ASSIGNMENT #1 Special Theory of Relativity UNIT I
1.
What was the objective of conducting the Michelson-Morley experiment? How is the negativeresult of the experiment interpreted?
2.
State Eintein’s postulates of special theory of relativity. Explain why Galilean transformation failed to explain the actual results of Michelson Morley experiment.3.
Derive an expression for time dilation and give an example to show that time dilation is a real effect4.
Show that the circle x2 + y2 = a2 in frame S appears to be an ellipse in a frame S’ which is movingwith velocity v relative to S.
5.
Show that x2 + y2 + z2 – c2t2 is invariant under Lorentz transformation.6.
If at the time t = t’ = 0, the origin of the systems S and S’ just coincide and a spherical pulse of light is produced at the common origin, show that the speed of propagation of the spherical wave front is the same in both systems i.e c, where S’ is moving relative to stationary system S.7. Deduce the relativistic velocity addition theorem. Show that it is consistent with Einstein’s second postulate.
8. Verify the statement that no material particle can attain speed of light c.
9. Calculate the length of a rod of length 5m in a frame of reference which is moving with a velocity 0.6 c in a direction making an angle of 30° with the rod.
10. With what velocity should a spaceship fly so that every day on it may correspond to three days on the earth’s surface?
ASSIGNMENT #2 Special Theory of Relativity UNIT I
1. Establish Einstein’s mass energy relation. Show that if the variation of mass with thevelocity is taken into account , the kinetic energy of the particle of rest mass mo and
moving with velocity v is given by
k = moc2 [ ( 1- v2/c2) -1/2 -1]
2. Find the kinetic energy and momentum of an electron moving with speed 0.6 c.
3. What is the length of a meter-stick appear to an observer at rest when it moves parallel to its length and its mass is measured as 3/2 times that of its rest mass?
4. An experimenter observes a radioactive atom moving with a velocity of 0.25c. The atom then emits a beta particle, which has a velocity of 0.9c, relative to the atom in the direction of its motion. What is the velocity of the beta particle as observed by the experimenter?
5. A man leaves the earth in a rocket that makes a round trip to the nearest star which is 4
light years away from Earth, at a speed of 0.8c in the direction of the star. How much younger will he be, w.r.t. to an observer on Earth, on his return than his twin brother who preferred to stay back?
6. Get an expression for Einstein’s mass-energy relationship. How will this expression get modified when the speed of the particle ‘v’ is very small in comparison to the speed of light?
7. The total energy of a moving meson is exactly twice its rest energy. Find the speed of the meson.
8. Show that E2/c2 – P2 is invariant under Lorentz transformation.
9. An air force rocket is chasing enemy’s spaceship. From earth, it is found that the speed of rocket is 2.55 X 1010 cm s-1 while that of enemy’s ship is 2.25 X 1010 cm s-1. What is the relative velocity of:
(i) Enemy’s ship as seen by air force rocket. (ii) Air force rocket as seen by enemy’s ship.
(iii) Air force rocket with respect to enemy’s ship as seen from earth. (iv) Enemy’s ship with respect to air force rocket as seen from earth.
ASSIGNMENT #3 Interference UNIT II
1. Define coherent sources. Discuss why two independent sources of light of samewavelength cannot show interference.
2. State and discuss the basic condition for observing the phenomena of interference of light.
3. In an interference pattern with two coherent sources the amplitude of intensity variation is found to be 5% of the average intensity. Calculate the relative intensities of the interfering sources.
4. How does the interference pattern by reflection in thin films differ from that of refraction? Why?
5. Explain why a thin transparent film appears coloured when observed in reflected light. Why an extended source is required for observing the bands? Why are the colours not observed in case of thick film?
6. Discuss the effect of introducing a thin plate in the path of one of the interfering beam in a biprism experiment. Deduce an expression for the displacement of the fringes.
7. Fresenal’s fringes are produced with light of wavelength 6× 10-5 cm. A thin glass plate ( =1.5) isμ
introduced in the path of one of the interfering beams. The central bright band is shifted to the position previously occupied by the 5th bright band. Calculate the thickness of the plate.
8. A soap film of = 1.33 is illuminated by white light incident at an angle of 45μ °. The light refracted by
it is examined by a spectrometer and a bright band is found corresponding to a wavelength 6000Å. Find the thickness of the film.
9. Describe and explain the formation of Newton’s tings in reflected monochromatic light. Why Newton’s rings are circular?
10. Newton’s rings are formed in reflected light of wavelength 6000Å with a liquid between the plane and curved surfaces. If the diameter of 6th bright ring be 3.1 mm and the radius of curvature of the
curved surfaces is 100 cm. Calculate the refractive index of the liquid.
