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# Electromagnetic theory and optics

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### Electromagnetic theory and optics

1. 1. Birla Institute of Technology Department of Applied Physics B.E. Semester I Physics I - Question Bank ( August- 2006) (New Syllabus-2005)1. First Unit – Waves and Oscillations:Types of waves, one-dimensional waves, progressive and stationary waves, Derivation of wave equation inone-dimension, phase velocity, group velocity, dispersion. Damping, forced vibrations & resonance .QUESTIONS:1. What is a wave? Define a periodic wave. What is the difference between a traveling wave and standing wave?2. How could you prove experimentally that energy is associated with a wave? When two waves interfere, does one alter the progress of the other? When waves interfere, is there a loss of energy? Explain your answer.3. Energy can be transferred by particles as well as by waves. How can we experimentally distinguish between these methods of energy transfer?4. Write mathematical expressions which represent waves traveling to the right and to the left. Write down a expression describing a transverse wave traveling along a string (a)in the positive x-direction with wavelength 11.4 cm, frequency 385 Hz, and amplitude 2.13 cm, (b)in the negative x-direction and having amplitude of 1.12 cm, frequency 548 Hz, and a speed of 326 m/s.5. The equation of a transverse wave traveling along a very long string is y = 6.0 sin (0.020 π x + 4.0 π t) where x and y are expressed in cm and t in seconds. Determine (a) the amplitude, (b) the wavelength,(c) the frequency, (d) the speed ,(e) the direction of propagation of the wave ,(f) the maximum transverse speed of a particle in the string, (g) the transverse displacement at x = 3.5 cm when t = 0.26 seconds.6. Prove that the slope of a string at any point is numerically equal to the ratio of the particle speed to the wave speed at that point.7. A uniform circular hoop of string is rotating clockwise in the absence of gravity. The tangential speed is vo. Find the speed of waves on this string. (Note that the answer is independent of the radius of the hoop and the linear mass density of the string).8. Two sinusoidal waves of the same frequency and of the same amplitude are propagating along a string in opposite directions. Obtain an expression for standing waves produced in the string. Find out the location of nodes and antinodes. Does energy propagate along the string? Comment on the energy distribution in the string.9. What are the three lowest frequencies for standing waves on a wire 9.88-m long having a mass of 0.107 kg, which is stretched under a tension of 236 N?10. In an experiment on standing waves, a string 92.4 cm long is attached to the prong of an electrically driven tuning fork, which vibrates perpendicular to the length of the string at a frequency of 60.0 Hz. The mass of the string is 44.2 g. How much tension must the string be under (weights are attached to the other end) if it is to vibrate with four loops? 1 6 August. 2006
2. 2. 11. How do the amplitude and the intensity of surface water waves vary with the distance from the source? 12. A line source emits a cylindrical expanding wave. Assuming the medium absorbs no energy, find how (i) the intensity and (ii) the amplitude of the wave depend on the distance from the source.13. Calculate the energy density in a sound wave 4.82 km from 47.4 kW siren, assuming the waves to be spherical, the propagation isotropic with no atmospheric absorption and the speed of sound to be 343 m/s.14. Write down the wave equation in one-dimension and explain the meaning of different symbols. Suggest a solution.15. (a) Define phase velocity and group velocity. (b) Make diagram and explain how group velocity is different from phase velocity. Write the formula for each in terms of the wave propagation constant and angular frequency.16. What is dispersion? What are dispersive and non-dispersive media? What has dispersion to do with group velocity? 217. Consider waves whose wavelength is ( h 1 − v ) / mo v and frequency is c2 2 mo c 2 / h 1 − v , where mo, c and h are constants and v is a variable. Show that the group c2 velocity of these waves is v . Also find the phase velocity as a function of v .18. Establish the equation of a damped harmonic oscillator, which is subjected to a resistive force proportional to the first power of its velocity. Assuming the damping to be less than critical answer the following:(i) Show that the motion of the system is oscillatory with its amplitude decaying exponentially with time.