3 = 0 +q (b) B1 6= 0, B2 = 0, B3 6= 0 1 2 3 (c) B1 = 0, B2 6= 0, B2 = 0 (d) B1 6= 0, B2 6= 0, B3 6= 0

Q5.40: I

The magnitude of magnetic fields at the center of the semicircular loops are B1 (at P1 ) and B2 (at P2 ), respectively, for a current I (shown in the figure). The linear parts are 2 −B1 ) assumed to be very long. The value of π(BB is 1 (a) 0 (b) 1 (c) 2 (d) π

. P1 D

D . P2 I

Q5.41: The magnitude of magnetic field produced by two L-shaped very long wire (each carrying current I) at the mid point of their shortest distance (= R) is B1 . In case of a uniform star-shaped wire (inscribed in a circle of radius R) carrying the same current as shwon in the figure it is B2 B2 is at the centre. The ratio B 1 √ √ (a) π/2 (b) 2( 2 − 1) (c) 3( 3 − 1) (d) 2π

I

I R R

2

R +d /4

2

(b) √µ02IπR2


R +d /4

(c)

BB2 I

Q5.42: Consider a pair of identical circular rings, each of radius R, separated by a distance d as shown in the figure. If both of them carry same current I in same direction, what is the magnetic field at the point midway between them ? (a) √µ02IR 2

B1

I

IπR2 4µ0 (R2 +d2 /4)3/2

I

R

d

(d)

µ0 IR2 2 (R +d2 /4)3/2


51

Q5.43: A constant current I flows down the surface of the inner cylinder (radius r), and back along the outer cylinder (radius R > r) of a long coaxial cable as shown in the figure. The self-inductance of the coaxial cable per unit length is µ0 I R µ0 µ0 µ0 R (a) 4π r (b) 2π ln Rr (c) 4π ln Rr (d) 4π ln R+r Q5.44: A square loop of egde a having n turns is rotated with a uniform angular speed ω about one of its diagonals which is kept fixed in a horizontal position. A uniform magnetic field B exists in the vertical direction. The emf induced in the coil is (a) nBa2 sin(2ωt) (b) n2 Ba2 cos(ωt) (c) nB 2 aωcos(ωt) (d) nBa2 ωsin(ωt)

I I

r R

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

ω X

Q5.45: A metal rod of length l rotates with uniform angular speed ω about its perpendic~ (parallel to the axis of rotation). The potential ular bi-sector in a magnetic field, B difference between the center of the rod and an end is (c) ωBl2 /2 (d) ωBl2 (a) 0 (b) ωBl2 /8 ~ which Q5.46: A metal disk of radius R rotates with angular speed ω in a uniform field B is perpendicular to the plane of the disk. The emf produced between the centre and the edge of the disk is (a) ωBR2 /4π (b) ωBπR2 (c) ωB/πR2 (d) ωBR2 /2 Q5.47: A metal rod rotates with constant angular speed ω in a ~ with one end sliding over ω uniform vertical magnetic field B a metal ring of radius R as shown in the figure. Neglect the resistance and inductance of the circuit and also friction between the rod and the ring,. The emf (E) generated is 2 2 2 ωR (a) ωBR (b) ωBR (c) ωBR (d) 4πB 2 2π 4π

Q5.48: A metal bar of mass m slides on two parallel, frictionless, conducting rails a distance d apart as shown in the figure. A battery of emf E is connected across the rails, and a uniform ~ perpendicular to the plane of the rails, fills magnetic field B, the entire region. Assume that the rails have zero resistance and the bar has a resistance R (ignore self-inductance of the circuit). At time t = 0, the switch (S) is closed and the bar starts to move. What will be the terminal speed of the bar ? (a)

E Bd

(b)

B2 d mR

(c)

mRE Bd

B

E

R

X

X BX

X

X

X

X

X

X

X

Xd X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

E X

S X

X

(d)


mdE BR

52

My Physics Tutor

Q5.49: A square-shaped metal loop of side a is drawn away with a constant speed v from a long, straight wire carrying a current I. The wire and the square loop both lie in the same plane. If the resistance of the loop is R, then the induced current in the loop varies with the distance from the wire (r) as µ0 Iav µ0 Iv µ0 Ia2 v µ0 Ia2 v (a) 2π(r+a/2)R (b) 2πra (c) 2πr(r+a)R (d) 4πr(r+a)R 2R

Q5.50: A straight conducting rod is pulled with a constant speed v along y-axis keeping it parallel to the x-axis over a parabola made of metallic wire in presence of a vertical uniform magnetic B. The equation of the parabola is y = ax2 (a > 0 is a constant). The rod has resistance ρ per unit length whereas the parabolic wire has no resistance. Assume that at t = 0, the rod was on the x-axis. If the current in the circuit I(t) varies with time as I(t) ∝ tα , the value of the constant α is (a) -1/2 (b) 0 (c) 1 (d) 2