11. Derive an expression for wavelength of light in Newton ׳s ring experiment for transmitted light if the plane glass plate is replaced by an another convex lens in such a way that both lenses are in contact with their curved surfaces .
12. Derive an expression for the path difference produced by a thin wedge - shaped film , in case of reflected and transmitted light.
ASSIGNMENT #4 Diffraction UNIT II
1. Determine the width of the mth order maxima for the double slit Fraunhofer ׳ s diffraction pattern.2. Give the theory of formation of diffraction patterns using a plane transmission grating. How would you use it to determine the wavelength of light.
3. Get the expression of and distinguish between dispersive power and resolving power of transmission grating.
4. Discuss the phenomena of Fraunhofer diffraction at a single slit and show that the relative intensities of the successive maxima are given as
1:4/9π2:4/25π2 : 4/9π2
:---5. Define the limit of resolution and Resolving power of an optical apparatus. Derive an expression for the Resolving power of a telescope and a microscope.
6. A parallel beam of mono-chromatic light is allowed to be incident on a plane grating
having 500 lines/ cm and the second order spectral line is found to be diffracted through 300. Calculate
the wavelengths of light used.
7. Light of wavelength 5000 Å is incident normally on a slit. The first minimum of diffraction pattern is observed to lie at a distance of 5 mm from the central maximum on a screen placed at a distance of 2 m from the slit . Calculate the width the width of the slit.
8. A microscope objective gathers light over a cone of semi-angle 30˚ and uses visible light ( =λ
5500Ǻ).Estimate its resolving limit.
9. Find the minimum number of lines that a diffraction grating would need to have in order to resolve in first order the red doublet given by a mixture of hydrogen and deuterium. The wavelength difference is 1.8 Ǻ at =6553 Å.λ
10. A diffraction grating is just able to resolve two lines of wavelength 5140.34 Ǻ and 5140.85 Å in the first order .Will it resolve the lines 8037.20 Ǻ and 8037.50 Å in the second order.
11. A telescope of a certain objective has diameter of 100 inches .Estimate the smallest angle between two stars that can be separated by it .
12. In a grating spectrum, which spectral line in 4th order will overlap with 3rd order line of 5461 Å?
ASSIGNMENT #5 Polarization UNIT III
1. Define double refraction, Describe construction and working with use of a Nicole prism.2. Describe half shade polarimeter find specific rotation of a lane sugar solution if the plane of polarization is turned by 26. 4˚. Length of tube is 20cm and concentration of sugar is 20℅.
3. Discuses Huygens’s explanation of double refraction .Give the construction and theory of half wave plate.
4. How would you produce and detect the following with the help of a nicol prism and quarter wave plate. (1) Plane polarized light (ii) circularly polarized light (iii) elliptically polarized light.
5. Explain Brewster’s law. Show that when light is incident on a transparent medium at polarizing angle, the reflected and refracted rays are at right angles.
6. Obtain expression for minimum thickness of a quarter wave plate .Find thickness of a quarter ware plate for = 589 nm λ μ0 = 1۠55; μe = 1.54.
7. A tube 20 cm long with a solution of 15 gm of cane sugar in 100cc, of water is placed in the path of polarized light. Find the angle of rotation of plane of polarization if the specific rotation of cane sugar is 660.
8. A 20 cm long tube containing sugar solution rotates the plane of polarization by 110. If the specific
rotation is 660, calculate the strength of the solution.
9. 80 gm of impure sugar is dissolved in a litre of water. The solution gives an optical rotation of 9.90
when placed in a tube of length 20 cm. If the specific rotation of pure sugar is 660 dm-1(gm/cc)-1.
Calculate the percentage purity of the sugar sample.
ASSIGNMENT #6 LASER UNIT III
1. Write short notes on(i) Spontaneous absorption (ii) Stimulated emission (iii) Spontaneous absorption (iv) Optical pumping
(v) Population inversion (vi) Resonant cavity
2. What are Einstein’s coefficients? Derive Einstein relation.
3. Describe the construction and action of the ruby laser.
4. Explain the action of a helium-neon laser. How is it superior to a ruby laser?
5. Explain the principle of optical pumping and stimulated emission of radiation. Discuss the properties of laser radiation and mention some of its applications.
6. What are differences between spontaneous and stimulated emission.
7. Why is spontaneous radiation incoherent?
8. A certain ruby laser emits 1.00 J pulses of light whose wavelength is 694nm. What is the minimum number of Cr3 + ions in the ruby crystal?