(ii) Find out the frequency of oscillation. Comment on whether the frequency of oscillation is greater or less than the natural frequency of oscillation.19. (i)Define quality factor Q and relaxation time τ. Find out a relationship between Q and τ. Show that relaxation time is directly proportional to the mass of the oscillator. (ii) Show that the fractional change in frequency of the damped oscillator is = 1/(8Q2 )20. For the oscillator of Q 20, calculate following: (i) What is the total energy of the oscillator? (ii) Show that the amount of power loss is proportional to both the damping force and the velocity of the particle21. A mass less spring, suspended from a rigid support, carries a flat disc of mass 100g at its lower end. It is observed that the system oscillates with a frequency 10 Hz and the amplitude of the damped oscillations reduces to half its undamped value in one minute. Calculate the resistive force constant and the relaxation time of the system.22. (a) What do you understand by free and forced vibrations? (b) What do you mean by resonance? When does it occur? (c) Write a note on sharpness of resonance23. Give the theory of forced vibrations and deduce the condition for amplitude and velocity resonance. Write a note on sharpness of resonance. Give two examples of common phenomenon in which resonance plays an important role. 2 6 August. 2006
3. 3. 24. In a damped oscillator let m=250gm, k=85N/m and b=0.070 kg/s. In how many periods of oscillation will the mechanical energy of the oscillator drop to half of its initial value?25. A mass less spring of spring constant 10 N m is suspended from a rigid support and carries a mass of 0.1 kg at its lower end. The system is subjected to a resistive force -Lv, where L is the resistive force constant and v is the velocity. It is observed that the system performs damped oscillatory motion and its energy decays to 1/e of its initial value in 50 s. (i) What is the value of resistance force constant? (ii) What is the Q value of the oscillator? (iii) How long does it take for the amplitude to fall one half of its initial value? (iv) What is the percentage change in frequency due to damping?26. An object of mass 0.1kg is hung from a spring whose spring constant is 100 Nm-1. A resistive force -Lv acts on the object where v is the velocity in meters per second and L = 1 Nsm-1. The object is subjected to harmonic driving force of the form Fo cos qt where Fo = 2N and q = 50 radians/sec. In the steady state what is the amplitude of the oscillations and the phase relative to applied force.27. (a) Show that in the steady state forced vibration the power supplied by the driving force averaged mbp 2 F02 over a cycle is given by P = and the power is maximum when p = ω 0 . (ω 2 0 − p2 ) 2 + 4b 2 p 2 (b) Find two values of p , namely p1 and p 2 at which the power P is half of that of resonance and show that p1 p 2 = ω 0 22. Second Unit – Fields:Vector and scalar fields, physical and mathematical concepts of gradient, divergence and curl, Gauss’stheorem and Stokes’ theorem .QUESTIONS:28. Explain vector and scalar fields with suitable examples of each. Explain the physical meaning of Divergence, Curl and Gradient.29. State and explain the physical significance of Gauss’s divergence theorem and Stoke’s theorem.30. Prove the following r v r r r r r r r r r r r r r(a) ∇ ⋅ (φA) = (∇φ ) ⋅ A + φ (∇ ⋅ A) , (b) ∇ ⋅ ( AxB ) = B ⋅ (∇xA) − A ⋅ (∇xB ) v r r r r r(c) ( A ⋅ ∇) r = A , (d) curl ( gradφ ) = ∇x(∇φ ) = 0 r r r r(e) div (curlA) = ∇ ⋅ (∇xA) = 031. For a position vector r show that: (a) curl r =0 (b) div r =332. (a) Find the gradient of r = x2 + y2 + z2 r r r r r r (b) Explain the nature of following vector fields (i) ∇xA = 0, (ii )∇xA ≠ 0 , (iii) ∇ ⋅ A = 0 , r r (iv) ∇ ⋅ A ≠ 0 3 6 August. 2006
