Chapter #30 Solutions - Optics - Ajoy Ghatak - 1st Edition

Chapter #29 Solutions - Optics - Ajoy Ghatak - 1st Edition

Chapter #28 Solutions - Optics - Ajoy Ghatak - 1st Edition

Chapter #27 Solutions - Optics - Ajoy Ghatak - 1st Edition

Chapter #26 Solutions - Optics - Ajoy Ghatak - 1st Edition

Chapter #25 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. (a) Calculate the number of photons emitted per second by a 5 mW laser assuming that it emits light of wavelength 6328 Å.[Ans: 1.6 × 1016](b) The beam is allowed to fall normally on a plane mirror. Calculate the force acting on the mirror.[Ans: 3.3 × 10 – 11 N] Get solution

2. Assume a 40 W sodium lamp (λ ≈ 5893Å) emitting light in all directions. Calculate the rate at which the photons cross an unit area placed normally to the beam at a distance at a distance of 10 m from the source.[Ans: ≈ 1017 photons/m2-sec] Get solution

3. In the photoelectric effect, a photon is completely absorbed by the electron. Show that the laws of conservation of energy and momentum cannot be satisfied simultaneously if a free electron is assumed to absorb the photon. (Thus the electron has to be bound to an atom and the atom undergoes a recoil when the electron is ejected. However, since the mass of the atom is much larger than that of the electron, the atom picks up only a small fraction of the energy, this is somewhat similar to the case of a tennis ball hitting a heavy object, the momentum of the ball is reversed with its energy remaining almost the same.) Get solution

4. If photoelectrons are emitted from a metal surface by using blue light, can you say for sure that photoelectric emission will take place with yellow light and with violet light? Get solution


Chapter #24 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Show that in the limit of ...(i.e. at normal incidence) the reflection coefficient is the same for parallel and perpendicular polarizations. Get solution

2. Consider a magnetic dielectric with a permeability such that ... Show that for such a material the reflection coefficient for normal incidence is identically equal to zero. This realization is equivalent to the situation where the impedance is matched at the junction of two transmission lines. (The quantity ...can be considered as the intrinsic impedance of the medium.) Get solution

3. A right –circularly polarized beam is incident on a perfect conductor at 45°. Show that the reflected beam is left –circularly polarized. Get solution

4. Assume n1 = 1.5 and n2 = 1.0 (see Example 24.6)(a) for θ1 = 45° show that ...Similarly calculate r⊥ and t⊥.(b) On the other hand, for θ1 = 33.69° show that ... Get solution

5. Consider a right – circularly polarized beam incident on a medium of refractive index 1.6 at an angle of 60°. Calculate ... and ... and show that the reflected beam is right elliptically polarized with its major axis much longer than its minor axis. What will happen at 58°? [ Ans. ... = -0.0249, ...= -0.4581] Get solution

6. Consider a y-polarized wave incident on a glass –air interface ...at ...and at .... Write the complete expressions for the transmitted field and show that in the latter case it is an evanescent wave with depth of penetration ...equal to about 8.8 ×10 –8 m; assume λ = 6000Å. Get solution

7. For gold, at ... the complex refractive index is given by n2 = 0.166 + 3.15i. Calculate ... and show that the reflectivity at normal incidence is approximately 94%. [ Hint : Use Eq. (75) directly]. On the other hand at ... show that the reflectivity is only 39%. Get solution

8. Show that for ..., Eq. (97) takes the form ...   (98)as it indeed should be. Get solution

9. Using the various equations in Sec. 24.4 calculate the transmittivity and show that... Get solution

10. Assume the third medium in Fig. 24.14 to be identical to the first medium, i.e., ...Thus...Using Eq. (97), show that...   (99)where...   (100)is called the coefficient of finese. Equation (99) is identical to the result derived in Sec. 16.2 while discussing the theory of the Fabry – Perot interferometer. Get solution

11. When the angle of incidence is equal to the Brewster’s angle, show that ... [as given by Eq. (21)] is equal to unity. Get solution


Chapter #23 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. On the surface of the earth we receive about 1.37 kW of energy per square meter from the sun. Calculate the electric field associated with the sunlight (on the surface of the earth) assuming that it is essentially monochromatic with l= 6000Å.[Ans. ~1000 V/m] Get solution

2. (a) On the surface of the earth we receive about 1370 W m-2 of energy. Show that the radiation pressure is about 4.6 μPa (1 Pa = 10-5 N m-2 ).(b) A 100 W sodium lamp (λ ≈ 5890 Å) is assumed to emit waves uniformly in all directions. What is the radiation pressure on a plane mirror at distance of 10 m from the bulb? Get solution

3. A 1 kW transmitter is emitting electromagnetic waves of (of wavelength 40 m) uniformly in all directions. Calculate the electric field at a distance of 1 km from the transmitter. Get solution

4. Ocean water can be assumed to be a non –magnetic dielectric with ... and ...mhos/m. (a) Calculate the frequency at which the penetration depth will be 10 cm. (b) Show that for frequencies less than 108 s-1, it can be considered as a good conductor.[Ans (a) ~6×106 s–1] Get solution

5. For silver one may assume ...and ...mhos/m. Calculate the skin depth at 108 s–1.[Ans. ...cm] Get solution

6. Show that for frequencies ≲ 108 sec-1, a sample of silicon will act like a good conductor. For silicon one may assume ...and ...mhos/cm. Also calculate the penetration depth for this sample at .... Get solution

7. In a conducting medium show that H also satisfies an equation similar to Eq. (94). Get solution

8. Using the analysis given in Sec. 23.7 and assuming ... (which is valid for an insulator) show that... and... where... Get solution

9. For the glass used in a typical optical fiber at ...Å, ......mhos/m. Calculate ...and show that we can use the formulae given in the previous problem. Calculate β and loss in dB/km. [Hint : the power would decrease as exp (–2 βz); loss in dB/km is defined in Sec. 24.8][Ans. ...m-1 ; loss ≈ 3.7 dB/km] Get solution


Chapter #22 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Discuss the state of polarization when the x and y components of the electric field are given by the following equations:(a) ... (b) ...(c) ... (d)   ... In each case, plot the rotation of the tip of the electric vector on the plane z = 0.[Ans: (a) Linearly polarized, (b) Right-circularly polarized, (c) Left-circularly polarized, and (d) Left-elliptically polarized.] Get solution

2. The electric field components of a plane electromagnetic wave are Ex = 2E0 cos (ωt – kz + ϕ) ; Ey = E0 sin (ωt – kz)Draw the diagram showing the state of polarization (i.e., circular, plane, elliptical or unpolarized) when (a) ϕ = 0  (b) ϕ = π/2  (c) ϕ = π/4 Get solution

3.  Using the data given in Table 22.1, calculate the thickness of quartz half wave plate for λ0 = 5890Å.            [Ans: 32.34 μm] Get solution

4. A right-circularly polarized beam is incident on a calcite half-wave plate. Show that the emergent beam will be left-circularly polarized. Get solution