r

v

I a

y

x

~ = Q5.51: The electric field of an electromagnetic wave propagating in air is given as E ˆ i(ωt+bx−ay) . The wave incidents on a flat perfectly conducting slab (aî + bˆj + ck)e positioned at x = 0. The electric field of the reflected beam is ˆ i(ωt+bx+ay) (a) (−aî + bˆj + ck)e ˆ i(ωt+bx+ay) (b) (−aî − bˆj + ck)e ˆ i(ωt−bx−ay) (c) (−aî + bˆj + ck)e ˆ i(ωt−bx+ay) (d) (−aî − bˆj + ck)e Q5.52: For an electromagnetic wave propagating in vacuum, the electric field is given by ~ = (aî+bˆj)ei(ωt−kz) . The energy flux per unit time crossing unit area perpendicular E to the direction of propagation is (a) 0 √ (b) 12 c0 a2 + b2 (c) c0 (a2 + b2 )cos2 (ωt − kz) (d) 2c1 0 (a2 + b2 )cos2 (ωt − kz) Q5.53: The initial volume charge density ρ(r, t = 0) inside a material of conductivity σ and permittivity is ρ0 . At subsequent times ρ(r, t) is given by σt σt σt − σt (a) ρ0 e (b) ρ0 cosh (c) ρ0 [1 − e ] (d) ρ0 /[1 − e ]

Q5.54: While passing through a conducting medium (with permeability µ, permittivity and conductivity σ) the amplitude of an electromagnetic wave (of frequency ω) gets gradually attenuated which is quantified by the skin depth. If the complex wave number k in such case obeys the relation: k 2 = µω 2 + iµσω, then for a metal of high conductivity (σ ω), which of the following relations is correct for skin depth d ? Bhramar Chatterjee, Sirshendu Gayen


53

 2 2  x − y  for z = x + iy, these relations may be useful: x2 + y 2   

(a) d ∝

p σ/ω

(b) d ∝

p 1/σω

 = Re(z 2 ) p  = [Re(z 2 )]2 + [Im(z 2 )]2  (= |z|2 = |z 2 |) p √ (c) d ∝ ω/σ (d) d ∝ σω

Q5.55: The ratio of conduction current density to displacement current density (|Jc |/|Jd|) for a material of conductivity σ and permittivity when exposed to an electric field varying sinusoidally with frequency ω is (a) σ/ω (b) ω/σ (c) /σ (d) ωσ/

Q5.56: The electromagnetic radiation of wave-length of one micron (1 µ m ≡ 1.24 eV ≡ 300 THz) lies in the range of (a) x-ray (b) ultra-violate (c) near infra-red (d) micro-wave Q5.57: An em-wave traveling in air has normally incident on an air-dielectric interface. The dielectric media is isotropic and non-magnetic with relative permittivity of 9. What percentage of incident power will be reflected from the interface ? (a) 0 (b) 25% (c) 33.3% (d) 50% Q5.58: An em-wave in normally incident on an air-dielectric interface. The dielectric media is isotropic and non-magnetic. The magnetic field of the em-wave in the ~ = 4 × 10−8 (2î + 3ˆj)ei(3x+4y−5×108 t) Wb/m2 . What dielectric medium is given by B percentage of incident power will be reflected from the interface ? (a) 25% (b) 33.3% (c) 50% (d) 100% Q5.59: An em-wave traveling in an isotropic dielectric medium is specified by ~ = î80πcos(4 × 108 t − 2z) V/m; E ~ = ˆj0.5cos(4 × 108 t − 2z) A/m H The relative permeability of the medium is: (a) 1.25 (b) 2 (c) 2.25

(d) 4

Q5.60: If light is incident at the Brewster angle on a dielectric, (a) the reflected light is completely circularly polarized. (b) the transmitted light is completely circularly polarized. (c) the transmitted light is completely polarized parallel to the plane of incidence. (d) the reflected light is completely polarized perpendicular to the plane of incidence. Q5.61: For maximum polarization by reflection from a plane dielectric the angle, φ between reflected and refracted ray is (a) 0 < φ < π/2 (b) π/4 (c) π/2 < φ < π (d) π/2 Q5.62: For maximum polarization by reflection from air-dielectric (refractive index n > 1) interface the angle between refracted ray and the normal to the interface is (a) tan−1 (n − 1) (b) tan−1 n (c) cot−1 n (d) π/2 Q5.63: A perfectly reflecting sphere of radius R is kept in the path of a parallel beam of light of large aperture. If F be the force exerted by the beam for an intensity I, Bhramar Chatterjee, Sirshendu Gayen

54

My Physics Tutor then what will be the force when the sphere is a perfect absorber ? (a) 0 (b) F/2 (c) F