9. A pulsed laser is constructed with ruby crystal as active element. Ruby rod contains typically a total of 3X1019 Cr3+ ions .If the laser emits light at 6943Å. find the energy of one emitted photon and the
TUTORIAL # 1 Special Theory of Relativity UNIT I
1. Prove that x2 + y2 + z2 = c2t2 is invariant under Lorentz transformation.
2. What will the length of a meter rod appear to be for a person traveling parallel to the length of the rod at a speed of 0.8c relative to the rod?
3. How fast should a rocket move relative to an observer in order that its length may appear to the observer to be 99% of its proper length?
4. Calculate the percentage contraction of a rod moving with a velocity 0.8c in a direction inclined at 600 of its own length.
5. A man leaves the earth in rocket ship that makes a round trip to the nearest star which is 4 light years away at a speed of 0.8c.How much younger will he be on his return than his twin brother who preferred to stay behind.
6. Find the shape of a circle at rest in a frame S when viewed from a frame S`, when S` is moving with speed v along x-axis with respect to frame S.
7. Observer on the earth (assumed to be an inertial frame of ref.) sees a spaceship P receding from him at 2.0×108m/s and overtaking another spaceship Q receding at 1.5×108 m/s. Find
relative velocity of -
(i) spaceship Q as observed by P.
(ii) spaceship P as observed by Q
8. The mass of a moving electron is 11 times its rest mass. Find its kinetic energy and momentum. 9. A moving electron collides with a stationary electron and an electron position
10. pair comes into being as a result. When all four particles have the same velocity after the collision, the kinetic energy required for this process is minimum. Use a relativistic calculation to show that Vmin = 6m0c2, where ‘m0’ is the rest mass of the electron.
10. A body moving at 0.5c with respect to an observer disintegrates into two fragments that move in opposite direction relative to their e.m. along the same line of motion as the original body. One fragment has a velocity of 0.6c in the backward direction relative to the e.m. and the other has a velocity of 0.5c in the forward direction. What velocities will the observer find?
TUTORIAL # 2 Special Theory of Relativity UNIT I
1. Get an expression for the number of fringe –shifts in Michelson – Morley Experiment. Calculate this number when l = 11m, = 6000Å, c = 3×10λ 8m/s, and v = 3×104m/s.
2. If the frame S` moves along the y – axis with a speed ‘v’ with respect to ‘O’ an observer in S – get the expression for all the components of linear velocity as measured by O for a particle in S` in terns of the quantities measured w. r.t. S`.
3. A certain particle called meson has a life time 2×10-6sec.
(a) What is the mean life time when the particle is traveling with a speed of 2.994×108m/s?
(b) How for does it go during one mean life?
4. Find the speed of a 0.1MeV electron according to classical non- relativistic and relativistic mechanics.
5. Get an expression for Einstein mass – energy relationship. How will this expression get modified when the speed of the particle ‘v’ is very small in comparison to the speed of light ‘c’? 6. “Provided the rest mass of a particle is zero, the particle is always Devoid of any energy”
analyze the statement.
7. A rocket ship ‘A’ moving with a speed 0.9c relative to an observer on earth, if the spacecraft ‘B’ is to pass rocket ‘A’ at a relative speed of 0.5c in the same direction, what speed must ‘B’ have with respect to the earth?
8. Show that (E2/c2 – p2) is invariant.
9. A man leaves the earth in a rocket in a rocket ship that carries a clock in it, with a speed of 0.8c. How much slower or faster will his clock run? On his return, then a clock on earth?
10. Show that a circle in a rest frame will look like an ellipse from a frame moving away from the rest frame with a speed ‘v’.
11. An airplane in flying at 300m/s. How much time must elapse before a clock in the air plane and one on the ground differ by 1 S?
TUTORIAL # 3 Interference UNIT
1. In interference pattern with two coherent sources, the amplitude of intensity variation is found to be 5% of the average intensity, calculate the relative intensities of the interfering sources. 2. On introducing a thin sheet of mica thickness 1.2×10-4cm, in the path of interfering beams in a
bi-prism experiment, the central fringe is shifted through a distance equal to the spacing between successive bright fringes. Calculate the μmica when = 6 ×10λ -7 m.
3. The distance between the slit and bi-prism and that between the bi-prism and the screen are each 50cm. The obtuse angle of the bi-prism is 1700 and its R.I. is 1.5. The width of the fringes is 0.0135cm, calculate the wavelength of the light used.
4. Show the fringe due to a thin film which appears bright in reflected light, appears dark in transmitted light. Deduce the necessary operation.
