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 