5. What will be the Brewster angle for a glass slab (n = 1.5) immersed in water (n = 4/3).          [Ans: 48.4°] Get solution

6. Consider the normal incidence of a plane wave on a quartz quarter wave plate whose optic axis is parallel to the surface (see Fig. 22.24). Thus the optic axis is along the z-axis and the propagation is along the x-axis. Show that Ey propagates as an o-wave and Ez as an e-wave.(a) Assuming... at x = 0 show that the emergent light would be right circularly polarized.(b) On the other hand, if one assumes... at x = 0 show that the emergent beam is linearly polarized. Get solution

7. Show that the angle between the vectors D and E is the same as between the Poynting vector S and the propagation vector k. Get solution

8. Consider the propagation of an extra-ordinary wave through a KDP crystal. If the wave vector is at an angle of 45° to the optic axis, calculate the angle between S and k. Repeat the calculation for LiNbO3. The values of no and ne for KDP and LiNbO3 are given in Table 22.1.        [Ans: 1.56° and 2.25°] Get solution

9. Prove that when the angle of incidence corresponds to the Brewster angle, the reflected and refracted rays are at right angles to each other. Get solution

10. (a) Consider two crossed polaroids placed in the path of an unpolarized beam of intensity I0 (see Fig. 22.6). If we place a third polaroid in between the two then, in general, some light will be transmitted through. Explain this phenomenon. (b) Assuming the pass axis of the third polaroid to be at 45° to the pass axis of either of the polaroids, calculate the intensity of the transmitted beam. Assume that all the polaroids are perfect.          [Ans: 1/8 I0] Get solution

11. A quarter-wave plate is rotated between two crossed polaroids. If an unpolarized beam is incident on the first polaroid, discuss the variation of intensity of the emergent beam as the quarter-wave plate is rotated. What will happen if we have a half-wave instead of a quarter-wave plate? Get solution

12. In Problem 22.11, if the optic axis of the quarter-wave plate makes an angle of 45° with the pass axis of either polaroid, show that only a quarter of the incident intensity will be transmitted. If the quarter-wave plate is replaced by a half-wave plate, show that half of the incident intensity will be transmitted through. Get solution

13. ...For calcite, the values of no and ne for λ0 = 4046Å are 1.68134 and 1.49694 respectively; corresponding to λ0 = 7065Å the values are 1.65207 and 1.48359 respectively. We have a calcite quarter-wave plate corresponding to λ0 = 4046Å. A left-circularly polarized beam of λ0 = 7065Å is incident on this plate. Obtain the state of polarization of the emergent beam. Get solution

14. A HWP (half wave plate) is introduced between two crossed polaroids P1 and P2. The optic axis makes an angle 15° with the pass axis of P1 as shown in Fig. 22.39(a) and (b). If an unpolarized beam of intensity I0 is normally incident on P1 and if I1, I2, and I3 are the intensities after P1, after HWP and after P2 respectively then calculate I1/I0, I2/I0 and I3/I0.[Ans: ½, ½, ⅛] Get solution

15. Two prisms of calcite (no > ne) are cemented together as shown in Fig. 22.40, so as to form a cube. Lines and dots show the direction of the optic axis. A beam of unpolarized light is incident normally from region I. Assume the angle of the prism to be 12°. Determine the path of rays in regions II, III & IV indicating the direction of vibrations (i.e., the direction of ).... Get solution

16. A λ/6 plate is introduced in between the two crossed polarizers in such a way that the optic axis of the λ/6 plate makes an angle of 45° with the pass axis of the first polarizer (see Fig. 22.41). Consider an unpolarized beam of intensity I0 to be incident normally on the polarizer. Assume the optic axis to be along the z-axis and the propagation along the x-axis. Write the y and z components of the electric fields (and the corresponding total intensities) after passing through (i) P1 (ii) λ/6 plate and (iii) P2 .... Get solution

17. A beam of light is passed through a polarizer. If the polarizer is rotated with the beam as an axis, the intensity I of the emergent beam does not vary. What are the possible states of polarization of the incident beam? How to ascertain its state of polarization with the help of the given polarizer and a QWP? Get solution

18. Consider a Wollaston prism consisting of two similar prisms of calcite (no = 1.66 and ne = 1.49) as shown in Fig. 22.29, with angle of prism now equal to 25°. Calculate the angular divergence of the two emerging beams. Get solution

19. (a) Consider a plane wave incident normally on a calcite crystal with its optic axis making an angle of 20° with the normal [see Fig. 19.18(a)]. Thus ψ = 20°. Calculate the angle that the Poynting vector will make with the normal to the surface. Assume no ≈ 1.66 and ne ≈ 1.49. (b) In the above problem assume the crystal to be quartz with no ≈ 1.544 and ne ≈ 1.553.         [Ans: (a) 4.31°] Get solution

20. Consider the incidence of he following REP beam on a sugar solution at z = 0:Ex = 5 cos ωt ; Ey = 4 sin ωt with λ = 6328Å. Assume  nl – nr = 10 –5 and nl = 4/3 study the evolution of the SOP of the beam. Get solution

21. Consider the incidence of the above REP beam on an elliptic core fiber with  ... 1.506845 and ... 1.507716 Calculate the SOP at z = 0.25 Lb, 0.5 Lb, 0.75 Lb and Lb. Get solution

22. When the optic axis lies on the surface of the crystal and in the plane of incidence, show (by geometrical considerations) that the angles of refraction of the ordinary and the extra-ordinary rays (which we denote by ro and re respectively) are related through the following equation:  ... Get solution

11&12. Get solution


Chapter #21 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Consider the reconstruction of the hologram as formed in the configuration of Example 21.2 by a plane wave traveling along a direction parallel to the z-axis. Show the formation of a virtual and a real image. Get solution

2. In continuation of Example 21.2, calculate the interference pattern when the incident plane wave makes an angle θ with the z-axis [see Fig. 14. 13]. Assume B ≈ A/d.[Ans: ...] Get solution

3. Figure 21.6 corresponds to the reconstruction of a doubly exposed hologram, the objects corresponding to the unstrained and strained positions of an aluminum bar of width 4 cm, thickness 0.2 cm and length 12 cm. If the strained position corresponds to a load of 1 gm force applied at the end of the bar, calculate the Young’s modulus of aluminum. Assume θ1 ≈ θ2 ≈ 0 and λ = 6328 Å. [Hint: N represents the number of fringes produced over the length of the cantilever.][Ans: 0.7 × 10 11 N/m2] Get solution


Chapter #20 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Consider a plane wave of wavelength 6 × 10-5 cm incident normally on a circular aperture of radius 0.01 cm. Calculate the positions of the brightest and the darkest points on the axis. Get solution

2. What would happen if the circular aperture in Problem 20.1 is replaced by a circular disc of the same radius? Get solution