5. In a Newton’s ring experiment, the diameters of 4th and 12th dark rings are 0.400cm and 0.7cm respectively. Calculate the diameter of the 20th dark ring.
6. Newton’s rings are formed in reflected light of wavelength of of wavelength 6000Å with a liquid in between the frame and curved surfaces. In the diameter of the 6th bright ring is 3.1 mm and
the radius of curvature of the curved surface is 100cm, calculate the R.I. of the liquid.
7. In a Newton’s ring arrangement with a film observed with light of wavelength 6×10-5 cm, the
difference of squares of diameters of successive rings are 0.125cm2. What will happen to this
quantity if:
(i) Wavelength of light is changed to 4.5×10-5 cm.
(ii) A liquid of R.I. 1.33 is introduces between the lens and the plate
(iii) The radius of curvature of the convex surface of the Plano convex lens is doubled. 8. Light of wavelength 5500Å falls normally on a thin, wedge-shaped film of R.I. 1.4 forming
fringes that are 2.5mm apart. Find the angle of the wedge in seconds.
TUTORIAL # 4 Interference UNIT II
1. Two plane rectangular pieces of glass are in contact at one edge and separated by a hair at the opposite edge so that a wedge formed. When light of wavelength 6000Å falls normally on the wedge, nine interference fringes are observed. What is the thickness of the hair?
2. Calculate the thickness of a soap bubble film (R.I. = 1.46) that will result in constructive interference in the reflected light if the film is illuminated with light whose wavelength in free apace is 6000Å.
3. In an arrangement of double slit, the slits are illuminated by light of wavelength 600mm. Find the distance of the first point on the screen from the central maximum where intensity is 75% of the central maximum.
4. A transparent paper of refractive index 1.45 and of thickness 0.02mm is pasted on one of the slits of a young’s double slit experiment which uses monochromatic light of wavelength 620mm. How many fringes will cross through the center is the paper is removed?
5. Newton’s rings are formed with reflected light ( = 5890Å) using a Plano-convex lens and aλ
plane plate with a liquid between them. The diameter of the 10th ring is 4.2mm and radius of
curvature of the lens is 100cm. Find the refractive index of the liquid if the ring is (a) Dark and (b) Bright
6. Newton’s rings are formed by reflection in the air film between the plane surface and a convex surface of radius 100cm. If the squares of the radii of successive bright rings be 100, 152, 198, 243, 302 and 350×10-4 cm2, make the best possible calculation of the wavelength of light.
7. Fringes are produced by a Fennel’s bi-prism in the focal plane of a reading microscope which is 100cm from the slit. A lens inserted between the bi-prism and the eye-piece gives two images of the slit in two positions. In one case, the two images of the slits 4.05mm apart and in the other case 2.90mm apart. If sodium light of wavelength 5893Å is used, find the width of the interference fringes. If the distance between the slit and the bi-prism is 10cm and refractive index of the material of the prism is 1.5, calculate the angle in degrees which the inclined faces of the bi-prism make with the base.
TUTORIAL # 5 Diffraction UNIT II
1. Light of wavelength 5000Å is incident normally on a slit. The first minimum of the diffraction pattern is observed to lie at a distance of 5mm from the central maximum on a screen placed at distance of 2mm from the Slit Calculate its width.
2. Calculate the angles at which the first dark band and the next bright band are formed in the fraunhofer diffraction pattern of a slit 0.3 mm wide. The wavelength of the light used is given as 5890Å.
3. Microwaves of wavelength = 2.0cm are incident normally on a slit 5.0cm wide. Determine theλ
angular width of the central maximum.
4. A screen is placed 200cm away from a narrow slit which is illuminated with light of wavelength 6×10-5cm. If the first minima lie 5mm on either side of central maximum, calculate the slit width.
5. The parallel slits have width 0.01mm each and opaque space between thin is 0.02mm. If the slits are illuminated with light of wavelength 5×10-5cm, the pattern is obtained on a screen placed at
a distance 50cm from the slits. Gets
(i) spread of the central diffraction maximum
(ii) the spacing between the consecutive interference fringes
(iii) the angular position of first four interference maxima on one side
(iv) the linear distances of first four interference maxima on one side from the central maximum.
6. A plate transmission grating having 5000 lines per cm is being used under normal incidence of light. Answer the following questions:
(a) What is the longest wavelength of light for which a spectrum can be seen? (b) What is the highest order spectrum that can be seen for the light of = 6000Å.λ
(c) The above mentioned spectral line in the second order spectrum overlaps with another spectral line in the next higher order. Find the wavelength of the other line.
