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curvature from waves coming from F and render them plane, by subtracting the same curvature from the flatter waves from P it must render them concave; that is, the rays after passing through the lens are converging and intersect at P'. If the source is placed at P', obviously the rays will meet at P. Points such as P and P', so related that one is the image of the other, are called conjugate foci.


439. Formula for conjugate foci; secondary foci. Since in Fig. 405 the curvature of the wave when it emerges from the lens is opposite in direction to its curvature when it 1 1 reaches the lens, the sum of these curvatures, + , repreD; sents the power of the lens to change the curvature of the 1 incident wave, which by § 437 is. Hence


that is, the sum of the reciprocals of the distances of the conjugate foci from the lens is equal to the reciprocal of the focal length. If D ̧= D1, then the equation shows that both D and D are equal to 2ƒ.


1 1 1







The two conjugate foci S and S' which are at equal distances from the lens are called the secondary foci, and their distance from the lens is twice the focal distance.


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FIG. 406. Formation of a real image by a lens

440. Images of objects. Let a candle or electric-light bulb be placed between the principal focus F and the secondary focus S at PQ (Fig. 406), and let a screen be placed at P'Q'. An enlarged inverted image will be seen upon the screen.

This image is formed as follows: All the light which strikes the lens from the point P is brought together at a point P'. The location of this image P' must be on a straight line drawn from P through C; for any ray passing through C will remain parallel to its original direction, since the portions of the lens through which it enters and leaves may be regarded as small parallel planes (see § 423). The image P'Q' is therefore always formed between the lines drawn from P and Q through C. If the focal length ƒ and the distance of the object D. are known, the distance of the image D, may be obtained easily from formula (1).

The position of the image may also be found graphically as follows: Of the cone of rays passing from P to the lens, that

FIG. 407. Ray method of constructing an image

ray which is parallel to the principal axis must, by § 437, pass through the principal focus F. The intersection of this line with the straight line through C locates the image P' (see Fig. 407). Q', the image of Q, is located similarly.

441. Size of image. Since the image and object are always between the intersecting straight lines PP' and QQ', the similar triangles PCQ and P'CQ' show that

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distance of object from lens

distance of image from lens



may be seen from Fig. 407, as well as from formulas (1) and (2), that

1. When the object is at S the image is at S', and image and object are of the same size.


2. As the object moves out from S to a great distance the image moves from S' up to F", becoming smaller and smaller. 3. As the object moves from S up to F the image moves out to a very great distance to the right, becoming larger and larger.

4. When the object is at F the emerging waves are plane (the emerging rays are parallel), and no real image is formed.


FIG. 408. Virtual image formed by a convex lens



442. Virtual image. We have seen that when the object is at F the waves after passing through the lens are plane. If, then, the object is nearer to the lens than F, the emerging waves, although reduced in curvature, will still be convex, and, if received by an eye at E, will appear to come from a point P' (Fig. 408). Since, however, there is actually no source of light at P', this sort of image is called a virtual image. Such an image cannot be projected upon a screen as a real image can, but must be observed by an eye.

The graphical location of a virtual image may be accomplished precisely as in the case of a real image (§ 440). It will be seen that in this case (Figs. 408 and 409) the image is enlarged and erect.

FIG. 409. Ray method of locating a virtual image in a convex lens


FIG. 410. Virtual focus of a concave lens

443. Image in concave lens. When a plane wave strikes a concave lens, it must emerge as a divergent wave, since the middle of the wave is retarded by the glass less than the edges (Fig. 410). The point F from which plane waves appear to come after passing through such a lens is the principal focus of the lens. For the same

reason as in the case of the convex lens the centers of the transmitted waves from P and Q (Fig. 411), that is, the images P' and Q', must lie upon the lines PC and QC; and since the



FIG. 411. Image in a concave lens

FIG. 412. Ray method of locating
an image in a concave lens

curvature is increased by the lens, they must lie closer to the lens than P and Q. Fig. 411 shows the way in which such a lens forms an image. This image is always virtual, erect, and diminished. The graphical method of locating the image is shown in Fig. 412.


444. Image of a point in a plane or a curved mirror. We are all familiar with the fact that to an eye at E (Fig. 413), looking into a plane mirror mn, a pencil point at P appears to be at some point P' behind the mirror. We are able in the laboratory to find experimentally the exact location of this image P' with respect to P and the mirror, but we may also obtain this location from theory as follows: Consider a light wave which originates in the point P (Fig. 413) and spreads in all directions. Let aob be a section of the wave at the instant at which it reaches the reflecting surface mn. An instant later, if there were no reflecting surface, the wave would have reached the position of the dotted line cod.


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FIG. 413. Wave reflected
from a plane surface

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Since, however, reflection took place at mn, and since the reflected wave is propagated backward with exactly the same velocity with which the original wave would have been propagated forward, at the proper instant the reflected wave must have reached the position of the line co,d, so drawn that 00, is equal to 00. Now the wave co,d has its center at some point P', and it will be seen that P' must lie just as far below mn as P lies above it, for cod and cod are arcs of equal circles

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having the common chord cd. For the same reason, also, P' must lie on the perpendicular drawn from P through mn. When, then, a section of this reflected wave co,d enters the eye at E, the wave appears to have originated at P' and not at P, for the light actually comes to the eye from P' as a center rather than from P. Hence P' is the image of P. We learn, therefore, that the image of a point in a plane mirror lies on the perpendicular drawn from the point to the mirror and is as far back of the mirror as the point is in front of it.

Precisely the same construction applied to curved mirrors shows at once (Fig. 414 and Fig. 415) that the image of a point in any mirror, plane or curved, must lie on the perpendicular drawn from the point to the mirror; but if the mirror

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