Question Bank For Part I (Hons)

Transcription

1 Question Bank For Part I (Hons) Paper I Unit - I MATHEMATICAL METHODS I (25 Marks) 1. What are absolute convergence and conditional convergence? (1 + 1) 2. (a) Show that the exponential series : 1+ x + x 2 /2! + x 3 /3! + is absolutely convergent for all finite values of x. (3) (b) What is the condition for absolute convergence of the series : 1+ x + x 2 /2 + x 3 /3 +? (3) 3. Expand 1/(x 2) about x =1 in Taylor series. What is the radius of absolute convergence of the series? (3 + 3) 4. In a certain month, the probability of having a thunder storm on a day is 30%. What is the probability of having exactly 3 such storms, (ii) at least 3 such storms in a week? (3 + 3) 5. A Gaussian distribution function is given by f(x) = N exp { (x a) 2 /b 2 }. Calculate the normalization constant N, the mean value of x and the variance. Plot f(x) vs. x for two different values of b but a fixed value of a, on the same graph. ( ) 6. Prove that the mean and the variance have equal values for a poisson distribution. (5) 7. Prove that : (A + B) C + (B + C) A + (C + A) B = 0. (3) 8. Prove that : (A B) C + (B C) A + (C A) B = 0. (3) 9. Calculate the shortest distance of the straight line y = 2x + 3 from the origin, by vector method. 10. Prove by vector method, that the line joining the mid points of two sides of a triangle is parallel to the third side and half of it in length. 11. Prove that the vector is normal to the surface (x, y, z) = constant. Find out the unit normal to the surface : z 3 = x 2 y + y 2 x at the point (1, 1, 4). (3 + 3) 12. (i) Prove that :, (ii). (3 + 3) 13. Show that if a vector field is both solenoidal and irrotational, then it must satisfy Laplace s equation. (3) 14. (a) Calculate the divergence of r/r n, for r 0, where r r. (3) (b) Calculate the curl of the vector v = r, where is a constant vector. (3) 15. State Gauss divergence theorem. Prove that : ⅓ S r ds = V, where S is the surface enclosing the volume V. (2 + 2) 16. Prove that : (i) dv = S A ds, where S is the surface enclosing the volume V. (3) (ii) dv = S ds, where S is the surface enclosing the volume V. (3) 17. Define a Symmetric, a Skew-symmentric, a Hermitian, a Skew-Hermitian an orthogonal and a unitary matrix. (6) 18. Prove that the eigenvalues of a Hermitian matrix are all real and its eigen vectors corresponding to different eigenvalues are orthogonal. (3 + 3) 19. Prove that the eigenvalues ( ) of a unitary matrix are uni-modular, i.e., =1. (3) 20. Find the eigenvalues and eigenvectors of the matrix : (2 + 4)

2 0 i i 0 MATHEMATICAL METHODS II (25 Marks) 1. Solve this equation : d 2 x/dt 2 + x = 0, by Frobenius method (about x = 0). 2. Solve this equation : (1 x 2 ) d 2 y/dx 2 2x dy/dx + y = 0, by Frobenius method (about x = 0) : find the indicial equations, the recursion relation and finally, two linearly independent series solutions (first three terms of each). For what sort of values of l can we have a polynomial solution of this equation? ( ) 3. Solve this equation : d 2 y/dx 2 2x dy/dx + ny = 0, by Frobenius method (about x = 0) : find the indicial equations, the recursion relation and finally, two linearly independent series solutions (first three terms of each). For what sort of values of l can we have a polynomial solution of this equation? ( ) 4. Establish the orthogonality property of the Legendre polynomials. (4) 5. Establish the orthogonality property of the Hermite polynomials. (4) 6. Solve the equation : 2 = 0 in two dimensions, by the separation of variables method and obtain the general solution satisfying the conditions : = 0 for x = 0, x = a, and y = 0. (7) 7. Write down Laplace s equation in spherical polar coordinates in three dimensions and separate the variables to obtain three ordinary differential equatios. (6) 8. Write down Laplace s equation in spherical polar coordinates in three dimensions and separate the variables to obtain three ordinary differential equatios. (6) 9. Prove that : 0 2 sin mx sin nx dx = 0 2 sin mx sin nx dx = mn. 10. f(x) = 0, for x < 0, f(x) = x, for 0 x and f(x + 2 ) = f(x) otherwise. Expand f(x) in a Fourier Series. What will be the value of f( ) if calculated from the series. (7 + 1) 11. Expand the following functions in Fourier series : (i) f(x) = x, for x and f(x + 2 ) = f(x) otherwise. (6) (ii) f(x) = x, for x and f(x + 2 ) = f(x) otherwise. What will be the value of f( ) if calculated from the series. (4 + 1) (iii) f(x) = x 2, for x and f(x + 2 ) = f(x) otherwise. (6) (iv) f(t) = sin t, for 0 t <, f(t) = 0, for t 2 and f(t + 2 ) = f(t) otherwise. (6) (v) f(t) = sin t, for 0 t <, and f(t + ) = f(t) otherwise. (5)