(d) if the width of the opaque part be double than the transparent part of the grating then which orders of the spectra will be absent?
(e) If 90% of the width of the ruled portion of the grating is covered, what will happen to the observed spectrum?
(f) If the number of line per cm is kept unchanged but the shape of the groove is changed, then how the Dispersive Power of the grating and the relative intensity of the spectra of different orders get affected?
TUTORIAL # 6 Diffraction UNIT II
7. A plane diffraction grating having 6000 line / cm is used to photograph a spectrum. Calculate the angular dispersion in the region of the second order spectrum of wavelength 5.9 ×10-5 cm. If
the camera has focal length of 25cm, calculate the linear dispersion in the spectrograph and also the separation between the spectral lines 5890Å and 5896Å in the second order. (1.7×104
red/cm, 4.25 ×105, 2.55×10-2cm).
8. A diffraction grating used at normal incidence gives a green light (line) of = 5400Å in certainλ
order, superimposed on the violet line of = 4050Å of the next higher order. If the angle ofλ
diffraction is 300 how many lines per cm are there in the grating?
9. A telescope of aperture 5.0cm views a wire gauge from a distance of 50m. What is the smallest structure in the wire gauge which it can show clearly? The wavelength of the illuminating light is 5.0×10-5cm.
10.A microscope is used to resolves two self luminous objects separated by distance of 4×10-5cm.
(a) If the wavelength of light is 5461Å, calculate the numerical aperture (N.A.) of the objective. (b) Find the angle subtended at the eye by there objects when viewed from the minimum distance of distinct vision as 25cm.(e) calculate the minimum overall magnifying power of the microscope to see two resolved images, taking the resolving power of the eye to be 1.5 minutes.
11.A diffraction grating is just able to resolve two lines of = 5140.34Å and5140.85Å in the firstλ
order. Will it resolve the lines 8037.20Å and 8037.50Å in the second order?
TUTORIAL # 7 Polarization UNIT III
1. The refractive index of glass is 1.5. Calculate the polarization angle for it. Also calculate the angle of refraction.
2. The critical angle of light in a certain substance is 400. What is the polarizing angle for it?
3. If the plane of vibration of the incident beam makes an angle 300 with the optic axis, compare
the intensities of ordinary and extraordinary rays.
4. Two Nicol’s are crossed to each other. Now one of them is rotated through 600. What
percentage of incident unpolarised light will pass through the system?
5. A tube of sugar solution 20cm long is placed between crossed Nicol’s and illuminated with light of wavelength 6×10-5cm. If the optical rotation produced is 130 and the specific rotation is
650/dm/gm/cm3, determine the strength of the solution.
6. A certain length of 5% solution causes the optical rotation of 200. How much length of 10%
solution of the same substance will cause 350 rotations?
7. 80 gm of impure sugar is dissolved in 1 litre of water. The solution gives an optical rotation of 9.90 when placed in a tube of length 20cm. If the specific rotation of pure sugar solution is
660/dm/gm/cm3, find the percentage purity of sugar sample.
8. The specific rotation of quartz at 5086Åis 29.73 deg/mm. Calculate the difference in refractive indices.
TUTORIAL # 8 LASER UNIT IV
1. Find the intensity of a laser beam of 10mW power and having a diameter of 1.3mm. Assume the intensity to be uniform across the beam.
2. Coherence length for sodium light is 2.945x10-2 and the wavelength is 5890Ao. Calculate
a. The number of oscillations corresponding to the coherent length and b. Coherence time.
3. A laser beam of pulse power 1012 watt is focused on an object of area 10-4cm2. What is the
energy flux in watt/ cm2 at the point of focus?
4. A certain ruby laser emits 0.8 J pulses of light. There are 4x1018 ions of cr3+ in the ruby laser.
5. Assume that a continuous perfectly monochromatic laser beam of wavelength 623.8 nm is chopped into 10-10s pulses. Calculate resultant coherence length and band width.
6. Write short notes on
a. Spontaneous absorption b. Stimulated emission c. Optical pumping d. Population inversion e. Resonant Cavity
7. Explain the construction and working principle of Ruby laser. 8. Describe the construction and working principle of He-Ne laser.
9. Compare and contrast the ruby laser and He-Ne laser. In what ways would semiconductor laser be considered superior over solid/ gas lasers?
10. Define coherence and hence explain spatial and temporal coherence. Deduce expression for coherence length and coherence time.
11. Define coherence and hence explain spatial and temporal coherence. Deduce expression for coherence length and coherence time.