4. 4. 33. A semi cylindrical surface of radius R and length L is kept on the xoy plane with its axis along x axis. r r A uniform field A makes an angle β units with the z axis in the zox plane. Calculate the flux of A through the given surface.3. Third Unit – Electromagnetic Theory3.1. Gauss’s law in integral and differential form, electric potential and relation with E ( SS*- capacitanceelectric energy density), dielectrics, three electric vectors, dielectric susceptibility boundary conditions onE and DQUESTIONS: r r r r34. Show that for a static electric field E, ( ∫ E.dl = 0 ), and ∇xE = 035. Obtain Gauss’s law of electrostatics in differential form. r r36. Use the divergence theorem and the integral form of Gauss law to show ∇⋅D = ρ37. For an electric field E = 2A [ (xyz + y2z + yz2 ) i + ( x2z + xyz + xz2) j + ( x2y +xy2 + xyz) k ] volt/m, calculate the volume charge density at the point (0,2,-1)38. Sketch the lines of force for the following systems of charges: a) Point charges +q and –q separated by a distance d. b) Point charges +q and +q separated by a distance d. c) Three equal point charges +q, +q, and +q at the vertices of an equilateral triangle. d) A finite line of charge e) A thin circular uniformly charged disc of radius R39. A point charge q is fixed at the tip of a cone of semi-vertical angle θ. Show that the electric flux through the base of the cone is q (1 - cosθ) / 2ε0.40. State and explain Gauss’s law of electrostatics in integral form.41. The electrostatic potential due to certain charge distribution is given by the expression ϕ (x,y,z) = - (Vo/a4) (x2yz + xy2z + xyz2) volts where Vo and a are constants. Calculate: (a) the electric field at the points A(0, 0, a), B(0, a, a), and C(a, a, a) (b) the magnitude of the field at C (c) the charge density at points A, B, and C.42. A particle of mass m and charge – Q is constrained to move along the axis of a ring of radius a. The ring carries a uniform charge density +λ along its length. Initially the particle is in the plane of the ring where the force on it is zero. Show that the period of oscillation of the particle when it is displaced slightly from its equilibrium position is given by 2ε 0ma 2 T = 2π λQ43. Electric charge is uniformly distributed throughout an infinitely long cylinder of radius R, having a linear density of charge λ. Obtain an expression for the electric field at a distance r from the cylinder axis when (i) r > R. (ii) r < R. 4 6 August. 2006
5. 5. 44. Obtain an expression for the potential difference between two points in an electric field as a path integral of the field. Explain (i) conservative field (ii) equipotential surface. 3.2. Amperes law in integral and differential form, applications. Hall effect, three magnetic vectors, magnetic permeability and susceptibility, boundary conditions on B and H. & 3.3 .Faraday’s law in integral and differential form ( SS - Inductance, magnetic energy density , continuity equation for charge , displacement current, Maxwell’s equations in free space , electromagnetic wave r r r equation for plane waves in dielectric medium and free space, relation between E , B and k , Poyinting vector .45. (a) State and explain the Biot-Savart law. (b) Use the Biot-Savart law to determine the magnetic field at a distance r from a straight finite wire segment carrying a steady current I. What form does B take when the wire is very long?46. r r (a) Prove that ∇ ⋅ B = 0 . (b) Using above relation, explain whether a magnetic monopole can exist or not. Also comment on whether lines of B are closed or open.47. (a) Write down Ampere’s law in integral and differential form. (b) Derive the differential form from the integral form.48. Use Ampere’s law to find the magnetic field B at a distance r from an infinite long straight wire. Why can you not use this law to find B due to a finite length of wire?49. Is the flow of electrons in fillamental shape different than the flow of electrons in a conductor, if in both cases the value of current is i ? How?50. An infinite long, straight cylindrical conductor of radius R is carrying a current I uniformly distributed over its cross-section. Find the magnetic field at points at distance r from its axis, located outside ( r > R) and inside ( r < R ) the conductor.51. A circular loop of radius R is carrying a current I. Determine B at a point P on the axis and at a distance z from the center of the coil.52. A wire carrying a current of 100 A is bent into the form of a circle of radius 5.08 cm. Calculate the flux density at the center of the coil. Also calculate the flux density on the axis of the coil at a distance of 12 cm from the center.53. A long straight wire carries a current I. It lies in the plane of a rectangular coil such that its two sides of length L are parallel to the first wire, one at a distance a, and the other at a distance b μ o IL Φ= ln(b / a ) from the wire. Show that the magnetic flux Φ through the circuit is 2π .54. A square loop of wire of edge a carries a current I. Show that the value of the magnetic induction B at the center of the loop is given by 2 2μ o I B= πa55. You are given a length L of wire in which a current I may be established. The wire may be formed into a circle or a square each of one turn. Which yields the larger value of B at the central point and why? 5 6 August. 2006