3 Unit-II WAVES & OPTICS I (25 Marks) 1. What is called simple harmonic motion? Calculate the kinetic, potential and hence the total energy of a particles oscillating simple harmonically. ( ) 2. (i) Two simple harmonic motions of same frequency are along the same directions. Is the resultant motion simple harmonic? (3) (ii) What is called beats? How is beat formed? Two simple harmonic vibrations of different frequencies are superimposed. Is the resultant vibration simple harmonic? (4) 3. A particle is subjected to two simple harmonic motions at right angles to each other, of same frequency. Show that the resultant locus of the particle is an ellipse. What will be the locus when the two motions are in the same phase, opposite phase and out of phase by 90 0? ( ) 4. A simple harmonic oscillator is moving along X-axis under the action of a damping force proportional to its velocity. Construct the differential equation describing the motion of the oscillator. Solve the differential equation for under damped, over damped and critically damped cases. ( ) 5. What is forced vibration? What are the differences between free and forced vibrations? A particle in simple harmonic motion is subject to a damping force proportional to its velocity and an external harmonic force. Construct the differential equation of the particle. Solve the differential equation to get the position and velocity of the particle. ( ) 6. What are amplitude resonance and velocity resonance? At what frequencies of the applied force the velocity and amplitude resonance occur? ( ) 7. Show that : for a forced vibrating particle the average instantaneous rate of work done by the applied force in steady state is equal to the average power spent in overcoming the resisting force. (4) 8. What is power factor? What is the value of power factor at resonance? (1 + 2) 9. What do you mean by sharpness of resonance? What is quality factor? (2 + 2) 10. What is coupled vibration? What are eigenfrequencies and normal modes? ( ) 11. What are differences between vibrations and wave motion? What do you mean by longitudinal and transverse waves? What are the characteristics of plane progressive harmonic waves? ( ) 12. The equation of a plane progressive harmonic wave is given by y(x,t) ASin( t ) c where the symbols have their usual significance. Construct the differential equation of wave motion. Show that the excess pressure y p K satisfies x the same differential equation. (3 + 2) 13. A plane progressive wave is passing through a fluid medium of bulk modulus K and density ρ, Show that velocity of the wave is given by K/. Write down the assumptions made while deriving the expression. (6 + 2) 14. Prove that the velocity of sound waves in a solid bar is Y /, where Y is the Young s modulus and ρ is the density of the material of the bar. (6)

4 15. (a) For a simple harmonic progressive wave, calculate the instantaneous kinetic energy and potential energy. Show that time average of kinetic and potential energy over a complete period is equal. ( ) (b) How do you define intensity of a sound wave? What are bel and phon? ( ) 16. What are group velocity and phase velocity? What is the relation between group and phase velocity? ( ) 17. What is Doppler effect? A source of sound of frequency n is moving toward/away from a stationary obser with velocity v s, what will be the apparent frequency to the observer? An observer is moving toward/away from a source of sound of frequency n with velocity v o, what will be apparent frequency to the observer? ( ) 18. State Fermat s principle of least time. How does it reduce to a principle of least path? (2 + 1) 19. (a) Starting from Fermat s principle, prove that the angle of incidence equals the angle of reflection in case of reflection on a plane mirror. (3) (b) Starting from Fermat s principle, obtain Snell s law for refraction at a plane surface. (4) 20. (a) Starting from Fermat s principle, derive the relation between object and image distance, for reflection at a concave spherical surface. Also obtain the second law of reflection (law of angles). (5 + 3) (b) Starting from Fermat s principle, derive the relation between object and image distance, for reflection at a convex spherical surface. Also obtain the second law of reflection (law of angles). (5 + 3) 21. (a) Starting from Fermat s principle, derive the relation between object and image distance, for refraction at a concave spherical surface. Also obtain the Snail s law for this case. (5 + 3) (b) Starting from Fermat s principle, derive the relation between object and image distance, for refraction at a convex spherical surface. Also obtain the Snail s law for this case. (5 + 3) 22. (a) What is an equivalent lens? Derive an expression for the equivalent focal length for two thin lenses in contact. (1 + 4) (b) Is it possible to achieve complete equivalence in case of two thin lenses separated by a distance? Derive, in this case, the expression for focal length of the equivalent lens, that produces the same magnification for the same object distance. Find also the position of the image in this case. ( ) 23. What are Cardinal points for an optical system. Define each pair of them. (1 + 3) 24. Define linear and angular magnification. State Langrange-Helmholtz law. ( ) 25. Derive expressions for the translation and the refraction matrix for a paraxial ray. (3 + 3) 26. (a) Derive the relation between object and image distance, for a thin lens, using matrix method. (4) (b) Derive the relation between object and image distance, for a system of two thin lenses, separated by a distance, using matrix method. (5) (c) Derive an expression for the system matrix for a sphere of radius R, made of glass (refractive index = 1.5). (4) 26. (a) State and explain Huygen s principle. (3) (b) Obtain the second law of reflection (law of angles), using Huygen s construction. (4)