3. A plane wave (λ = 6 × 10-5 cm) is incident normally on a circular aperture of radius a.(a) Assume a = 1 mm. Calculate the values of z (on the axis) for which maximum intensity will occur. Plot the intensity as a function of z and interpret physically. Repeat the calculations for λ = 5 × 10-5 cm and discuss chromatic aberration of a zone plate.(b) Assume z = 50 cm. Calculate the values of a for which minimum intensity will occur on the axial point. Plot the intensity variation as a function of a and interpret physically. Get solution

4. Consider a circular aperture of diameter 2 mm illuminated by a plane wave. The most intense point on the axis is at a distance of 200 cm from the aperture. Calculate the wavelength.     [Ans: 5 × 10-5 cm] Get solution

5. If a zone-plate has to have a principle focal length of 50 cm corresponding to λ = 6 × 10-5 cm, obtain an expression for the radii of different zones. What would be its principle focal length for λ = 5 × 10-5 cm? [...mm, 60 cm] Get solution

6. In a zone-plate, the second, fourth, sixth…zones are blackened; what would happen if instead the 1st, 3rd, 5th, etc., zones were blackened? Get solution

7. (a) A plane wave is incident normally on a straight edge (see Fig. 20.24). Show that the field at an arbitrary point P is given by...where ....(b) Assume λ0 = 5000 Å and d = 100 cm. Write approximately the values of I/I0 at the points O, P (y = 0.5 mm), Q (y = 1 mm) and R (y = -1 mm) where O is at the edge of the geometrical shadow.... Get solution

8. Consider a straight edge being illuminated by a parallel beam of light with λ = 6 × 10-5 cm. Calculate the positions of the first two maxima and minima on a screen at a distance of 50 cm from the edge. Get solution

9. In a straight edge diffraction pattern, one observes that the most intense maximum occurs at a distance of 1 mm from the edge of the geometrical shadow. Calculate the wavelength of light, if the distance between the screen and the straight edge is 300 cm.   [Ans. ≈ 4480 Å] Get solution

10. In a straight edge diffraction pattern, if the wavelength of the light used is 6000 Å and if the distance between the screen and the straight edge is 100 cm, calculate the distance between the most intense maximum and the next maximum. Find approximately the distance in centimeters inside the geometrical shadow where I /I0 = 0.1.[Ans.y ≈ 0.027 cm] Get solution

11. Consider a plane wave falling normally on a narrow slit of width 0.5 mm. If the wavelength of light is 6 × 10-5 cm, calculate the distance between the slit and the screen so that the value of v1 would be 0.5, 1.0, 1.5 and 5.0 (see Fig. 20.19 – 20.22). Discuss the transition to the Fraunhofer region. Get solution

12. Consider the Fresnel diffraction pattern produced by a plane wave incident normally on a slit of width b. Assume λ = 5 × 10-5 cm, d = 100 cm. Using Table 20.1, approximately calculate the intensity values (for b = 0.1 cm) at y = 0, ± 0.05 cm, ± 0.1 cm. Repeat theanalysis for b = 5 cm. Get solution

13. In Sec. 19.7 we obtained the diffraction pattern of a circular aperture of radius a. Obtain the diffraction pattern of an annular aperture bounded by circles of radii a1 and a2 (> a1). [This Problem is already given as Problem 19.5]. Get solution

14. Consider a rectangular aperture of dimensions 0.2 mm × 0.3 mm. Obtain the positions of the first few maxima and minima in the Fraunhofer diffraction pattern along directions parallel to the length and breadth of the rectangle. Assume λ = 5 × 10-5 cm and that the diffraction pattern is produced at the focal plane of a lens of focal length 20 cm. Get solution

15. The Fraunhofer diffraction pattern of a circular aperture (of radius 0.5 mm) is observed on the focal plane of a convex lens of focal length 20 cm. Calculate the radii of the first and the second dark rings. Assume λ = 5.5 × 10-5 cm.[Ans. 0.13 mm, 0.18 mm] Get solution

16. In the above problem, calculate the area of the patch (on focal plane) which will contain 95% of the total energy. Get solution

17. (a) The output of a He-Ne laser (λ = 6328 Å) can be assumed to be Gaussian with plane phase front. For w0 = 1 mm and w0 = 0.2 mm, calculate the beam diameter at z = 20 m.(b) Repeat the calculation for λ = 5000 Å and interpret the results physically. Get solution

18. A Gaussian beam is coming out of a laser. Assume λ = 6000 Å and that at z = 0, the beam width is 1 mm and the phase front is plane. After traversing 10 m through vacuum what will be (a) the beam width and (b) the radius of curvature of the phase front.... Get solution

19. A plane wave of intensity I0 is incident normally on a circular aperture as shown in Fig. 20.25. What will be the intensity on the axial point P? [Hint: You may use Eq. (25)] Get solution

20. Show that a phase variation of the type ...represents a diverging spherical wave of radius R. Get solution

21. Consider a resonator consisting of a plane mirror and a concave mirror of radius of curvature R (see Fig. 20.26). Assume λ = 1 μm, R = 100 cm and the distance between the 2 mirrors to be 50 cm. Calculate the spot size of the Gaussian beam.... Get solution

22. The output of a semiconductor laser can be approximated described by a Gaussian function with two different widths along the transverse (wT) and lateral (wL) directions as...where x and y represent axes parallel and perpendicular to the junction plane. Typically wT ≈ 0.5 μm and wL = 2 μm. Discuss the far field of this beam (see Fig. 20.27).... Get solution


Chapter #19 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Consider a rectangular aperture of dimensions 0.2 mm × 0.3 mm with a screen placed at a distance of 100 cm from the aperture. Assume a plane wave with λ = 5 × 10-5 cm incident normally on the aperture. Calculate the positions of maxima and minima in a region 0.2 cm × 0.2 cm of the screen. Show that both Fresnel and Fraunhofer approximations are satisfied. Get solution

2. In Problem 19.1 assume a convex lens (of focal length 20 cm) placed immediately after the aperture. Calculate the positions of the first three maxima and minima on the x-axis (implying ϕ = 0) and also on the y-axis (implying θ = 0). Get solution

3. The Fraunhofer diffraction pattern of a circular aperture (of radius 0.5 mm) is observed on the focal plane of a convex lens of focal length 20 cm. Calculate the radii of the first and the second dark rings. Assume λ = 5.5 × 10-5 cm.[Ans. 0.13 mm, 0.18 mm] Get solution

4. In Problem 19.3, calculate the area of the patch (on focal plane) which will contain 95% of the total energy. Get solution

5. Obtain the diffraction pattern of an annular aperture bounded by circles of radii a1 and a2 (> a1). [Hint: The integration limits of ρ in Eq. (35) must be a1 and a2] Get solution


Chapter #18 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. A plane wave (λ = 5000 Å) falls normally on a long narrow slit of width 0.5 mm. Calculate the angles of diffraction corresponding to the first three minima. Repeat the calculations corresponding to a slit width of 0.1 mm. Interpret physically the change in the diffraction pattern[ Ans. 0.057°, 0.115°, 0.17°; 0.29°, 0.57°, 0.86°] Get solution