6. 6. 56. A thin plastic disk of radius R has a charge q uniformly distributed over its surface. If the disk rotates at an angular frequency ω about its axis, show that the magnetic field at the center of the disc is: μ o ωq B= 2πR (Hint: The rotating disc is equivalent to an array of current loops)57. Find the magnetic field on the axis of and inside a long solenoid, having n turns per unit length. Derive the expression for B inside a toroid of N turns carrying a current I.58. A toroid of mean radius 12 cm has 800 turns of wire wound on it. If the wire can safely carry a maximum current of 3.6A, calculate the maximum value of magnetic intensity H that can be generated in the volume of the toroid. What is the corresponding value of magnetic induction B for air core toroid of the above description?59. (a) What is the Lorentz force? (b) A 1.5 MeV proton is moving vertically downwards. A field of magnetic induction 2.5 x 103 Wb/m2 acts horizontally from south to north. Find the magnitude and direction of the magnetic force exerted on the proton60. (a)Magnetic field does not do work on a charge particle. Why? (b) Out of electric and magnetic field, which would use if you want to just steer a moving charged particle. Why?61. Describe Hall Effect. Define the terms ‘Hall Voltage’ and ‘Hall Coefficient’ and derive the suitable expressions for them. What are the important applications of Hall Effect?62. Show in terms of the Hall electric field E and the current density J that the number of charge carriers per unit volume is given by the expression n=J B / (e E).63. r A copper strip of 100μm width is placed in a uniform magnetic field B perpendicular to the strip.A current of 20 Amp is sent through the strip such that a Hall potential difference V appears across the width of the strip. Calculate V . (The number of charge carriers per unit 28 volume for copper is 8.47 × 10 electrons/m3, B = 0.01 Tesla )64. (a) What is the relationship between B and H? (b) Obtain the boundary conditions on B and H across a surface between two magnetic media.65. A certain element has density 1.74x103 kg/m3 and atomic weight 24.31. If every atom of this element contributes one Bohr magneton, calculate the magnetic moment of 1 cm3 of this element assuming perfect alignment of the atomic dipoles.66. Consider a magnetic field, B = i2 – k, in a medium with μr = 1.1. Find the value of B in another medium with μr=1.5 if the boundary between the media is the plane z = 0.67. State Faraday’s laws of electromagnetic induction. Explain how the direction of the induced emf may be found.68. State the integral form of Faraday’s law. From this derive the differential form. Explain whether the electric field produced by a changing magnetic field is conservative or not.69. (a) Explain what you understand by motional EMF. (b) A metallic disc of radius ‘a’ is rotating with an angular velocity ω about its axis. A uniform field of magnetic induction B is present parallel to the axis, that is, perpendicular to the disc. Show by considering the magnetic force on free electrons, that the emf between the center of the disc and its circumference is ωBa2/2. Verify that the same result is obtained by applying Faraday’s law. 6 6 August. 2006
7. 7. 70. A coil of n turns and radius R carrying a current I is placed on a horizontal table. A small conducting ring of radius r is placed at a distance yo from the center and vertically above the coil. Find an expression for the emf induced in the ring when it is allowed to fall freely.71. A large conducting loop of radius R is fixed in the xoy plane centered on the origin. A small magnet with its axis parallel to the z-axis passes through the loop at uniform velocity. Sketch the variation of the induced current with time.72. The magnetic energy stored in a certain inductor is 25.3 mJ when the current is 62.0 mA. (a) Calculate the self inductance (b) What current is required for the magnetic energy to be four times as much?73. A solenoid 85.3cm long has a cross-sectional area of 17.2 sq cm. It has 950 turns of wire carrying a current of 6.57 A. Calculate (a) the magnetic energy density inside the solenoid (b) the total energy stored in the magnetic field inside the solenoid. (Neglect end effects).74. A length of copper wire carries a current of 10 A, uniformly distributed. Calculate (a) the magnetic energy density and (b) the electric energy density at the surface of the wire. The wire diameter is 2.5mm and its resistance per unit length is 3.3 Ω /km.75. A uniform magnetic field B is changing in magnitude at a constant rate dB/dt. You are given a mass m of copper wire which is to be drawn into a wire of radius r and formed into a circular loop of radius R. Show that the induced current in the loop does not depend on the size of the wire of the loop and assuming B perpendicular to the loop, is given by m dB i= where ρ is the resistivity and δ is the density of copper. 