(d) 2F

Q5.64: The approximate force exerted on a perfectly reflecting mirror by an incident laser beam of power 10 mW at normal incidence is (a) 10−15 N (b) 10−13 N (c) 10−11 N (d) 10−9 N Q5.65: The energy stored in a 30 cm length of laser beam operating at 5 mW is (in picoJoule) (a) 0.1 (b) 5 (c) 150 (d) 800 Q5.66: Consider the propagation of em wave from a point source in a non-dispersive medium in one dimension. The energy contained between two wave fronts (separated by a constant phase) varies with distance from the source (r) as (a) 1/r (b) 1/r 2 (c) r (d) independent of r Q5.67: Consider the propagation of em wave radiated from an infinitely long line source in a non-absorbing medium in three dimension. If the power radiated per unit length from the line source is constant, how does the amplitude of electric field vector decay with the distance (r) ? (a) r −2 (b) r −1 (c) r −1/2 (d) r 0 Q5.68: Consider the propagation of em wave from a point source in a non-absorbing medium in d-dimension. If the power radiated from the point source is constant, how does the intensity decay with the radial distance from the source (r) ? (a) r −1 (b) r −2 (c) r −(d−1) (d) r −(d+1)/2 Q5.69: When the phase velocity varies inversely proportional to wavelength, which one is correct relation between the group velocity, vg and the phase velocity, vp ? (a) vg = vp /2 (b) vg = vp (c) vg = 2vp (d) vg = 4vp Q5.70: For a parabolic dispersion relation (ω ∝ k 2 ) which one of the following is correct relation between the group velocity, vg and the phase √ velocity, vp ? (d) vg = 2vp (a) vg = vp (b) 2vg = vp (c) vg = 2vp Q5.71: Which of the following magnetic vector potentials correspond to the uniform ~ = B0 kˆ magnetic field B ~ = B0 xˆj (a) A ~ = −B0 yî (b) A ~ (c) A = B0 [−yî + xˆj] ˆ ~ = B0 [zî + (z + x)ˆj + (x + y)k] (d) A Q5.72: Two magnetic vector potentials A~1 = −B0 (yî − xˆj) and A~2 = −2B0 yî correspond to same uniform magnetic field. Which of the following gauge functions connects the two potentials ? (a) ±B0 xy (b) ±B0 xy/z (c) ±B0 (x + y) (d) ±B0 (yz + zx) Bhramar Chatterjee, Sirshendu Gayen


55

Q5.73: Electro-magnetic field (Fµν ) is an anti-symmetric tensor of second rank. The number of independent components is (a) 3 (b) 6 (c) 10 (d) 16


56


My Physics Tutor

SOLUTIONS

CHAPTER

13

Solution to introductory problems

S0.1: The eigenvalues of a triangular matrix are its diagonal elements. So the eigen-values of A are 1, 2, 3. n Again, if λi ’s are the eigen-values of A, then λni will be Qthe eigen-values of A for integral values of n provided that A is non-singular i.e. λi 6= 0 when n is negative. i

Obviously, the given matrix is non-singular, so the eigen-values of A−1 are 1, 1/2, 1/3. Hence, the trace of A−1 is 11/6. Remark 13.0.1 Why the eigenvalues of a triangular matrix (Am×m ) are its diagonal elements is obvious from the characteristic equation: (λ − a11 )(λ − a22 ) · · · (λ − amm ) = 0, obtained for an upper triangular or a lower triangular matrix simply by expanding a11 − λ a12 ··· a1m 0 a22 − λ · · · a2m detAm×m = = 0, .. .. .. .. . . . . 0 0 · · · amm − λ a11 − λ 0 ··· 0 a21 a22 − λ · · · 0 or, = 0 .. .. .. .. . . . . am1 am2 · · · amm − λ

S0.2: The Fourier transformation (FT) of a Hermite-Gauss function is also Hermite-Gauss function. The zeroth order Hermite polynomial is 1, implying the most common example that a Gaussian function is its own FT. This coincidence has a nice resemblance with the energy eigen-function of a SHO. Both the momentum and the coordinate has the quadratic dependance in the Hamiltonian of a SHO, so solving the Schrödinger equation either in coordinate representation or in momentum representation will give the same functional from of the energy eigen-function. And obviously, the wave function in coordinate space and momentum space are related by FT. 115

116

My Physics Tutor

S0.3: The change of velocity is 2V . It is simple to consider the scattering process in rest frame of the heavy mass where the light mass comes with some speed and bounces back with the same speed after elastic scattering. One may prefer to calculate it straightforward using the conservation of linear momentum and energy as the collision is elastic. From conservation of momentum: MV + 0 = MV 0 + mv

⇒ M(V − V 0 ) = mv

(13.1)

And from conservation of energy: 1 1 1 MV 2 + 0 = MV 02 + mv 2 2 2 2

⇒ M(V 2 − V 02 ) = mv 2

(13.2)

Equation (13.2)÷(13.1) implies, v = V + V 0 . Speed of the heavy mass after collision remains practically the same i.e. V ' V 0 , implies v = 2V . S0.4: If l is the length of the ladder, then at any instant of time: x2 (t) + y 2(t) = l2 ⇒ (x/2)2 (t) + (y/2)2(t) = (l/2)2 . So the locus of the mid-point will be part of a circle. Let us inspect the locus of an arbitrary point P (X, Y) on the ladder which divides the ladder in a : b (= 1 − a) ratio. y y x Y Then X = cosθ = and = sinθ = . Again from the al l bl l 2 2 (0, y) (ax)2 (by)2 2 2 2 constrain: x + y = l ⇒ a2 + b2 = 1 i.e. Xa2 + Yb2 = 1 implying that the locus of P will be part of an ellipse in general. It may be noted that the trajectories have no explicite dependence on how fast or slow and uniformly or non-uniformly the end is pulled. The schematic shows the locus.

y

P (X, Y) θ

(x, 0) x

y

y

P

x

x

x

P a : 1-a ( 0a> 0.5)

Figure 13.1: Trajectories of different points on a sliding ladder. S0.5: The angle between any pair of springs is γ = cos−1 (− 31 ) = 109◦ 280 . Bhramar Chatterjee, Sirshendu Gayen