5 (c) Obtain the Snell s law of refraction, using Huygen s construction. (4) ELECTRONICS I (25 Marks) 1. What are semiconductors? (1) 2. Differentiate between intrinsic and extrinsic semiconductor. (2) 3. Define drift current? (2) 4. Give the expression for drift current density. (3) 5. Define the term diffusion current? (2) 6. Give the expression for diffusion current density. (2) 7. Differentiate between drift and diffusion currents. (2) 8. What is depletion region in PN junction? (2) 9. What is barrier potential? (2) 10. What is reverse saturation current? (2) 11. What is the total current at the junction of pn junction diode? (2) 12. Give the diode current equation? (2) 13. What is recovery time? Give its types. (3) 14. Define storage time. (2) 15. Define transition time. (2) 16. Define PIV. (2) 17. Draw V-I characteristics of p-n diode. (3) 18. Write the application of p-n diode. (2) 19. What is a metal semiconductor contact? (2) 20. Define contact potential in metal semiconductor contact. (2) 21. Give the symbol and structure of schottky diode. (2) 22. Give the applications of schottky diode. (2) 23. Compare between schottky diode, and conventional P-N junction diode. (2) 24. Why zener diode is often preferred than PN diode. (2) 25. Draw the V-I characteristics curve for zener diode. (2) 26. What is zener breakdown? (2) 27. What is avalanche breakdown? (2) 28. What is tunneling phenomenon? (2) 29. Give the application of tunnel diode. (2) 30. Give the advantages and disadvantages of tunnel diode. (2) 31. Draw equivalent circuit of tunnel diode. (2) 32. What is varactor diode? (2) 33. Why an ordinary transistor is called bipolar? (2) 34. Collector region of transistor is larger than emitter. Why? (2) 35. Why is BJT is called current controlled device? (2) 36. Define Early Effect. (2) 37. Draw the characteristics of CE configuration. (4) 38. Among CE, CB, CC which one is most popular. Why? (2) 39. Compare CE, CB and CC configurations. (3) 40. Why h parameter model is important for BJT. (4)

6 41. Define current amplification factor. (2) 42. What do you meant by multi emitter transistor. (2) 43. In a CR connection, the value of IE is 6.28 ma and the collector current Ic is 6.20 ma. Determine d.c. current gain. (3) 44. The transistor has I E = 10 ma and α = Find the value of base and collector currents. (3) 45. If a transistor has a α of 0.97 find the value of β. If β = 200, find the value of α. (3) 46. Give some applications of BJT. (2) 47. Why it is called field effect transistor? (2) 48. Why FET is called voltage controlled device. (2) 49. Define the term threshold voltage. (2) 50. What is channel length modulation? (2) 51. Compare JFET with BJT. (2) 52. Draw the transfer characteristics curve for JFET. (2) 53. Differentiate between N and P channel FETs. (2) 54. Write some applications for JFET. (2) 55. Compare MOSFET with JFET. (2) 56. Compare N channel MOSFET with P channel MOSFET. (2) 57. Differentiate between current voltage relationships of the N channel and P channel MOSFET. (2) 58. Draw the V-I characteristics curve of MOSFET. (2) Paper IIA Unit-I CLASSICAL MECHANICS I (25 Marks) 1. (a) Find out the radial and cross-radial (trans-radial) components of velocity and acceleration of a particle moving in a plane, in plane polar coordinate system. (3+3) (b) Prove that the acceleration of a particle in uniform circular motion in a plane is directed towards the centre with a magnitude v 2 /r, where v is the linear speed. What happens if the circular motion is not uniform? (3 + 2) 2. Find out the components of velocity and acceleration of a particle in cylindrical polar coordinate system. (3+3) 3. (a) Find out the components of velocity of a particle in spherical polar coordinate system. Hence derive an expression for its kinetic energy. (5+2) (b) Find also the components of acceleration of the particle. (3) 4. A particle is moving in a plane under a force field : F = k(x 2 y i + xy 2 j) Calculate the work done as the particle describes a square (0,0) (1,0) (1,1) (0,1) (0,0). (4) 5. (a) A particle is moving in a plane under a force field : F = k(x j y i)/( x 2 + y 2 ). Calculate the work done as the particle describes a unit circle about the origin in the anticlockwise direction. (4) (b) Calculate F. Is the result consistent with Stoke s theorem? Comment. ( ) 6. (a) A particle is moving in a plane under a force field : F = k(x j y i)/ ( x 2 + y 2 ). Calculate the work done as the particle describes a unit circle about the origin in the anticlockwise direction. (4) (b) Calculate F. Check consistency with Stoke s theorem. (3 + 2)