4πρδ dt76. Around a cylindrical core of cross-sectional area 12.2 sq cm are wrapped 125 turns of insulated copper wire. The two terminals are connected to a resistor. The total resistance in the circuit is 13.3 Ω. An externally applied uniform longitudinal magnetic field in the core changes from 1.57 T in one direction to 1.57 T in the opposite direction in 2.88 ms. How much charge flows through the circuit?77. A circular coil of radius 10.3 cm consists of 34 closely wound turns of wire. An externally produced magnetic field of 2.62 mT is perpendicular to the coil. (a ) If no current is in the coil, what is the number of flux linkages? (b) When the current in the coil is 3.77 A in a certain direction, the net flux through the coil is found to vanish. Find the inductance of the coil.78. A solenoid 126 cm long is formed from 1870 windings carrying a current of 4.36A. The core of the solenoid is filled with iron of relative permeability constant 968. Calculate the inductance of the solenoid, assuming that it can be treated as ideal. Radius = 2.725 cm.79. State Ampere’s law. Explain why this law is inadequate in describing a situation where electric field is varying with time.80. What is displacement current? Explain its physical significance. What are the consequences of the introduction of displacement current density in Maxwells equation.81. Show that the displacement current density between the capacitor plates is given by dE jd = ε o dt82. Show that the displacement current between the capacitor plates in a parallel plate capacitor is equal to the conduction current in the connecting leads.83. Derive the equation of continuity for charge using the law of charge conservation and the divergence theorem.84. Write down Maxwell’s four equations in integral and differential form. 7 6 August. 2006
8. 8. 85. From Maxwell’s equations show that any charge distribution in a conductor of conductivity σ is reduced to 1/e times the original one in a time ε0 /σ .86. Derive the wave equations for E and B from Maxwell’s equations. Discuss that Maxwell’s 1 equations lead to the existence of electromagnetic waves, having speed c = . (μ oε o ) (a) Suggest simple solutions of above wave equations. (b) Would we get wave equation if displacement current were not introduced?87. r For plane electromagnetic waves propagating in free space, show that E, B and k are mutually perpendicular. (Hint: use Maxwell’s equations). (a) Show that in an electromagnetic wave E and B are in phase. Eo ω 1 (b) Prove that = =c= Bo k μ oε o88. A nonconducting medium has refractive index n. Show that for EM waves the group velocity is c given by v g = ⎛ dn ⎞ ⎜n + ω ⎟ ⎝ dω ⎠89. Write the expression for Poynting vector. Explain the physical significance90. Derive an expression for Poynting vector.91. Show that for a plane electromagnetic wave, the average value of the Poynting vector c N= ε o E o2 292. A laser beam has a diameter of 2 mm. What is the amplitude of the electric field in the beam in vacuum if the power of the laser is 1.5 watts?93. A plane electromagnetic wave is traveling in the negative y direction. At a particular position and time, the magnetic field is along the positive z-axis and has a magnitude of 28 nT. What is the direction and magnitude of the electric field at that position and at that time?94. The intensity of direct solar radiation that was unabsorbed by the atmosphere on a particular summer day is 130 Wm-2. How close would you have to stand to a 1.0-kW electric heater to feel the same intensity? Assume that the heater radiates uniformly in all directions.95. The magnetic field equations for an electromagnetic wave in free space are B x = B sin (ky + wt), B y = B z = 0. (a) What is the direction of propagation? (b) Write the electric field equation. (c) Is the wave polarized? If so, in what direction?96. A space probe is radiating electromagnetic signals isotropically, the total power radiated over a solid angle 4 π being 20W. Calculate the amplitude of the electric field in the signal received by a detector at a distance of 2 million km from the probe. You may assume that the waves incident on the detector are plane waves. 8 6 August. 2006