2. A convex lens of focal length 20 cm is placed after a slit of width 0.6 mm. If a plane wave of wavelength 6000 Å falls normally on the slit, calculate the separation between the second minima on either side of the central maximum.[ Ans. ≈ 0.08cm] Get solution

3. In Problem 18.2 calculate the ratio of the intensity of the principal maximum to the first maximum on either side of the principal maximum.[ Ans. ~ 21] Get solution

4. Consider a laser beam of circular cross-section of diameter 3 cm and of wavelength 5×10-5 cm. Calculate the order of the beam diameter after it has traversed a distance of 3 km.[ Ans. ~ 14 cm. This shows the high directionality of laser beams] Get solution

5. A circular aperture of radius 0.01 cm is placed in front of a convex lens of focal length of 25 cm and illuminated by a parallel beam of light of wavelength 5×10-5 cm. Calculate the radii of the first three dark rings.[Ans. 0.76, 1.4, 2.02 mm] Get solution

6. Consider a plane wave incident on a convex lens of diameter 5 cm and of focal length 10 cm. If the wavelength of the incident light is 6000 Å, calculate the radius of the first dark ring on the focal plane of the lens. Repeat the calculations for a lens of same focal length but diameter 15 cm. Interpret the results physically.[Ans. 1.46 × 10-4 cm, 4.88 × 10-5 cm] Get solution

7. Consider a set of two slits each of width b = 5 × 10-2 cm and separated by a distance d = 0.1 cm, illuminated by a monochromatic light of wavelength 6.328 × 10-5 cm. If a a convex lens of focal length 10 cm is placed beyond the double slit arrangement, calculate the positions of the maxima inside the first diffraction minimum.[Ans. 0.0316 mm, 0.094 mm] Get solution

8. Show that when b = d, the resulting diffraction pattern corresponds to a slit of width 2b. Get solution

9. Show that the first order and second order spectra will never overlap when the grating is used for studying a light beam containing wavelength components from 4000 Å to 7000Å. Get solution

10. Consider a diffraction grating of width 5 cm with slits of width 0.0001 cm separated by a distance of 0.0002 cm. What is the corresponding grating element? How many orders would be observable at λ = 5.5 × 10-5 cm? Calculate the width of principal maximum. Would there be any missing orders? Get solution

11. For the diffraction grating of Problem 16.10, calculate the dispersion in the different orders. What will be the resolving power in each order? Get solution

12. A grating (with 15,000 lines per inch) is illuminated by white light. assuming that white light consist of wavelengths lying between 4000 and 7000 Å, calculate the angular widths of first and the second order spectra. [ Hint : You should not use Eq. (65); why] Get solution

13. A grating (with 15,000 lines per inch) is illuminated by sodium light. The grating spectrum is observed on the focal plane of a convex lens of focal length 10 cm. Calculate the separation between the D1 and D2 lines of sodium. (The wavelengths of D1 and D2 lines are 5890 and 5896 Å respectively.) [Hint : You may use Eq. (65).] Get solution

14. Calculate the resolving power in the second order spectrum of a 1 inch grating having 15,000 lines. Get solution

15. Consider a wire grating of width 1 cm having 1000 wires. Calculate the angular width of the second order principal maximum and compare the value with the one corresponding to a grating having 5000 lines in 1 cm. Assume λ = 5.5 × 10-5 cm Get solution

16. In the minimum deviation position of a diffraction grating the first order spectrum corresponds to an angular deviation of 30°. If λ = 6 × 10-5 cm, calculate the grating element. Get solution

17. Calculate the diameter of a telescope lens if a resolution of 0.1 seconds of arc is required at λ = 6 × 10-5 cm. Get solution

18. Assuming that the resolving power of the eye is determined by diffraction effects only, calculate the maximum distance at which two objects separated by a distance of 2 m can be resolved by the eye. (Assume pupil diameter to be 2 mm and λ = 6000 Å.) Get solution

19. (a) A pinhole camera is essentially a rectangular box with a tiny pinhole in front. An inversted image of the object is formed on the rear of the box. Consider a parallel beam of light incident normally on the pinhole. If we neglect diffraction effects then the diameter of the image will increase linearly with the diameter of the pinhole. On the other hand, if we assume Fraunhofer diffraction, then the diameter of the first dark ring will go on increasing as we reduce the diameter of the pinhole. Find the pinhole diameter for which the diameter of the geometrical image is approximately equal to the diameter of the first dark ring in the Airy pattern. Assume λ = 6000 Å and a separation of 15 cm between the pinhole and the rear of the box.[Ans. (a) 0.47mm] Get solution

20. Copper is an FCC structure with lattice constant 3.615 Å. An X-ray powder photograph of copper is taken. The X-ray beam consists of wavelengths 1.540 Å and 1.544 Å. Show that diffraction maxima will be observed at θ = (21.64°, 21.70°), (25.21°, 25.28°), (37.05°, 37.16°), (44.94°, 45.09°), (47.55°, 47.71°), (58.43°, 58.67°), (68.20°, 68.58°), (72.29°, 72.76°). Get solution

21. Tungsten is a BCC structure with lattice constant 3.1648 Å. Show that in the powder photograph of tungsten (corresponding to an X-ray wavelength of 1.542 Å) one would observe diffraction maxima at θ = 20.15°, 29.17°, 36.64°, 43.56°, 50.39°, 57.55°, 65.74° and 77.03°. Get solution

22. (a) In the simple cubic structure if we alternately place Na and Cl atoms we would obtain the NaCl structure. Show that the Na atoms (and the Cl atoms) independently form FCC structures. The lattice constant associated with each FCC structure is 5.6402 Å. Corresponding to the X-ray wavelength 1.542 Å, show that the diffraction maxima will be observed at θ = 13.69°, 15.86°, 22.75°, 26.95°, 28.97°, 33.15°, 36.57°, 37.69°, 42.05°, 45.26°, 50.66°, 53.98°, 55.10°, 59.84°, 63.69°, 65.06°, 71.27°, 77.45° and 80.66°.(b) Show that if we treat NaCl as a simple cubic structure with lattice parameter 2.82 Å then the maxima at θ = 13.69°, 26.95°, 36.57°, 45.26°, 53.98°, 63.69° and 77.45° will not be observed. Indeed in the X-ray diffraction pattern of NaCl, the maxima corresponding to these angles will be very weak. Get solution

23. Show that the mth order reflection from the planes characterized by (hkl) can be considered as the same as the first order reflection from the planes characterized by (mh mk ml). Get solution

24. Calculate the Fraunhofer diffraction pattern produced by a double slit arrangement with slits of widths b and 3b, with their centers separated by a distance 6b. Get solution

25. Consider the propagation of a 1 kW laser beam (λ = 6943 Å, beam diameter ≈ 1 cm) in CS2. Calculate fd and fnl and discuss the defocusing (or focusing) of the beam. Repeat the calculations corresponding to a 1000 kW beam and discuss any qualitative differences that exist between the two cases. The data for n0 and n2 are given in Sec. 18.11. Get solution

