the lens irrespective of sign. Remembering that is negative, we have pp-36, and now substituting in equation 2 (p. 155) we have

A distinct image will therefore be produced when the candle is placed either I 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 be either real or virtual. the other side of the lens; in the second case on the same side as the object.

this problem, for the image may In the first case it is formed on

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 p' in terms of p.

and the focal length of the lens is – 2 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 - 4 inches.

60. Explain the difference between real and virtual images, and give examples of each. If you had a convergent lens of 1 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 ʼn 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

lengths f1 and ƒ1⁄2 in contact.

Prove that for any number of such lenses placed in contact 1/F = X(1/f).

78. What is the focal length of a lens which is equivalent to two thin convergent lenses of focal lengths 15 cm. and 30 cm. placed in contact?

79. A concave lens of 8 cm. focal length is combined with a convex lens of 6 cm. focal length: what is the focal length of the combination?

80. A convex lens of focal length 16 cm, was placed in contact with a concave lens, and the focal length of the combination was found to be 48 cm. Calculate the focal length of the concave lens.

81. A candle is held 1 foot in front of a convex lens, and a distinct image of the flame is formed on a screen placed 4 inches behind it. A concave lens is now placed in contact with it, and it is found that the screen has to be moved 8 inches farther off in order to receive the image. What is the focal length of the concave lens?

82. Explain the action of a condensing lens when used as a magnifying glass. Give a sketch showing how the image is produced, and prove that the magnifying power is approximately equal to Alf, where A is the

distance of most distinct vision.

83. Describe the action of the eye, considered as an optical instrument, and explain the causes of abnormal vision. Will the magnifying effect of a given readinglens be greater when used by a long-sighted or a shortsighted person?

84. A person whose distance of most distinct vision is 20 cm. uses a lens of 5 cm. focal length as a readingglass at what distance from a book must he hold it? Also what will be its magnifying power?

85. A long-sighted person can only see distinctly objects which are at a distance of 48 cm. or more: by how much will he increase his range of distinct vision if he uses convex spectacles of 32 cm. focal length?

86. A short-sighted man can read printed matter distinctly when it is held at 15 cm. from his eyes: find the focal length of the glasses which he must use if he wishes to read with ease a book at a distance of 60 cm.

87. A convex lens produces an image of a candleflame upon a screen whose distance from the candle is 7; the lens is displaced through a distance d, when it is found that a distinct image is again produced upon the screen. Show that the focal length of the lens is (12 – d2)/4l.

88. Prove that the size of the object in the last question is a geometrical mean between the sizes of the two images produced.

89. In an experiment made according to the method of Ex. 82, the distance between the candle and screen was 255 cm. and the lens had to be shifted through a What was its focal length ?

distance of 73.7 cm.

90. Calculate the

mean value of the focal length of

a convex lens which gave the following results by the method of displacement :

91. Describe a method of measuring the velocity of light (a) in air, (b) in glass, its velocity in air being known. In using Foucault's method it was observed that when the mirror was turning 257 times per second the displacement of the image was 113 metre, the distance between the slit and the moving mirror being 8.58 metres, and between the two mirrors 605 metres. Show that the velocity of light is 296,000,000 metres per second. Camb. Schol. 1883. 92. State the principles on which the illuminating