9. 9. 97. Calculate the following quantities for a plane waves traveling in vacuum having an electric field amplitude, Eo = 40 μ V/m; (a) Average energy density in the wave. (b) Peak energy density (c) Average value of the Poynting vector (d) Peak value of the Poynting vector (e) Total electromagnetic energy in a cube of side 5 Km in the path of the wave (f) Obtain an expression for the inductance of a solenoid. 4. Plasma Physics: Plasma state, types of plasma, applications of plasma.98. What is plasma? Explain what you understand by collective behavior. Give examples of different kinds of plasmas. Write down briefly about hot and cold plasmas.99. What do you understand by Debye Shielding? Explain the conditions of plasma formation.100. Discuss various industrial applications of plasmas. 5. Physical Optics 5.1 Interference: Two-Beam Interference, Interference in Thin Films and Wedge-Shaped Layers, Reflection and Anti-Reflection Coatings, Applications of Interferometry: Newton’s rings, Michelson Interferometer. 101 What is interference? Explain why coherent light is essential to observe stationary interference pattern. State the conditions for good contrast observable in interference pattern in the Young double slit experiment. 102 Monochromatic light from a distant source falls on two closely spaced pin holes and interference fringes are obtained on a screen. Calculate the spacing between the consecutive maxima and minima. Discuss about the shape of the fringes. 103 Find the ratio of intensity at the center of a bright fringe to the intensity at a point one quarter of the distance between two fringes from the center in Young double slit experiment. 104 One of the two slits in Young double slit experiment is wider than the other, so that the amplitudes of light reaching the center point of the screen from wider slit, acting alone, is twice that from the other narrower slit, acting alone. Find the expression for resultant intensity Iθ at direction θ on the screen. 105 In the Young double slit experiment, interference fringes are formed using sodium light which predominately comprises two wavelengths (589.0 nm and 589.6 nm). Obtain the regions on the screen where the fringe pattern will disappear. The separation between the two slits is 0.5 mm and the distance between the double slit and screen is 100 cm. 106 In the Young double slit experiment a thin mica sheet (n = 1.5) is introduced in the path of one of the beams. If the central fringe gets shifted by 0.2 cm, calculate the thickness of the mica sheet. The separation between the two slits is 0.1 cm and the distance between the double slit and screen is 50 cm. 9 6 August. 2006
10. 10. 107 Discuss the formation of colors in thin films.108 Fringes of equal thickness are observed in a thin glass wedge of refractive index 1.52. The fringe spacing is 1 mm and wavelength of light is 589.3 nm. Calculate the angle of wedge in seconds of an arc.109 If the angle of a wedge is 0.25 degree and the wavelengths of incident sodium lines are 589.0 nm and 589.6 nm, find the distance from the apex at which the maxima due to two wavelengths first coincide when observed in reflected light.110 Draw the experimental set up to observe Newton’s interference rings. Show that the radius of m th bright ring is given by xm = [(m + 1 / 2)λR )]1 / 2 , where λ is the wavelength of the light source, R is the radius of the convex lens. What is the nature of the central fringe? How will be the pattern appear in the transmitted light?111 In costume jewelry, rhinestones (made of glass with n = 1.5) are often coated with silicon monoxide (n = 2.0) to make them more reflective. How thick should the coating be to achieve strong reflection for 560 nm light, incident normally?112 (a) In a Newtons rings experiment, the radius of the curvature of the plano-convex lens is 5.0 m and the lens diameter is 20 mm. If the wavelength of light used is 589 nm then find how many bright rings are produced. Also find the number of bright rings produced if the arrangement were immersed in water. (b)In a Newtons rings experiment the diameter of the 10th ring changes from 1.40 cm to 1.27 cm when a liquid is