Chapter 13. Solution to introductory problems

117

Let the forces acting on the ball through four springs are F~1 , F~2 , F~3 , F~4 . As the springs are identical and connected symmetrically all the forces are same in magnitude, say F . For equilibrium of the ball under these forces, F~1 + F~2 + F~3 + F~4 = 0. So, F~1 .(F~1 + F~2 + F~3 + F~4 ) = 0 ⇒ F 2 + 3F 2 cosγ = 0 implies the result. Remark 13.0.2 In methane (CH4 ) molecule, the carbon atom at the centre of a tetrahedron is bonded to four hydrogen atoms at the vertices of the tetrahedron through four hybridized sp3 orbitals with equilibrium bond angle of cos−1 (− 31 ) = 109◦ 280 . This is the basic structural unit of the ‘Diamond structure’ as well which appears in covalent bonded group-IV crystals such as diamond (carbon), silicon, and germanium. It also appears in ‘zinc-blende structure’ (the bonding in this case is partially covalent and partially ionic or electrostatic) of III-V binary semiconductors, such as GaAs, GaP, GaN, InAs, InP, InSb and some of the II-VI compounds, such as ZnS, ZnSe, CdS, and CdSe.

Cos -1 (− ) = 109028′

(a)

(b)

Figure 13.2: (a) Tetrahedral structure of methane (CH4 ) molecule, and (b) typical zincblende structure.

S0.6: The direction of the Poynting vector is along 51 (3bi + 4b j). Poynting vector is the direction along which energy flows for a plane electromagnetic wave. So, it is along ~ =E ~ 0 ei(~k.~r−ωt) . the direction of propagation and obtained by comparing with E S0.7: For an engine working between two reservers, the entropy will be maximum when work done is zero. This implies the two reservers are directly connected and final 2) temperature will be (T1 +T . 2 Following is the detailed calculation: If Tf is the final temperature and W amount of work is obtained then T1 W = Q1 − Q2 = C(T1 − Tf ) − C(Tf − T2 ) where C is the heat capacity T1 + T2 W =0 = C(T1 + T2 − 2Tf ) =⇒ Tf = 2 p p p p W is max 2 = C[( T1 − T2 ) − 2(Tf − T1 T2 )] =⇒ Tf = T1 T2

Q1 W Q2 T2


118

My Physics Tutor T1 + T2 2 } | {z

So, the final temperature lies in the range

arithmetic mean

The change in entropy of the system,

∆S =

≥ Tf ≥

ZTf

CdT + T

geometric mean

CdT T

T2

T1

= Cln

ZTf

p T T | {z1 }2

Tf2 T1 T2

T1 +T2 Obviously, the change in entropy, ∆S √ is maximum for Tf = 2 i.e. when no work is done (and ∆S = 0 for Tf = T1 T2 i.e. when the work done is maximum).

S0.8: The entropy in the temperature limit T → ∞ will be NkB ln(1 + g) i.e. just NkB times the logarithm of total number of microstates. For a system with finite number of states which are low in energy, when the thermal energy is appreciably high all the states can be considered as degenerate states (as the energy difference between them are negligibly small with respect to kB T ) hence expected to be occupied with equal probability. One may obtain the same result throughPformal calculation as follows: The partition function defined as Z = gi e−i β where i runs over all the states i

and β = 1/kB T , for this system: Z = e−1 β + ge−2 β = e−1 β (1 + ge−β )

where = 2 − 1 . One can choose the energy reference with 1 = 0 to make the calculations simple. The Helmhotz free energy, F = −kB T lnZ = 1 − kB T ln(1 + ge−β )

∂F and the entropy, S = − ∂T

N, V

1 ge−β T (1 + ge−β ) 0 1 g T →∞ −→ kB ln(1 + g) + β T e + g

= kB ln(1 + ge−β ) +

Remark 13.0.3 Consider the simplest case when g = 1 i.e. a system with only two states. It is interesting to note that at very high temperature the population of two states will be equal. In other words, one can not make an equilibrium situation with the excited state having higher population even at infinite temperature! That means, a situation with higher population in excited state corresponds to a temperature more than infinity what is called the negative temperature (in absolute scale)! In thermodynamics, temperature is defined by the relation: T1 = ∂U ∂S N, V implying that temperature goes negative when the Bhramar Chatterjee, Sirshendu Gayen

Chapter 13. Solution to introductory problems

119

slope of U − S curve is negative. Another point to mention here is that the concept of states at high enough temperature may be overlooked and a system can be described classically.

S0.9: Its 1/2 because obtaining |+iz from either of |+ix or |−ix (whatever comes out after first measurement) is half as |±ix = √12 (|+iz ± |−iz ). One may obtain the same result considering the detail sequential measurements as follows: P=1/2

In the first measurement, the probability of obtaining |+ix from |+iz is 12 . In the second measurement, the probability of obtaining |+iz from |+ix is 12 . So, the probability of obtaining |+iz starting from initial state |+iz via |+ix is 12 × 21 = 14 .