7 7. Show that the total linear momentum of a system of particles equals that of a particle having a mass equal to the total mass of the system and moving with the centre of mass of the system. (2) 8. Show that the rate of change of total linear momentum of a system of particles equals the total external force acting on the system. (3) 9. (i) Show that the rate of change of angular momentum of a single particle equals the torque acting on it. (2) (ii) Show that under suitable condition, the rate of change of total angular momentum of a system of particles equals the total external torque acting on the system. What is the restriction on the nature of the internal forces for the result to hold good? (4 + 1) 10. Show that the total angular momentum of a system of particles equals the angular momentum due to the motion of the center of mass and that due to the motion about the center of mass. (4) 11. Show that the total kinetic energy of a system of particles equals the kinetic energy due to the motion of the center of mass and that due to the motion about the center of mass. (4) 12. Define rigid body. Prove that, for a rigid body, L = I, where L is the angular momentum and I is the moment of inertia matrix. (2 + 5) 13. Prove that, the kinetic energy of a rigid body is given by : E = ½ I, where I is the moment of inertia matrix and is the angular velocity vector. (5) 14. (a) Show that, the moment of inertia of a rigid body about any axis is given by : I = I xx l 2 + I yy m 2 + I zz n 2 + 2I xy lm + 2I yz mn + 2I xz ln, where I is the moment of inertia matrix and (l, m, n) are the direction cosines of the axis. (5) (b) Prove that : if the moment of inertia of a planar body is same about any two mutually perpendicular axes in its plane, then it is same about any axis in that plane. (3) (c) The moment of inertia of square of mass m and side a, equals m a 2 /12, about an axis passing through its centre and parallel to any of its side. Calculate its moment of inertia about a diagonal. (2) 15. (a) State and prove the parallel axis theorem for momentum of inertia. Given that the moment of inertia of a rod of mass m and length a, equals m a 2 /12, about an axis passing through its centre and perpendicular to any of its length, calculate its moment of inertia about a parallel axis passing through an end. Calculate the same moment of inertia by direct integration and check the consistency with the parallel axis theorem. ( ) (b) State and prove the perpendicular axis theorem for momentum of inertia. Given that the moment of inertia of a circular lamina of mass m and radius r, equals m r 2 /2, about an axis passing through its centre and normal to any of its plane, calculate its moment of inertia about (i) a diameter, (ii) a tangent in its plane and (iii) a tangent normal to its plane. ( ) 16. (i) Calculate the moment of inertia of a sphere about a diameter. (5) (ii) Calculate the moment of inertia of a solid cylinder about an axis passing through its centre of gravity and perpendicular to its length. (5) 17. Establish the relation between the time derivative operators in the space-fixed and the rotating frame. Hence derive the expressions for the pseudo forces appearing in the bodyfixed frame of a rotating body. (3 + 3)