introduced between the lens and plate. Calculate the refractive index of the liquid.5.2. Diffraction: Fraunhofer Diffraction by Single Slit , Double Slit and Grating, Limit of Resolution, Rayleigh Criterion , Fresnel Diffraction (Qualitative)113 (a) What do you understand by diffraction of light? Distinguish clearly between interference and diffraction of light. (b)Distinguish between Fresnel and Fraunhoffer classes of diffraction. (c)Why is the diffraction of sound waves more evident in daily experience than that of light waves?114 Discuss the Fraunhofer diffraction at a single slit. Extend the theory to the case of a plane transmission grating. Explain what is meant by diffraction spectra of different orders and state the condition under which the grating spectra of even order are absent.115 What is plane diffraction grating. Describe how would you employ it for determining the wavelength of light. Deduce expression for its dispersive power.116 In case of a plane transmission grating, what would be the effects if the distances between the rulings are (i) very close, (ii) very small, compared with the wavelength of the radiation used?117 Show that the first order and second order spectra will never overlap when the diffraction grating is used for studying a light beam containing wavelength components from 400 nm to 700 nm118 Could you construct a diffraction grating for sound? If so, what grating spacing is suitable for a wavelength of 0.5 m?119 Explain why increasing the number of slits N in a diffraction grating sharpens the maxima. Why does decreasing the wavelength do so? Why does increasing the grating spacing do so? 10 6 August. 2006
11. 11. 120 (a)Calculate the minimum number of lines per cm in a 2.5 cm wide diffraction grating which will just resolve the sodium lines (589.0 nm and 589.6 nm) in the second order spectrum. (b)A diffraction grating is just able to resolve two lines of wavelengths 514.034 nm and 514.085 nm in the first order. Will it be able to resolve the lines 803.720 nm and 803.750 nm in the second order?121 In the second order spectrum of a plane transmission grating a certain spectral line appears at angle of 10 degrees, while another line of wavelength 0.05 nm greater appears at an angle 3 seconds greater. Find the wavelength of the lines and the minimum grating width required resolving them.5.3 Polarization :Polarization of light, Maluss law, polarization by reflection, Brewsters law, Double refraction, Analysis oflinearly and circularly polarized light Fresnels equations and their applicationsQUESTIONS:122 (a) Explain what is polarized light. Define plane of polarization. (b) List three methods to produce plane polarized light from unpolarized light. (c) Explain why polarization shows that light waves are transverse.123 (a) State and explain Malus’s law. (b) Three polarizing plates are stacked. The first and third are crossed; the one between has its axis at 45o to the axes of the other two. What fraction of the intensity of an incident unpolarized beam does the stack transmit?124 (a) State and prove Brewster’s law, given that the reflected and refracted beams are at right angles for complete polarization of the former. (b) When light is incident on a glass plate with n = 1.54, the reflected light is plane polarized. Find the angle of polarization and the angle of refraction. What is the polarization state of the refracted light? (c) When the angle of incidence for a glass plate is 60o, reflected light is linearly polarized. Find refractive index of the glass and angle of refraction.125 Light traveling in water of index of refraction 1.33 is incident on a plate of glass of index of refraction 1.53. At what angle of incidence is the reflected light completely polarized?126 Derive Fresnel’s equations for reflected and refracted wave amplitudes (a) for the case of E parallel to the plane of incidence. (b) for the case of E perpendicular to the plane of incidence.127 From Fresnel’s equations show that when light undergoes reflection from a denser medium it suffers a phase change of π.128 What is double refraction? Give examples of doubly refracting crystals. Explain the phenomenon of double refraction using Huygen’s theory.129 What would be the action of a half-wave plate (that is, a plate twice as thick as a quarter-wave plate) on (a) linearly polarized light (assume the plane of vibration to be at 45o to the optic axis of the plate)(b) circularly polarized light, and (c) unpolarized light? 11 6 August. 2006