+xS

P=1/2

+z

Z

+z

measurement

-z

SX measurement

P=1/2 P=1/2

-x

SZ measurement

+z -z

In the same way, the probability of obtaining final state |+iz starting from initial state |+iz via |−ix is 12 × 21 = 14 . So, the total probability is 14 + 14 = 12 S0.10: The answer is 2bcosθ = λ. In the well-known Braggs’ law 2dsinΘ = nλ, the inter-planar spacing d is measured along the direction of momentum transfer ~q = k~f − k~i and 2Θ is the angle of deviation of the incident beam from its initial direction of propagation (notice the difference of defining Θ in Braggs’ law with what is given in this problem!).

q = kf - ki kf

ki

2ϴ = angle of deviation


120


My Physics Tutor

Recommended books

Introductory books: 1. H. C. Verma, Concepts of physics (vols. I & II), Bharati Bhawan, 1999. 2. D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, John Wiley, 2011. 3. A. Beiser, Concepts of Modern Physics, McGraw-Hill, 2003. 4. R. P. Feynman, Lectures in Physics (vols. I-VII), Addison-Wesley, 1970. 5. R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, John Wiley, 1985. Problems & Solutions: 6. Yung-Kuo Lim (ed.), Problems and Solutions (vols. I-VII), World Scientific, 1998. 7. I. E. Irodov, Problems in General Physics, Mir Publishers, 1981. 8. I. E. Irodov, Fundamental Laws of Mechanics, Mir Publishers, 1980. 9. I. E. Irodov, Basic Laws of Electromagnetism, Mir Publishers, 1986. 10. I. E. Irodov, Problems in Atomic and Nuclear physics, Mir Publishers, 1983. 11. B. S. Belikov, General Methods for Solving Physics Problems, Mir Publishers, 1989. 12. A. A. Pinsky, Problems in Physics, Mir Publishers, 1980. 13. L. A. Sena, A collection of Questions and Problems in Physics, Mir Publishers, 1988. 14. S. S. Krotov, Aptitude Test Problems in Physics, Mir Publishers, 1990. 121

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15. M. P. Shaskol’skaya, I. A. El’tsin, Selected Problems in Physics with Answers, Pergamon Press, 1963. 16. B. Bukhovtsev, V. Krivchenkov, G. Myakishev, V. Shalnov, Problems in Elementary Physics, Mir Publishers, 1978. 17. G. L. Kotkin, V. G. Serbo, Collection of Problems in Classical Mechanics, Pergamon Press, 1971. 18. M. Chen, University of California, Berkeley, Physics Problems, with Solutions, Prentice-Hall, 1974. 19. W. G. Rees, Physics by Example: 200 Problems and Solutions, Cambridge University Press, 1994. 20. L. Holics, 300 Creative Physics Problems with Solutions, Anthem Press, 2010. 21. S. B. Cahn, G. D. Mahan, B. E. Nadgorny, A Guide to Physics Problems (vols. I-II), Kluwer Academic Publishers, 2007. 22. N. Newbury, M. Newman, J. Ruhl, S. Staggs, S. Thorsett, Princeton Problems in Physics, Princeton University Press, 1991. 23. A. D. Giacomo, G. Paffuti, P. Rossi, Selected problems in Theoretical physics, World Scientific, 1994. 24. P. Gnädig, G. Honyek, K. F. Riley, 200 Puzzling Physics Problems, Cambridge University Press, 2001. 25. P. P. Dendy, R. Tuffnell, C. H. B. Mee (eds.), Cambridge Problems in Physics, Cambridge University Press, 1991. 26. J. A. Cronin, D. F. Greenberg, V. L. Telegdi, University of Chicago Graduate Problems in Physics, Addison-Wesley, 1967. 27. O. L. de Lange, J. Pierrus, Solved Problems in Classical Mechanics, Oxford University Press, 2010. 28. R. V. Giles, J. B. Evett, C. Liu, Fluid Mechanics and Hydraulics (Schaum’S Outlines), McGraw-Hill, 2014. 29. E. Mendelson, 3,000 Solved Problems in Calculus (Schaum’S Outlines), McGrawHill, 1988. 30. M. R. Spiegel, J. Schiller, R. A. Srinivasan, Probability and Statistics (Schaum’S Outlines), McGraw-Hill, 2011. 31. W.-H. Steeb, Problems in Theoretical Physics: Introductory problems, BI-Wiss.Verlag, 1990. 32. J-M. D. Konink, A. Mercier, 1001 Problems in Classical Number Theory, AMS, 2007. Bhramar Chatterjee, Sirshendu Gayen

