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a great distance, the image moves from C up to F, and is always real, inverted, and diminished. The most convenient way of finding the focal length is to find where
the image of a distant object is formed.
We learn, then, that a concave mirror has exactly the optical properties of a convex lens. This is because, like the convex lens, it always diminishes the curvature of
FIG. 422. Ray method of locating rea! image in a concave mirror
the waves. The same formulas hold throughout, and the same constructions are applicable (see Fig. 422).
449. Summary for lenses and spherical mirrors.* 1. Real images, inverted; virtual images, erect. The length of all images is given by
where L, and L; denote the length of object and image respectively, and D, and D; their distances from the lens or mirror. 2. Convex lenses and concave mirrors have the same optical properties (always diminish the curvature of the waves).
a. If object is more distant than principal focus, image is real and (1) enlarged when object is between principal focus and twice
(2) diminished when object is beyond two focal lengths.
b. If object is less distant than principal focus, image is virtual and always enlarged.
3. Concave lenses and convex mirrors have the same optical properties (always increase the curvature of the waves).
Image always virtual and diminished for any position of object.
1 1 D; f
* Laboratory experiments on the formation of images by concave mirrors and by lenses should follow this discussion. See, for example, Experiments
45 and 46 of the authors' Manual.
This formula may be used in all cases if the following points are borne in mind:
a. Do is always to be taken as positive.
b. D; is to be taken as positive for real images and negative for virtual images.
c. f is to be taken as positive for converging systems (convex lenses and concave mirrors) and negative for diverging systems (concave lenses and convex mirrors).
QUESTIONS AND PROBLEMS
1. Show from a construction of the image that a man cannot see his entire length in a vertical mirror unless the mirror is half as tall as he is. Decide from a study of the figure whether or not the distance of man from the mirror affects the case.
2. A man is standing squarely in front of a plane mirror which is very much taller than he is. The mirror is tipped toward him until it makes an angle of 45° with the horizontal. He still sees his full length. What position does his image occupy?
3. How tall is a tree 200 ft. away if the image of it formed by a lens of focal length 4 in. is 1 in. long? (Consider the image to be formed in the focal plane.)
4. How long an image of the same tree will be formed in the focal plane of a lens having a focal length of 9 in.?
5. What is the difference between a real and a virtual image?
6. When does a convex lens form a real, and when a virtual, image? When an enlarged, and when a diminished, image? When an erect, and when an inverted, one?
7. When a camera is adjusted to photograph a distant object, what change in the length of the bellows must be made to photograph a near object? Explain clearly why this adjustment is necessary.
8. Rays diverge from a point 20 cm. in front of a converging lens whose focal length is 4 cm. At what point do the rays come to a focus?
9. An object 2 cm. long was placed 10 cm. from a converging lens and the image was formed 40 cm. from the lens on the other side. Find the focal length of the lens and the length of the image.
10. An object is 15 cm. in front of a convex lens of 12 cm. focal length. What will be the nature of the image, its size, and its distance from the lens?
11. Why does the nose appear relatively large in comparison with the ears when the face is viewed in a convex mirror?
12. Can a convex mirror ever form an inverted image? Why?
450. The photographic camera. A fairly distinct, though dim, image of a candle flame can be obtained with nothing more elaborate than a pinhole in a piece of cardboard (Fig. 423). If the receiving screen is replaced by a photographic plate, the arrangement becomes a pinhole camera, with which good pictures may be taken if the exposure is sufficiently long. If we try to increase the brightness of the image by enlarging the hole, the
image becomes blurred, because the narrow pencils a,a',, a,a' etc. become cones whose bases a',, a',, overlap and thus destroy the distinctness of the outline.
It is possible, without sacrificing distinctness of outline, to gain the increased brightness due to the larger hole by placing
a lens in the hole (Fig. 424). FIG. 424. Principle of the photoIf the receiving screen is now a
sensitive plate, the arrangement
becomes a photographic camera (Fig. 425). But while with the pinhole camera the screen may be at any distance from the hole, with a lens the plate and the
object must be at conjugate foci of the lens.
Let a lens of, say, 4 feet focal length be placed in front of a hole in the shutter of a darkened room, and a semitransparent screen (for example, architect's tracing paper) placed at the focal plane. A per
fect reproduction of the opposite landscape FIG. 425. The photographic
451. The projecting lantern. The projecting lantern is essentially a camera in which the position of object and image have been interchanged; for in the use of the camera the object is at a considerable distance, and a small inverted image is formed on a plate placed somewhat farther from the lens than the focal distance. In the use of the projecting lantern the object P (Fig. 426) is placed a trifle farther from the lens L' than its focal length, and an enlarged inverted image is formed on
FIG. 426. The projecting lantern (stereopticon)
a distant screen S. In both instruments the optical part is simply, a convex lens, or a combination of lenses which is equivalent to a convex lens.
The object P, whose image is formed on the screen, is usually a transparent slide which is illuminated by a powerful light A. The image is as many times larger than the object as the distance from L' to S is greater than the distance from L' to P. The light A is usually either an incandescent lamp or an electric arc. The moving-picture projector employs a long film of small "positives" which moves swiftly between the condensing lens L and the projecting lens L' (see opposite p. 386).
The above are the only essential parts of a projecting lantern. In order, however, that the slide may be illuminated as brilliantly as possible, a so-called condensing lens L is always used. This concentrates light upon the transparency and directs it toward the screen.
In order to illustrate the principle of the instrument, let a beam of sunlight be reflected into the room and fall upon a lantern slide. When a lens is placed a trifle more than its focal distance in front of the slide, a brilliant picture will be formed on the opposite wall.
452. The eye. The eye is essentially a camera in which the cornea C (Fig. 427), the aqueous humor 7, and the crystalline lens o act as one single lens which forms an inverted image P'Q' on the retina, an expansion of the optic nerve covering the inside of the back of the eyeball.
FIG. 427. The human eye
In the case of the camera the images of objects at different distances are obtained by placing the plate nearer to or farther from the lens. In the eye, however, the distance from the retina to the lens remains constant, and the adjustment for different distances is effected by changing the focal length of the lens system in such a way as always to keep the image upon the retina. Thus, when the normal eye is perfectly
FIG. 428. The pupil dilates when the light is dim and contracts when it is intense
relaxed, the lens has just the proper curvature to focus plane waves upon the retina, that is, to make distant objects distinctly visible. But by directing attention upon near objects we cause the muscles which hold the lens in place to contract