8 18. (i) How do you explain the rotating nature of a cyclone? (2) (ii) Why does a north flowing river damages its right bank more? (2) 19. A particle is dropped from a height h above the earth surface. Derive an expression for its horizontal shift due to Corioli force, as it reaches the round. (5) 20. Establish Euler s equation of motion for a rotating rigid body. Prove that : in absence of any external torque (i) the total angular momentum and (ii) the kinetic energy of the body remains conserved in the body-fixed frame. ( ) 21. Show that in absence of any external torque, the angular velocity vector of an axially symmetric top precesses, but a spherically symmetric top does not. (5 + 2) THERMAL PHYSICS I (25 Marks) 1. (a) Derive an expression for the pressure of an ideal gas on the basis of the kinetic molecular theory, neglecting the possibility of molecule-molecule collision. (5) (b) The density of hydrogen gas at N.T.P. is 0.09 kg/m 3. Calculate the r.m.s. speed of the H 2 molecules. (3) 2. (a) Starting from the fundamental relation : P = 1/3 m n C 2, (notations having usual significances) relate the r.m.s. speed of gas molecules with temperature. (2) (b) Calculate the r.m.s. speed of the O 2 molecules at 27 o C. (3) 3. For a Maxwellian gas, prove that the fraction of molecules having their x-component of velocity lying between u u + du, is given by : A exp ( bu 2 ), where A and b are constants. (5) 4. From Maxwell s rule for distribution of molecular speed, derive expressions for the mean, the r.m.s. and the most probable speed of the gas molecules. Show their position on a velocity distribution graph. ( ) 5. From the expression for Maxwell s distribution of molecular speed, obtain the rule for the distribution of molecular kinetic energy and momentum. (3 + 2) 6. Show that, if we define a dimensionless parameter C r as C/C rms, then the distribution law for C r becomes independent of oth the molecular mass and temperature. (2) 7. State the equipartition thorem of energy. Calculate the (C P /C V ) for a mono-atomic, diatomic and a (non-linear) poly-atomic gas on the basis of this theorem (ignoring the internal degrees of freedom of the atoms). (2 + 6) 8. Which assumptions made for an ideal gas were discarded by Van-der-Waal? Discuss how he introduced the pressure and the volume correction for a real gas. ( ) 9. Define critical temperature, critical pressure and critical volume of a gas. Derive their expressions for a Van-der-Waal s gas. (6 + 5) 10. The equation of a gas in given by : P = RT/( V b) exp( a/rtv), Determine the expressions for critical temperature, critical pressure and critical volume of this gas. (6) 11. The equation of a gas in given by : (P + a/tv 2 )(V b) = RT. Determine the expressions for critical temperature, critical pressure and critical volume of this gas. (6) 12. Define Boyle temperature. Show how it is related to the second Virial coefficient? (2 + 2) 13. Derive an expression for the mean free path of a molecule in a gas, assuming all other molecules are static. How does this expression get modified if all the molecules move

9 and their speeds obey Maxwell distribution? (4 + 1) 14. (a) Prove that the fraction of molecules that do not suffer a collision upto a distance x on its path, is given by : exp( x/ ), where is the mean free path. (4) (b) If the mean free path of the particles in a molecular beam is, then find out the percentage of molecules that (i) suffer a collision within the distance from the beginning, (ii) suffer a collision between the distances and 2 from the beginning. (2 + 2) 15. Establish the expression : P = ⅓ m n C 2 rms for the pressure of a gas, taking moleculemolecule collision into consideration. (5) 16. Derive an expression for the coefficient of viscosity ( of a gas on the basis of the kinetic molecular theory. How do you expect to vary with pressure and temperature? How far your expectations tally with the reality? ( ) 17. Derive an expression for thermal conductivity of a gas on the basis of the kinetic molecular theory. (8) 18.* Prove the relation K = Cv (notations are standard) starting from the kinetic molecular theory of gases. (8) 19. Derive an expression for the diffusion coefficient of a gas on the basis of the kinetic molecular theory. (6) 20. Establish Fourier s heat equation in one dimension, taking radiation loss into account. What form does this equation take under steady state condition? (6 + 1) 21. Two ends of a straight rod are maintained at 30 o C and 100 o C. Solve Fourier s heat equation in steady state, neglecting radiation loss, to find the temperature distribution along the rod. (4) 22. One end of an infinite, straight rod are maintained at an excess temperature and the other end is at the atmospheric temperature. Solve Fourier s heat equation in steady state, taking radiation loss into account, to show that the distribution of excess temperature along the rod is exponential. (4) 23. Both the ends of a straight rod are maintained at an excess temperature. Solve Fourier s heat equation in steady state, taking radiation loss into account, to find the distribution of excess temperature along the rod. (4) 24. (a) Define spectral emissive and absorptive powers. State Kirchoff s law in this context. Show that the ratio of the said emissive and absorptive power of a body equals the emissive power of a black body at the same temperature ( ) (b) How do you explain the Fraunhofer lines in the solar spectrum on the basis of Kirchoff s law? (3) 25. Obtain Newton s law of cooling from Stefan s law as an approximation. (3) 26. The amount of solar radiation incident on earth surface 1.36 kw/m 2. If the angular diameter of the sun as observed form earth 0.5 o, then estimate the surface temperature of the sun. (Given Stefan s constant 5.7 X 10 8 W 2 K 4.) (3)