26. The values of ...and ... for benzene are 1.5 and 0.6 ×10-10 C.G.S. units respectively. Obtain an approximate expression for the critical power. Get solution


Chapter #17 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. The orange Krypton like (λ = 6058 Å) has a coherence length of ~20 cm. Calculate the line width and the frequency stability.[ Ans. ~ 0.01 Å, ~ 1.5 × 10-6] Get solution

2. Laser linewidths as low as 20Hz have been obtained. Calculate the coherence length and the frequency stability. Assume λ = 6328 Å. Get solution

3. In Sec. 17.4 we had mentioned that the lateral coherence width of a circular source is 1.22λ/θ. It can be shown that for good coherence (i.e. for a visibility of 0.88 or better), the coherence width should be ƒ 0.3λ/θ. Assuming that the angular diameter of the sun is about 30′, calculate the distance between two pinholes which would produce a clear interference pattern.[ Ans. ~ 0.02 mm] Get solution

4. Calculate the distance at which a source of diameter 1 mm should be kept from a screen so that two points separated by a distance of 0.5 mm may be said to be coherent. Assume λ = 6×10-5 cm. Get solution

5. In a Michelson interferometer experiment, it is found that for a source S, as one of the mirrors is moved away from the equal path length position by a distance of about 5 cm, the fringes disappear. What is the coherence time of the radiation emerging from the source? Get solution

6. If we perform the Young’s double-hole experiment using white light, then only a few coloured fringes are visible. Assuming that the visible spectrum extends from 4000 to 7000 Å, explain this phenomenon qualitatively on the basis of coherence length. Get solution

7. Using the stellar interferometer, Michelson observed for the star Betelgeuse, that the fringes disappear when the distance between the movable mirrors is 25 inches. Assuming λ ≈6×10-5 cm, calculate the angular diameter of the star. Get solution

8. Consider Young’s double-hole experiment as shown in Fig. 17.5. The distance SS1 ≈ 1 m. Calculate the angular diameter of the hole S which will produce a good interference pattern on the screen. Assume λ = 6000 Å. Get solution

9. Assume a Gaussian pulse of form... Show that the Fourier transform is given by... You will have to use the following integral [see Appendix A]...Show that the temporal coherence is ~τ. Assume τ >> (1/ω0), plot the Fourier transform A(ω) [as a function of ω] and interpret it physically. Show that the frequency spread Δω ~1/τ. Get solution

10. In Problem 17.9, assume λ0 = 6×10-5 cm and τ ~10-9 sec. Calculate the frequency components predominantly present in the pulse and compare it with the case corresponding to τ ~10-6 sec. Get solution


Chapter #16 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Calculate the resolving power of a Fabry-Perot interferometer made of reflecting surfaces of reflectivity 0.85 and separated by a distance 1 mm at λ = 4880 Å. Get solution

2. Calculate the minimum spacing between the plates of a Fabry-Perot interferometer which would resolve two lines with Δλ = 0.1 Å at λ = 6000 Å. Assume the reflectivity to be 0.8. Get solution

3. Consider a monochromatic beam of wavelength 6000 Å incident (from am extended source) on a Fabry-Perot etalon with n2 = 1, h = 1 cm and F = 200. Concentric rings are observed on the focal plane of a lens of focal length 20 cm(a) Calculate the reflectivity of each mirror.(b) Calculate the radii of the first four bright rings. What will be the corresponding value of m?(c) Calculate the angular width of each ring where the intensity falls by half and the corresponding FWHM (in mm) of each ring. Get solution

4. Consider now two wavelengths 6000 Å and 5999.9 Å incident on a Fabry-Perot etalon with the same parameters as given in the previous problem. Calculate the radii of the first three bright rings corresponding to each wavelength. What will be the corresponding values of m? Will the lines be resolved? Get solution

5.  Consider a monochromatic beam of wavelength 6000 Å incident normally on a scanning Fabry-Perot interferometer with n2 = 1 and F = 400. The distance between the two mirrors is written as h = h0 + x. With h0 = 10 cm, calculate(a) The first three values of x for which we will have unit transmittivity and the corresponding value of m.(b) Also calculate the FWHM Δh for which the transmittivity will be half.(c) What would be the value of Δh if F was 200?[Ans: (a) x ≈ 200 nm (m = 333334),500 nm(m = 333335); (b) Δh ≈ 8 nm]. Get solution

6. In continuation of Problem16.5, consider now two wavelengths λ0 (= 6000 Å) and λ0 + Δλ incident normally on the Fabry-Perot interferometer with n2 = 1, F = 400 and h0 = 10 cm. What will be the value of Δλ so that T = ½ occurs at the same value of h for both the wavelengths.... Get solution

7. Consider a laser beam incident normally on the Fabry-Perot interferometer as shown in Fig 16.15.(a) Assume h0 = 0.1 m, c = 3 x 108 m/s, ν = ν0 = 5 x 1014 s-1. Plot T as a function of x (–100 nm x F = 200 and F = 1000.(b) Show that if ν = (ν0 ± p 1500 MHz; p = 1,2,…) we will have the same T vs. x curve; 1500 MHz isknown as the free spectral range (FSR). What will be the corresponding values of δ? Get solution


Chapter #15 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. A glass plate of refractive index 1.6 is in contact with another glass plate of refractive index 1.8 along a line such that a wedge of 0.5° is formed. Light of wavelength 5000 Å is incident vertically on the wedge and the film is viewed from the top. Calculate the fringe spacing. The whole apparatus is immersed in an oil of refractive index 1.7. What will be the qualitative difference in the fringe pattern and what will be the new fringe width? Get solution

2. Two plane glass plates are placed on top of one another and on one side a cardboard is introduced to form a thin wedge of air. Assuming that a beam of wavelength 6000 Å is incident normally, and that there are 100 interference fringes per centimeter, calculate the wedge angle. Get solution

3. Consider a non-reflecting film of refractive index 1.38. Assume that its thickness is 9 × 10–6 cm. Calculate the wavelengths (in the visible region) for which the film will be non-reflecting. Repeat the calculations for the thickness of the film to be 45 × 10–6 cm. Show that both the films will be non – reflecting for a particular wavelength but only the former one will be suitable. Why? Get solution

4. In the Newton’s rings arrangement, the radius of curvature of the curved side of the plano –convex lens is 100 cm. For λ = 6 × 10–5 cm what will be the radii of the 9th and 10th bright rings? Get solution

5. In the Newton’s rings arrangement, the radius of curvature of the curved surface is 50 cm. The radii of the 9th and 16th dark rings are 0.18 cm and 0.2235 cm. Calculate the wavelength. [Hint: The use of Eq. (66) will give wrong results, why?) Get solution

6. In the Newton’s rings arrangement, if the incident light consists of two wavelengths 4000Å and 4002Å calculate the distance (from the point of contact) at which the rings will disappear. Assume that the radius of curvature of the curved surface is 400 cm. Get solution

7. In Problem 15.6 if the lens is slowly moved upward, calculate the height of the lens at which the fringe system (around the center) will disappear. Get solution