Recommended books

123

33. E. J. Barbeau, M. S. Klamkin, W. O. J. Moser, Five Hundred Mathematical Challenges, MAA, 1995. 34. W. Sierpinski, 250 Problems in Elementary Number. Theory, Elsevier, 1970. 35. V. V. Batygin, I. N. Toptygin, Problems in electrodynamics, Academic Press, 1964. 36. M. Rousseau, J. P. Mathieu, Problems in Optics, Pergamon Press, 1973. 37. M. Cini, F. Fucito, M. Sbragaglia, Solved Problems in Quantum and Statistical Mechanics, Springer, 2012. 38. D. A. R Dalvit, J. Frastai, I. Lawrie, Problems on Statistical Mechanics, CRC Press, 1999. 39. V. Galitski, B. Karnakov, V. Kogan, V. Galitski, Jr. Exploring Quantum Mechanics, Oxford University Press, 2013. 40. E. d’Emilio, L. E. Picasso, Problems in Quantum Mechanics, Springer, 2011. 41. K. Tamvakis, Problems & Solutions in Quantum Mechanics, Cambridge University Press, 2005. 42. F. Constantinescu, E. Magyari, Problems in Quantum Mechanics, Pergamon Press, 1971. 43. S. Fl¨ ugge, Practical Quantum Mechanics, Springer-Verlag, 1971. 44. I. I. Gol‘dman, V. D. Krivchenkov, Problems in Quantum Mechanics, Dover, 1993. 45. Y. Peleg, R. Pnini, E. Zaarur, E. Hecht, Quantum Mechanics (Schaum’S Outlines), McGraw-Hill, 2010. 46. A. Z. Capri, Problems and Solutions in Non-relativistic Quantum Mechanics, World Scientific, 2002. 47. D. ter Haar, Problems in Quantum Mechanics, Dover, 2014. 48. G. L. Squires, Problems in Quantum Mechanics, Cambridge University Press, 1995. 49. H. Mavromatis, Exercises in Quantum Mechanics, Springer-Science+Business Media, 1992. 50. L. Mihály, M. C. Martin, Solid State Physics, Wiley, 2009. 51. J. Cazaux, Understanding Solid State Physics, CRC Press, 2016. 52. A. Rigamonti, P. Carretta, Structure of Matter, Springer, 2007. Mathematical Method: 53. M. L. Boas, Mathematical Methods in the Physical Sciences, Wiley, 2007. Bhramar Chatterjee, Sirshendu Gayen

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54. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, Academic Press, 2005. 55. I. N. Bronshtein, K. A. Semendyayev, G. Musiol, H. M¨ uhlig, Handbook of Mathematics, Springer, 2007. 56. M. R. Spiegel, S. Lipschutz, D. Spellman, Vector Analysis (Schaum’S Outlines), McGraw-Hill, 2009. 57. M. R. Spiegel, S. Lipschutz, J. J. schiller, D. Spellman, Complex Variables (Schaum’S Outlines), McGraw-Hill, 2009. 58. M. J. Ablowitz, A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, 2003. 59. B. S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 2014. 60. K. T. Tang, Mathematical Methods for Engineers and Scientists 1-3, SpringerVerlag, 2007. 61. E. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, 2003. 62. D. G. Zill, W. S. Wright, M. R. Cullen, Differential Equations with Boundary-Value Problems, Brooks/Cole, Cengage Learning, 2013. Classical Mechanics: 63. M. R. Spiegel, Theoretical Mechanics (Schaum’S Outlines), McGraw-Hill, 2006. 64. H. Goldstein, C. Poole, J. Safko Classical Mechanics, Pearson, 2013. 65. S. N. Biswas, Classical Mechanics, New Central Book Agency, 2000. 66. R. D. Gregory, Classical Mechanics, Cambridge University Press, 2007. 67. D. Morin, Introduction to Classical Mechanics: With Problems and Solutions, Cambridge University Press, 2008. 68. S. T. Thornton, J. B. Marion, Classical Dynamics of Particles and Systems, Cengage Learning, 2004. 69. L. D. Landau, E. M. Lifshitz, Classical Mechanics, Elsevier Butterworth-Heinemann, 2005. 70. D. Kleppner, R. Kolenkow, An Introduction to Mechanics, Cambridge University Press, 2014. 71. A. P. French, M. G. Ebison, Introduction to Classical Mechanics, Kluwer Academic Publishers, 1986. 72. A. P. French, Vibrations and Waves, CRC Press, 1971. Bhramar Chatterjee, Sirshendu Gayen

Recommended books

125

73. R. M. Dreizler, C. S. L¨ udde, Theoretical Mechanics, Graduate Texts in Physics, Springer-Verlag, 2011. Electromagnetic Theory: 74. D. J. Griffiths, Introduction to Electrodynamics, Prentice Hall, 2003. 75. J. D. Jackson, Classical Electrodynamics, Wiley, 1999. 76. A. Das, Lectures On Electromagnetism, World Scientific, 2013. 77. W. Nolting, Theoretical Physics 3: Electrodynamics, Springer, 2016. 78. L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, 2003. Optics 79. A. Ghatak, Optics, Tata McGraw-Hill, 2009. 80. E. Hecht, Optics, Addision-Wesley, 2016. 81. A. Lipson, S. G. Lipson, H. Lipson, Optical Physics, Cambridge University Press, 2011. 82. I. R. Kenyon, The Light Fantastic: A Modern Introduction to Classical and Quantum Optics, Oxford University Press, 2008. Electronics & Experimental Methods: 83. A. P. Malvino, Electronic Principles, McGraw-Hill, 1998. 84. R. L. Boylestad, L. Nashelsky, Electronic Devices and Circuit Theory, Pearson Education, 2013. 85. T. F. Bogart, J. S. Beasley, G. Rico, Electronic Devices and Circuits, Prentice Hall, 2001. 86. J. Millman, C. C. Halkias, Integradted Electronics, Tata Mc-Graw Hill, 1991. 87. S. M. Sze, Semiconductor Devices: Physics and Technology, John Wiley, 2012. 88. P. Horowitz, W. Hill, The Art of Electronics, Cambridge University Press, 2015. Thermodynamics & Statistical Physics: 89. H. B. Callen, Thermodynamics and an Introduction to Thermostatics, Wiley, 1985. 90. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965. 91. C. Borgnakke, R. E. Sonntag, Fundamentals of Thermodynamics, Wiley, 2016. Bhramar Chatterjee, Sirshendu Gayen

