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45. Find the refractive index of the same prism for lithium light, the minimum deviation produced being 26° 30'.
46. The minimum deviation produced by a hollow prism filled with a certain liquid is 30°; if the refracting angle of the prism is 60°, what is the index of refraction of the liquid ?
47. A prism is to be made of crown glass, the refractive index of which is known to be 1.526, and it is required to produce a minimum deviation of 17° 20': to what angle must it be ground ?
48. A ray of light is incident almost perpendicularly upon a prism of angle a and refractive index u: show that if u is small the deviation is given by
d=(u - 1)a. 49. The refractive index of rock-salt is 1.54: what deviation would be produced by a rock-salt prism of 1° 30' angle ? and what should be the angle of a rocksalt prism which is required to produce a deviation of
50. Taking the usual values for the refractive indices of water and glass, prove that the deviations produced by thin prisms of water and glass are one-third and one-half of the angles of the prisms respectively. Note.—The refractive index for water and air is 4/3 (1.3), and
for glass and air 3/2 (or 1.5). 51. Find the angle of a water prism which will produce the same deviation as that given by a glass prism of 2° angle.
52. In order to determine the refractive index of a double convex lens, its focal length and the radii of curvature of its faces were measured, and were found to be
f= 30.6 cm., r1= 30.4 cm., ra= 34.5 cm. What was the index of refraction of the glass ?
It can be proved 1 that the focal length of a lens, in terms of
its refractive index and the radii of curvature of its surfaces,
is given by the formula 1/f=(u - 1)(1/rı - 1/12). In applying this we must remember that the radius of curva.
ture of the first surface is negative, or ??=;- 30•4; and since the lens is convex, its focal length is also negative. Thus - 1/30.6=(x - 1)(-1/30-4 - 1/34.5),
M - 1 = 30.4 x 34.5/30.6 x 64.9=0.528, and
..H= 1.528. 53. Taking the usual values for the refractive indices of water and glass, prove that the focal length of a glass lens when immersed in water is four times its focal length in air.
54. Prove that the focal length of a plano-concave glass lens is equal to twice the radius of the concave surface.
55. A double convex lens is to be made of glass of refractive index 1.5, and the radius of curvature of one of its faces is 20 cm. If the lens is to have focal length of 30 cm., what must be the radius of curvature of the other face?
56. The radii of curvature of the two faces of an equiconvex lens are each equal to 46.5 cm., and it is made of glass of refractive index 1.532 : show that its focal length is 43.7 cm.
57. Explain what is meant by the optical centre of a lens, and prove that the optical centre of a plano-convex or plano-concave lens lies on the curved surface.
58. A candle stands at a distance of 3 ft. from a wall : in what position must a convex lens of 8 in. focal length be placed between them so as to produce upon the wall a distinct image of the candle ? Let p denote the distance (in inches) of the candle from the
lens ; then 36 - p will be the distance of the screen from
1 Aldis, Geometrical Optics, Art. 67.
the lens irrespective of sign. Remembering that ţ is
- 1/8 = 1/(-36)-1),
p2 – 36 p +288=0. This may be written in the form
(-24) (-12)=0, and
.'. P=24 or 12. A distinct image will therefore be produced when the candle
is placed either 1 ft. or 2 ft. from the lens. 59. If an object at a distance of 3 in. from a convex lens has its image magnified three times, what is the focal length of the lens ? There are two solutions to this problem, for the image may
be either real or virtual. In the first case it is formed on the other side of the lens; in the second case on the same
side as the object. In both cases, since the image is three times as large as the
object, its distance from the lens must be three times that of the object ; but this only gives us the numerical value
of ' in terms of p. (1) Image real
p is negative and - 3p or - 9 in. Thus
1/9 - 1/3= -4/9, and the focal length of the lens is – 24 inches. (2) Image virtual —
p' is positive and=3p= +9 in.
1/f=1/9-1/3= - 2/9, so that the focal length in this case is – 43 inches. 60. Explain the difference between real and virtual images, and give examples of each. If you had a convergent lens of i ft. focal length, where would you place an object so as to produce by means of the lens (1) a real and diminished image, (2) an erect and virtual image? Give sketches showing how the image is produced in each case.
61. Rays of light diverging from a point 6 in. before a lens are brought to a focus 18 in. behind it: what is the focal length of the lens ?
62. An object is placed at a distance of 60 cm. from a convex lens of 15 cm. focal length : where is the image formed ? Compare its size with that of the object.
63. An object whose length is 5 cm. is placed at a distance of 12 cm. from a convex lens of 8 cm. focal length : what is the length of the image ?
64. A candle is placed at a distance of 10 ft. from a wall, and it is found that when a convex lens is held midway between the candle and the wall a distinct image is produced upon the latter. Find the focal length of the lens and the relative sizes of the object and image.
65. A coin half an inch in diameter is held on the axis of a convergent lens, and i ft. in front of it: if the focal length of the lens is 8 in., find the position and magnitude of the image.
66. Draw figures, approximately to scale, showing the paths of the rays of light, and the positions of the images formed when a luminous object is placed at a distance of (1) 1 inch, (2) 6 inches from a convergent lens of 2 in. focal length.
67. An object is placed 8 in. from a convex lens, and its image is formed 24 in. from the lens on the other side. If the object were placed 4 in. from the lens, where would the image be ?
68. The distance of an object from a convergent lens is double the focal length of the lens : prove that the image and object are of the same size.
69. A candle stands at a distance of 2 metres from a wall, and it is found that when a lens is held half a metre from the candle a distinct image is produced upon the wall: find the focal length of the lens, and also state the relative sizes of image and object.
70. A lens of 9 in. focal length is to be used for the purpose of producing an inverted image of an object magnified three times : where must each be situated ?
71. A convex lens is held 5 ft. in front of a wall, and it is found that there is one position in which an object can be held in front of the lens such that an inverted image six times as large as it is thrown upon the wall. Determine this position, and also find the focal length of the lens.
72. At what distance from a convex lens must an object be placed so that the image may be half the size of the object ?
73. You are provided with a convex lens of 18 in. focal length, and are required to place an object in such a position that its image will be magnified three times : find the positions which will give (1) a real, and (2) a virtual image of the required size.
74. In order to find the focal length of a concave lens, it was blackened, with the exception of a circle 4 cm. in diameter at its centre. A beam of sunlight was allowed to pass through this, when it was found that an illuminated circle of 20 cm. diameter was formed on a screen held 64 cm. behind the lens and parallel to it. What was the focal length of the lens ?
75. A convex lens produces a real image n times as large as the object : prove that the latter must be at a distance (n + 1)f/n from the lens.
76. A glass scale, 4 cm. long, was held in front of a convergent lens, and on holding a screen 90 cm. behind the lens, an image of the scale, 20 cm. in length, was produced upon the screen : prove that the lens had a focal length of 15 cm.
77. Show how to find the focal length (F) of a combination obtained by placing two thin lenses of focal