8. An equiconvex lens is placed on another equiconvex lens. The radii of curvature of the two surfaces of the upper lens are 50 cm and those of the lower lens are 100 cm. The waves reflected from the upper and lower surface of the air film (formed between the two lenses) interfere to produce Newton’s rings. Calculate the radii of the dark rings. Assume λ = 6000 Å. Get solution

9. In the Michelson interferometer arrangement, if one of the mirrors is moved by a distance 0.08 mm, 250 fringes cross the field of view. Calculate the wavelength. Get solution

10. The Michelson interferometer experiment is performed with a source which consists of two wavelengths 4882 Å and 4886 Å. Through what distance does the mirror have to be moved between two positions of the disappearance of the fringes? Get solution

11. In the Michelson interferometer experiment, calculate the various values of θ′ (corresponding to bright rings) for d = 5 × 10–3 cm. Show that if d is decreased to 4.997 × 10–3 cm, the fringe corresponding to m = 200 disappears. What will be the corresponding values of θ′? Assume λ = 5 × 10–5 cm. Get solution


Chapter #14 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. In the Young’s double – hole experiment (see Fig 14.6), the distance between the two holes is 0.5 mm, λ = 5 × 10-5 cm and D = 50 cm. What will be the fringe width? Get solution

2. Figure 14.23 represents the layout of Lloyd’s mirror experiment. S is a point source emitting waves of frequency 6 × 1014 sec-1. A and B represent the two ends of a mirror placed horizontally and LOM represents the screen. The distances SP, PA, AB and BO are 1 mm, 5 cm, 5 cm and 190 cm respectively. (a) Determine the position of the region where the fringes will be visible and calculate the number of fringes. (b) Calculate the thickness of a mica sheet (n = 1.5) which should be introduced in the path of the direct ray so that the lowest fringe becomes the central fringe. The velocity of light is 3 × 1010 cm/sec.... Get solution

3. (a) In the Fresnel’s biprism arrangement, show that d = 2 (n –1) aαwhere a represents the distance from the source to the base of the prism (see Fig. 14.19), α is the angle of the biprism and n the refractive index of the material of the biprism.(b) In a typical biprism arrangement b/a = 20 and for sodium light (λ = 5893 Å) one obtains a fringe width of 0.1 cm; here b is the distance between the biprism and the screen. Assuming n = 1.5, calculate the angle α. Get solution

4. In the Young’s double hole 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. Assume d = 0.1 cm, and D = 50 cm. Get solution

5. In order to determine the distance between the slits in the Fresnel biprism experiment, one puts a convex lens in between the biprism and the eye piece. Show that if D > 4 f one will obtain two positions of the lens where the image of the slits will be formed at the eye piece; here f is the focal length of the convex lens and D is the distance between the slit and the eye piece. If d1 and d2 are the distances between the images (of the slits) as measured by the eye piece, then show that .... What would happen if D f ? Get solution

6. In the Young’s double hole experiment, interference fringes are formed using sodium light which predominantly comprises of two wavelengths (5890 Å and 5896 Å). Obtain the regions on the screen where the fringe pattern will disappear. You may assume d = 0.5 mm and D = 100 cm. Get solution

7. If one carries out the Young’s double hole interference experiment using microwaves of wavelength 3 cm, discuss the nature of the fringe pattern if d = 0.1cm, 1cm and 4 cm. You may assume D = 100 cm. Can you use Eq. (21) for the fringe width? Get solution

8. In the Fresnel’s two mirror arrangement (see Fig. 14.18) show that the points S, S1 and S2 lie on a circle and S1S2 = 2bθ where b = MS and θ is the angle between the mirrors. Get solution

9. In the double hole experiment using white light, consider two points on the screen, one corresponding to a path difference of 5000 Å and the other corresponding to a path difference of 40000 Å. Find the wavelengths (in the visible region) which correspond to constructive and destructive interference. What will be the colour of these points? Get solution

10. (a) Consider a plane which is normal to the line joining two point coherent sources S1 and S2 as shown in Fig. 14.12. If S1P – S2P = Δ, then show that  ...   ...   where the last expression is valid for D >> d. (b) For λ = 0.5 μm, d = 0.4 mm and D = 20 cm; S1O – S2O = 800 λ. Calculate the value of S1P – S2P for the point P to be first dark ring and first bright ring. Get solution

11. In continuation of the above problem calculate the radii of the first two dark rings for (a) D = 20 cm and (b) D = 10 cm. Get solution

12. In continuation of Problem 14.10 assume that d = 0.5 mm, λ = 5 × 10 –5 cm and D = 100 cm. Thus the central (bright) spot will correspond to n = 1000. Calculate the radii of the first, second and third bright rings which will correspond to n = 999, 998 and n = 997 respectively. Get solution

13. Using the expressions for the amplitude reflection and transmission coefficients [see Eqs. (67)-(72) in Chapter 24], show that they satisfy Stokes’ relations. Get solution

14. Assume a plane wave incident normally on a plane containing two holes separated by a distance d. If we place a convex lens behind the slits, show that the fringe width, as observed on the focal plane of the lens, will be f λ / d where f is the focal length of the lens. Get solution

15. In Problem 14.14, show that if the plane (containing the holes) lies in the front focal plane of the lens, then the interference pattern will consist of exactly parallel straight lines. However, if the plane does not lie on the front focal plane, the fringe pattern will be hyperbolae. Get solution

16. In the Young’s double hole experiment calculate I / Imax where I represents the intensity at a point where the path difference is λ/5. Get solution


Chapter #13 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Standing wave are formed on a stretched string under tension of 1 N. The length of the string is 30 cm and it vibrates in 3 loops. If the mass per unit length of wire is 10 mg/cm, calculate the frequency of the vibrations. Get solution

2. In the above problem, if the string is made to vibrate in its fundamental mode, what will be the frequency of vibration? Get solution

3. In the experimental arrangement of Wiener, what should be the angle between the film and the mirror if the distance between two consecutive dark bands is 7 × 10–3 cm. Assume λ = 6 × 10–5 cm. Get solution

4. Standing wave with five loops are produced on a stretched string under tension. The length of the string is 50 cm and the frequency of vibrations is 250 sec–1 . Calculate the time variation of the displacement of the points which are at distances of 2 cm, 5 cm, 15 cm, 18 cm, 20 cm, 35 cm and 45 cm from one end of the string. Get solution

5. The displacements associated with two waves (propagating in the same direction) having same amplitude but slightly different frequencies can be written in the form... and ...(such displacements are indeed obtained when we have two tuning forks with slightly different frequencies.) Discuss the superposition of the displacements and show that at a particular value of x, the intensity will vary with time. Get solution

6. In the above problem assume v = 330 m/sec, ν = 256 sec –1, Δν = 2 sec –1 and a = 0.1 cm. Plot the time variation of the intensity at x = 0 , ...and .... Get solution

7. Use the complex representation to study the time variation of the resultant displacement at x = 0 in Problems 13.5 and 13.6. Get solution