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92. D. J. Amit, Y. Verbin, Statistical Physics, World Scientific, 1999. 93. R. K. Pathria, Statistical Mechanics, Butterworth-Heinemann, 1996. 94. K. Huang, An introduction to Statistical Physics, Taylor and Francis, 2001. 95. S. Blundell, Concepts in Thermal Physics, Oxford University Press, 2010. 96. P. B. Pal, An Introductory Course of Statistical Mechanics, Narosa, 2008. 97. H. Gould, J. Tobochnik, Statistical and Thermal Physics, Princeton University Press, 2010. 98. M. Kardar, Statistical Physics of Particles, Cambridge University Press, 2007. 99. M. Plischke, B. Bergersen, Equilibrium Statistical Physics, Prentice-Hall, 1989. 100. L. D. Landau, E. M. Lifshitz, Statistical Physics Part-1, Elsevier ButterworthHeinemann, 2005. 101. M. Toda, R. Kubo, N. Saitô, Statistical Physics I: Equilibrium Statistical Mechanics, Springer-Verlag, 1983. 102. M. Toda, R. Kubo, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, Springer, 1998. Quantum Mechanics: 103. J. J. Sakurai. Modern Quantum Mechanics, Addison-Wesley, 1994. 104. C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, (vols. I & II), Wiley, 1977. 105. D. J. Griffiths, Introduction to Quantum Mechanics, Pearson Prentice Hall, 1995. 106. N. Zettili, Quantum Mechanics: Concepts & Applications, Wiley, 2009. 107. E. S. Abers, Quantum Mechanics, Prentice Hall, 2004. 108. B. H. Bransden, C. J. Joachain, Quantum Physics, Pearson Prentice Hall, 2000. 109. S. Gasiorowitcz, Quantum Physics, Wiley, 1996. 110. R. Shankar, Principles of Quantum Physics, Springer, 1994. 111. B. R. Desai, Quantum Physics with Basic Field Theory, Cambridge University Press, 2010. 112. L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Elsevier Butterworth-Heinemann, 2005. 113. J. L. Basdevant, J. Dalibard, Quantum Mechanics (& solver), Springer, 2006. Bhramar Chatterjee, Sirshendu Gayen

Recommended books

127

114. J. R. Boccio, Quantum Mechanics: Mathematical Structure and Physical Structure, Part-1 & 2, 2014. 115. A. B. Migdal, Qualitative Methods in Quantum Theory, Addison-Wesley, 1989. Atomic & Molecular Physics: 116. H. E. White, Introduction to Atomic Spectra, McGraw Hill Kogakusha, 1999. 117. B. H. Bransden, C. J. Joachain, Physics of Atoms and Molecules, Pearson Education, 2003. 118. C. N. Banwell, E. M. McCash, Fundamentals of Molecular Spectroscopy, Tata McGraw Hill, 1994. 119. H. Haken, H. C. Wolf, Physics of Atoms and Quanta, Springer-Verlag, 1993. Condensed Matter Physics: 120. N. W. Ashcroft, N. D. Mermin, Solid State Physics, Saunders College, 1976. 121. G. D. Mahan, Condensed Matter in a Nutshell, Princeton University Press, 2011 122. C. Kittel, Introduction to Solid State Physics, Wiley, 2008. 123. M. P. Marder, Condensed Matter Physics, Wiley, 2000. 124. S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2014. 125. J. Sólyom, Fundamentals of the Physics of Solids (vols. I-III), Springer, 2010. 126. J. R. Hook, H. E. Hall, Solid State Physics, Wiley, 2010. 127. S. Blundell, Magnetism in Condensed Matter, Oxford University Press, 2001. 128. M. Tinkham, Introduction to Superconductivity, McGraw-Hill, 2004. 129. A. C. Rose-Innes, E. H. Rhoderick, Introduction to Superconductivity, Pergamon Press, 1994. 130. J. F. Annett, Superconductivity, Superfluids and Condensates, Oxford University Press, 2004. 131. C. P. Poole, R. Prozorov, H. A. Farach, R. J. Creswick, Superconductivity, Elsevier, 2014. 132. C. Timm, Theory of Superconductivity, TU Dresden, 2011-12. Nuclear and Particle Physics: 133. K. S. Krane, Introductory Nuclear Physics, John Wiley, 1988. Bhramar Chatterjee, Sirshendu Gayen

128

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134. C. A. Bertulani, Nuclear Physics in a Nutshell, Princeton University Press, 2007. 135. W. E. Burcham, M. Jobes, Nuclear and Particle Physics, Pearson, 1995. 136. W. Greiner, J. A. Maruhn, Nuclear Models, Springer, 1996. 137. J. S. Lilley, Nuclear Physics: Principles and Applications, Wiley, 2001. 138. G. F. Knoll, Radiation Detection and Measurement, Wiley, 2010. 139. W. N. Cottingham, D. A. Greenwood, An Introduction to Nuclear Physics, Cambridge University Press, 2001. 140. D. Griffiths, Introduction to Elementary Particles, Wiley-VCH, 2004. 141. F. Halzen, A. D. Martin, Quarks & Leptons, Wiley, 1984. 142. D. H. Perkins, Introduction to High Energy Physics, Cambridge University Press, 2000. Special Theory of Relativity: 143. E. F. Taylor, J. A. Wheeler, Spacetime Physics, W. H. Freeman, 1992. 144. R. Resnick, Introduction to Special Relativity, Wiley, 1968.