8. Discuss the superposition of two plane waves (of the same frequency and propagating in the same direction) as a function of the phase difference between them. (Such a situation indeed arises when a plane wave gets reflected at the upper and lower surfaces of a glass slab; see Sec. 15.2). Get solution

9. In Example 11.1 we had discussed the propagation of a semicircular pulse on a string. Consider two semicircular pulses propagating in opposite directions. At t =0, the displacement associated with the pulses propagating in the +x and in the –x directions are given by...and ...respectively. Plot the resultant disturbance at t = R/v, 2.5 R/v, 7.5 R/v and 10 R/v; where v denotes the speed of propagation of the wave. Get solution


Chapter #12 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Use Huygens’ principle to study the reflection of a spherical wave emanating from a point on the axis at a concave mirror of radius of curvature R and obtain the mirror equation... Get solution

2. Consider a plane wave incident obliquely on the face of a prism. Using Huygens’ principle, construct the transmitted wavefront and show that the deviation produced by the prism is given by...where A is the angle of prism, I and t are the angles of incidence and transmittance. Get solution


Chapter #11 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. The displacement associated with a wave is given by(i) ...(ii) ...(iii) ...where in each case x and y are measured in centimeters and t in seconds. Calculate the wavelength, amplitude, frequency and the velocity in each case. Get solution

2. A transverse wave (λ = 15 cm, ν = 200 sec –1 ) is propagating on a stretched string in the +x –direction with an amplitude of 0.5 cm. At t = 0 the point x = 0 is at its equilibrium position moving in the upward direction. Write the equation describing the wave and if ρ = 0.1 g/cm, calculate the energy associated with the wave per unit length of wire. Get solution

3. Assuming that the human ear can hear in the frequency range 20 Get solution

4. Calculate the speed of longitudinal waves at NTP in (a) argon (γ = 1.67), (b) Hydrogen (γ =1.41).[Ans : (a) 308 m/s, (b) 1.26×105 cm/s] Get solution

5. Consider a wave propagating in the +x-direction with speed 100 cm/sec. The displacement at x =10 cm is given by the following equationy(x =10, t) = 0.5 sin (0.4t) where x and y are measured in centimeters and t in seconds. Calculate the wavelength and the frequency associated with the wave and obtain an expression for the time variation of the displacement at x = 0. Get solution

6. Consider a wave propagating in the – x-direction whose frequency is 100 sec –1. At t =5 sec the displacement associated with the wave is given by the following equation:y(x, t = 5) = 0.5 cos (0.1x)where x and y are measured in centimeters and t in seconds. Obtain the displacement (as a function of x) at t = 10 sec. What is the wavelength and the velocity associated with the wave? Get solution

7. Repeat the above problem corresponding toy(x, t = 5) = 0.5 cos (0.1x) + 0.4 sin (0.1x + π/3) Get solution

8. A Gaussian pulse is propagating in the +x-direction and at t = t0 the displacement is given by...Find y(x, t). Get solution

9. A sonometer wire is stretched with a tension of 1 N. Calculate the velocity of transverse waves if ρ =0.2 g/cm. Get solution

10. The displacement associated with a three –dimensional wave is given by...Show that the wave propagates along a direction making an angle 30° with the x –axis. Get solution

11. Obtain the unit vector along the direction of propagation for a wave, the displacement of which is given by...where x, y and z are measured in centimeters and t in seconds. What will be the wavelength and the frequency of the wave?... Get solution


Chapter #10 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Using the empirical formula given by Eq.(14) calculate the phase and group velocities in silica at λ0 = 0.7μm ,0.8μm, 1.0μm, 1.2μm and 1.4μm. Compare with the (more accurate) values given in Table 10.1. Get solution

2. For pure silica we may assume the empirical formula...where λ0 is measured in μm.(a) Calculate the zero dispersion wavelength.(b) Calculate the material dispersion at 800 nm in ps/km.nm.[1.32 μm; -101 ps/km.nm] Get solution

3. Let...where λ0 is the free space wavelength. Derive expressions for phase and group velocities.[Ans: vg = c/n0] Get solution

4. Consider a LED source emitting light of wavelength 850 nm and having a spectral width of 50 nm. Using Table 10.1 calculate the broadening of a pulse propagating in pure silica.[Ans: 4.2 ns/km] Get solution

5. In 1836 Cauchy gave the following approximate formula to describe the wavelength dependence of refractive index in glass in the visible region of the spectrum...Now (see also Table 12.2)  n(λ1) = 1.50883 ; n(λ2) = 1.51690 for borosilicate glass  n(λ1) = 1.45640 ; n(λ2) = 1.46318 for vitreous quartzwhere λ1 = 0.6563 μm and λ2 = 0.4861 μm.(a) Calculate the values of A and B.(b) Using the Cauchy formula calculate the refractive index at 0.5890 μm and 0.3988 μm and compare with the corresponding experimental values:(i) (1.51124 and 1.52546) for borosilicate glass and(ii) (1.45845 and 1.47030) for vitreous quartz. Get solution

6. The refractive index variation for pure silica in the wavelength region 0.5 μm 0 ...where C0 = 1.4508554, C1 = – 0.0031268, C2 = – 0.0000381, C3 = 0.0030270, C4= –0.0000779, C5 = 0.0000018, l = 0.035 and λ0 is measured in μm. Calculate and plot n(λ0) and d2n/dλ02 in the wavelength domain 0.5 0 Get solution

7. (a) For a Gaussian pulse given by...the spectral width is approximately given by...Assume λ0 = 8000 Å. Calculate ...for τ0 =1 ns and for τ0 = 1 ps.(c) For such a Gaussian pulse, the pulse broadening is given by ... where .... Using Table 8.1, calculate Δτ and interpret the result physically. Get solution

8. As a Gaussian pulse propagates the frequency chirp is given by...(a) where p is defined in Eq. (50). Assume a 100 ps (= τ0) pulse at λ0 = 1 μm. Calculate the frequency chirp ...at t – z/vg = –100 ps, –50 ps, +50 ps and +100 ps. Assume z = 1 km and other values from Table 8.1. Get solution

9. Repeat the previous problem for λ0 = 1.5 μm ; the values of τ0 and z remain the same. Discuss the qualitative difference in the results obtained in the previous problem. Get solution

10. The frequency spectrum of E(0,t) is given by the function A(ω). Show that the frequency spectrum of E(z,t) is simply ...implying that no new frequencies are generated – different frequencies superpose with different phases at different values of z. Get solution

11. The time evolution of a Gaussian pulse in a dispersive medium is given by...where .... Calculate explicitly the frequency spectrum of E(0,t) and E(z,t) and show that the results agree with that of the Problem 10.10. Get solution


Chapter #9 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Consider the Gaussian function ...Using Eq.(21 ) show that .... Plot ...for a = 2 and σ = 1.0, 5.0 and 10.0. Hence show that ...     (46)which is the Gaussian representation of the delta function. Get solution