Index

matrix, 99 of states, 97, 98, 105, 106, 108 deuteron, 112 dipolar interaction, 113 dipole moment, 7, 10, 49 dispersion relation, 11, 44, 54, 91, 92, 97, 104–107 displacement current, 53 double slit experiment, 69, 70, 73 Drude model, 107

δ-function, 6, 7, 23, 24, 45, 49 state excited, 86 Aharonov-Bohm effect, 65 annihilation/creation operator, 75, 78 band-gap, 64 baryon, 112 Bloch oscillation, 106 Bohr orbit, 10, 85 Boltzmann distribution, 97 Bose condensation, 92 Braggs’ law, 119 Brillouin zone/boundary, 102, 105 broken symmetry, 11, 80, 101, 102, 109

effective mass, 105–107 elastic collision, 1, 28, 29, 103 limit, 41 scattering, 101, 111 electric flux, 7, 46 entropy, 2, 8, 89, 91, 92, 94, 98 equation of state, 98 Dieterici’s, 90 Van der Waal’s, 90 exchange interaction, 108

Carnot engines, 90 Cerenkov radiation, 111 chemical potential, 89, 94, 99 Compton wavelength, 111 critical angle, 70 point, 90 crystal momentum, 104, 105, 108 cycloid, 16, 27, 28, 38

Fermi energy, 11, 93, 95, 97, 103, 104, 106 Fourier transform, 1, 23, 97 Fourier transformation, 115 Fraunhofer diffraction, 69, 71, 72 Fresnel diffraction, 69 full width at half maxima (FWHM), 22

de Broglie wavelength, 10, 103 degenerate Fermi gas, 95 density 129

130

My Physics Tutor

gauge transformation/symmetry, 10, 54, 80, 109 Gaussian function, 22 generating function, 24 group velocity, 44, 54, 104, 106 hadron, 113 half-life, 112 Hamiltonian, 9, 10, 77, 78, 80–83, 91, 95–97, 107, 108 heat capacity, 2, 8, 89, 91, 107 helicity, 79, 81 Hermite-Gauss function, 1, 23, 115 Hermitian matrix, 18 operator, 77, 79 Hohmann transfer, 35 hydrogen atom, 9, 10, 76, 77, 85, 86 impedance spectroscopy, 60 intensive variable, 89 isospin, 112 Koch snowflake, 26 Kronecker-δ, 19 Lagrangian, 7 Laguerre polynomial, 23 Landau theory, 11 Laplace transformation, 97 Laplace-Runge-Lenz vector, 34 Legendre polynomial, 1, 23, 24 Lenoir cycle, 90 lepton, 112 Lissajous figure, 61 Lorentzian, 1 magnetic dipole moment, 49 magnons, 93, 107, 108 mechanical momentum, 10 microstate, 93, 94, 96, 107 moment of inertia, 11, 38, 79 momentum transfer, 119 non-singular matrix, 115 Nyquist plot, 60 ocean wave, 44 oscillator, 100 Bhramar Chatterjee, Sirshendu Gayen

anharmonic, 91 charged harmonic, 78 dumbbell, 38 simple harmonic, 9, 75, 78, 81, 93, 107 parity, 9, 10, 23, 75, 80, 112 Peltier Effect, 65 pendulum, 36, 38 circular , 39 simple, 37 perturbation, 9 phase transition, 11 phase velocity, 54 phonon, 106–108, 112, 113 Poynting vector, 2, 117 quantum number, 76 azimuthal, 76 principal, 85 rotational, 86 quasi-particle, 107 random walk, 6, 20, 21 resolving power, 69 Reynolds number, 43 rotor planar, 79 spherical, 79 Rutherford scattering, 111 self-inductance, 51 semi-classical approximation, 76 simple harmonic, 36, 37, 39, 69, 78, 86, 100 specific heat, 8, 89, 91–94, 96, 97, 100, 104, 106–108 spherical capacitor, 49 spherical harmonics, 77 standard deviation, 20, 21 Stark effect, 10 state antisymmetric, 9 bound, 76, 77, 80 degenerate, 2, 78, 80, 82, 96, 98 eigen, 8–10, 75–83, 86, 98, 99, 104 excited, 2, 9, 76, 82, 85, 94–96

Index ground, 2, 9, 10, 75, 76, 78–82, 85–87, 91, 94–97 initial, 79 non-degenerate, 2, 8, 75, 80, 96, 99 ortho-normal, 77 rotational, 86 singlet, 107 spin, 79, 81 symmetric, 9 triplet, 107 vibrational, 11 Stern-Gerlach experiment/apparatus, 76, 81, 87 stopping potential, 86 sudden approximation, 9 superconductivity, 80, 109

131 tensor, 19, 55 terminal speed, 51 tight-binding, 105 time-reversal, 79, 80 torus, 25 triangular lattice, 21 number, 24 pulse, 64 uncertainty, 77, 112 vorticity, 25 WKB approximation, 77 Zeeman effect, 79, 80, 87, 109


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