2. Consider the ramp function defined by the following equation ...   (47)Show that  ..., where ...is the rectangle function defined by Eq.(4).Taking the limit ...show that ...where ...is the unit step function. Thus we get the following important result:If a function has a discontinuity of ...then its derivative (at x = a) is .... Get solution

3. Consider the symmetric function...Show that... Get solution

4. Consider the function  ... Calculate its Fourier spectrum ...and evaluate approximately ... Evaluate f(t)using the expression for F(ω). Get solution

5. Calculate the Fourier transform of the following functions(a)   ...(b) ...In each case make an estimate of ...and interpret physically. Get solution

6. Show that the convolution of two Gaussian functions is another Gaussian function:... Get solution


Chapter #8 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Consider a periodic force of the form:F(t) = F0sin ωtfor0 t T/2 = 0forT/2 t Tand   F(t + T) = F(t)where  ω = 2 π/TShow that  ...One obtains a periodic voltage of the above form in a half wave rectifier. What will be the Fourier expansion corresponding to full wave rectification? Get solution

2. In quantum mechanics, the solution of the one dimensional Schrödinger equation for a free particle is given by  ...where p is the momentum of the particle of mass m. Show that   ... Get solution

3. In continuation of the above problem, if we assume...then show that   ...Also show that   ...Indeed ...dx represents the probability of finding the particle between x and x + dx and ...dp represents the probability of finding the momentum between p and p + dp and we would have the uncertainty relation... Get solution

4. Get solution

4a. Use Eq. (54) to calculate the Fourier transform of the following functions(a) f(t) = A ...(b) f(t) = A ... t > 0  = 0       t Get solution


Chapter #7 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. The displacement in a string is given by the following equation:...where a, λ and ν represent the amplitude, wavelength and the frequency of the wave. Assume a = 0.1 cm, λ = 4 cm, ν = 1 sec –1. Plot the time dependence of the displacement at x = 0, 0.5 cm, 1.0 cm, 1.5 cm, 2 cm, 3 cm and 4 cm. Interpret the plots physically. Get solution

2. The displacement associated with a standing wave on a sonometer is given by the following equation:...If the length of the string is L then the allowed values of λ are 2L, 2L/2, 2L/3, … (see Sec. 13.2). Consider the case when λ = 2L/5; study the time variation of displacement in each loop and show that alternate loops vibrate in phase (with different points in a loop having different amplitudes) and adjacent loops vibrate out of phase. Get solution

3. A tunnel is dug through the earth as shown in Fig. 7.15. A mass is dropped at the point A along the tunnel. Show that it will execute simple harmonic motion. What will the time period be?... Get solution

4. A 1 g mass is suspended from a vertical spring. It executes simple harmonic motion with period 0.1 sec. By how much distance had the spring stretched when the mass was attached? Get solution

5. A stretched string is given simultaneous displacement in the x- and y- directions such that...Study the resultant displacement (at a particular value of z) as a function of time. Get solution

6. In the above problem, if...what will be the resultant displacement? Get solution

7. As mentioned in Sec.7.5, alkali metals are transparent to ultraviolet light. Assuming that the refractive index is primarily due to the free electrons and that there is one free electron per atom, calculate ...for Li, K and Rb; you may assume that the atomic weights of Li, K and Rb are 6.94, 39.10 and 85.48 respectively; the corresponding densities are 0.534, 0.870 and 1.532 g/cm3. Also, the values of various physical constants are: m = 9.109 × 10 – 31 kg, q = 1.602 × 10 – 19 C and ε0 = 8.854 × 10 – 12 C/N-m2.[Ans: 1550 Å, 2884 Å and 3214 Å; the corresponding experimentalvalues are 1551 Å, 3150 Å and 3400 Å respectively]. Get solution

8. (a) In a metal, the electrons can be assumed to be essentially free. Show that the drift velocity of the electron satisfies the following equation...  where ν represents the collision frequency. Calculate the steady state current density (J = – N q v) and show that the conductivity is given by   ...(b) If r represents the displacement of the electron, show that   ...which represents the polarization. Using the above equation show that   ...which represents the dielectric constant variation for a free electron gas. Get solution

9. Assuming that each atom of copper contributes one free electron and that the low frequency conductivity σ is about 6 ×107 mhos/meter, show that ν ≈ 4 × 10 13 s –1. Using this value of ν, show that the conductivity is almost real for ω 11 s –1. For ω = 10 8 s–1 calculate the complex dielectric constant and compare its value with the one obtained for infra-red frequencies.It may be noted that for small frequencies, only one of the electrons of a copper atom can be considered to be free. On the other hand, for X-ray frequencies all the electrons may be assumed to be free (see Problems 7.10, 7.11 and 7.12). Discuss the validity of the above argument. Get solution

10. Show that for high frequencies (ω ≫ ν) the dielectric constant (as derived in Problem 7.8) is essentially real with frequency dependence of the form...where ...is known as the plasma frequency. The above dielectric constant variation is indeed valid for X-ray wavelengths in many metals. Assuming that at such frequencies all the electrons an be assumed to be free, calculate ωp for copper for which the atomic number is 29, mass number is 63 and density is 9 g/cm3.        [Ans: ~ 9 × 10 16 sec – 1] Get solution

11. Obtain an approximate value for the refractive index of metallic sodium corresponding to λ= 1 Å. Assume all the electrons of sodium to be free. Get solution

12. In an ionic crystal (like NaCl, CaF2, etc.) one has to take into account infra-red resonance oscillations of the ions. Show that Eq. (68) modifies to...where M represents the reduced mass of the two ions and p represents the valency of the ion (p = 1 for Na+, Cl –; p = 2 for Ca++, F2– –). Show that the above equation can be written in the form*  ...where  ...  ... Get solution

13. The refractive index variation for CaF2 (in the visible region of the spectrum) can be written in the form...where λ is in meters.(a) Plot the variation of n2 with λ in the visible region.(b) From the values of A1 and A2 show that m/M ≈ 2.07 × 10– 5 and compare this with the exact value.(c) Show that the value of n∞ obtained by using the constants A1, A2, λ1 and λ2 agrees reasonably well with the experimental value. Get solution

14. (a) The refractive index of a plasma (neglecting collisions) is approximately given by (see Sec. 7.6)...where    ...s –1is known as the plasma frequency. In the ionosphere the maximum value of N0 is ≈ 10 10 – 10 12 electrons/m3. Calculate the plasma frequency. Notice that at high frequencies n2 ≈ 1; thus high frequency waves (like the one used in TV) are not reflected by the ionosphere. On the other hand, for low frequencies, the refractive index is imaginary (like in a conductor – see Sec. 24.3) and the beam gets reflected. This fact is used in long distance radio communications (see also Fig. 3.27).(b) Assume that for x ≈ 200 km, N = 10 12 electrons/m3 and that the electron density increases to 2 × 10 12 electrons/m3 at x ≈ 300 km. For x N, plot the corresponding refractive index variation. Get solution


Chapter #30 Solutions - Optics - Ajoy Ghatak - 1st Edition

1. Get solution 2. Get solution 3. Get solution 4. Get solution 5. Get solution