587. Spherical aberration. In the proof given in section 576, it was assumed that reflection took place only from a small part of the mirror near the center. It is true that sharp images can be formed by spherical mirrors only when a small portion of a mirror is used. The blurring which occurs when a relatively large portion is used is called spherical aberration. The effect of spherical aberration is readily demonstrated. An image of some bright, sharply defined object, such as the filament of an incandescent lamp, is formed on a screen by a concave mirror of moderate size. The mirror is moved until the sharpest image is obtained. If the mirror is now held still, and it is covered with a M FIG. 376 FIG. 377 piece of cardboard with a hole in it, the image becomes appreciably sharper. The remedy for spherical aberration is to cover up all the mirror except a relatively small portion. In Fig. 376 is shown the reflection of a parallel beam of light when a spherical mirror of large "aperture" is used. Only those rays reflected from a region lying near the center M converge to the principal focus; but when the mirror is parabolic, all the parallel beam converges to the principal focus (Fig. 377). If a small source is placed at the principal focus of the parabolic mirror, the reflected beam is a parallel beam. But parabolic mirrors are useful only in the case of parallel beams. If it is desired to form an accurate image of a small object near at hand by reflection from a large mirror, the surface of the mirror must be a portion of an ellipsoid (Fig. 378) with the object and image, F1 and F2, at the foci. It is one of the properties of the ellipsoid that light starting from one focus F2 will be converged to F1, no matter what part of 2 the surface reflects the light. But such a surface does not give sharp images for any other position of the object. Although the spherical mirror does have the defect of spherical aberration, it is better than any other form when it is desired for all sorts of use; that is, when the object is sometimes at one distance and sometimes at another. When the mirror is small compared with its radius of curvature, there is no practical difference in the shapes of the F F FIG. 378 parabolic, the ellipsoidal, and the spherical mirror; hence spherical mirrors of small aperture are accurate enough for all practical purposes. PROBLEMS 1. Find the position of the image formed by a concave mirror of radius 30 cm. when the object is placed (a) 60 cm. from the mirror; (b) 20 cm. from the mirror; (c) 10 cm. from the mirror. 2. Find by graphical construction the position of the image for the case of a concave mirror, radius of curvature 10 in., with an object 8 in. from the mirror. 3. Compute the position of the image formed by a convex mirror of radius 20 cm. when the object is (a) 40 cm. away; (b) 10cm. away. 4. Find the positions of the images of Problem 3 by graphical methods. 5. A concave mirror has a radius of curvature of 20 cm. An object 4 cm. high is placed 5 cm. from the mirror. Calculate (a) the position and (b) the size of the image. 6. An object 10 cm. high is placed 20 cm. from a convex mirror of radius 30 cm. Calculate (a) the position and (b) the size of the image. 7. It is desired to produce a real image 2 ft. high, of an object 4 in. high, by means of a concave mirror with a radius of 10 in. Where must the object be placed? 8. A concave mirror for shaving is designed to give a virtual image of one's face, at a distance of 15 in. from the eyes, with a magnification of 2. What is the radius of curvature? CHAPTER XLII REFRACTION Refraction, 588. The cause of refraction, 589. The sine of an angle, 590. Snell's law of refraction, 591. Proof of the sine law by the wave theory, 592. Total reflection; the critical angle, 593. Examples of total reflection, 594. The relative index and the absolute index, 595. The prism, 596. Dispersion, 597. Newton's experiment with colors, 598. Atmospheric refraction of light, 599. The mirage, 600. M 588. Refraction. When light passes from one medium into another, the rays are usually bent at the surface separating the two media. A simple method of illustrating this bending, or refraction, of light is to use a small tank, or aquarium, with sides and ends of glass. If a little fluorescein is added to the water in the tank, the path of the light through the water will be luminous. Fig. 379 indicates the appearance when a narrow beam of light is incident on the surface at an angle of about 45 degrees. As the beam enters the water it is bent toward the normal, or perpendicular, to the surface; that is, the angle of incidence is greater than the angle of refraction (the angle of refraction is the angle the refracted beam makes with a normal to the surface). As illustrated in the figure, the beam falls on a mirror M lying on the bottom of the tank, and is reflected back to the surface and again refracted. When the beam passes from water to air, it is bent away from the normal; that is, the angle of incidence is less than the angle of refraction. FIG. 379 When one looks at objects lying on the bottom of shallow water, they appear to be nearer the surface than they really are. In Fig. 380 let O1 be a stone or some other object. A ray of light from the stone will reach the eye at E by the path shown. Because 2 of the bending, the eye sees a virtual image of O1 at 11. An object O2, a little farther away, appears to be at 12; another, at O3, арpears to be at 13. It is very easy to observe in shallow water of uniform depth that the farther parts look shallower. This effect is shown in the figure by the fact that I2 is nearer the surface than 11, and I nearer than I2. In clear water one may actually wade into much deeper water thinking that he is going into more shallow water. 2 I 3 FIG. 380 E wave theory. It so happens that these two theories lead to opposite conclusions in explaining refraction. Hence it becomes possible, as we shall see, to choose between them. The corpuscular theory. According to the corpuscular theory, light consists of small particles, or corpuscles, which are shot off from a source at very high speeds. To explain refraction it seems necessary to suppose that these particles B A B A are attracted by matter, and, as they approach a denser medium, are given an acceleration toward the medium they are entering. In Fig. 381 is shown the path of a corpuscle as it enters a denser medium. The vectors A and B represent the tangential and perpendicular components of the velocity of the light-corpuscle before it enters the medium. A'and B'represent the components of the velocity of the light in the denser medium. The attraction of the medium for the corpuscle increases the perpendicular component of the velocity; hence B' is greater than B. But this attraction does not change the tangential component, so that A' is the same as A. This explanation leads to two conclusions: 1. Since the perpendicular velocity B' is greater than B, the resultant of A' and B' is a velocity which is different in direction FIG. 381 from the resultant of A and B; hence a change in the direction of the ray (refraction) must take place. 2. The resultant of A' and B' is greater in magnitude than the resultant of A and B; hence the velocity in the denser medium is greater than it is outside. This latter conclusion flatly contradicts that given by the wave theory, as we shall see later. B A A When a corpuscle emerges from the denser medium, the attraction causes a retardation of the corpuscle; hence it decreases the velocity and bends the path, as shown in Fig. 382. As before, the component tangential to the surface is not changed; that is, A' is equal to A. But owing to the attraction of the matter of the medium, B is less than B'. B The wave theory. According to the wave theory the refraction • as light passes from air to water or to glass is caused by a change in the velocity of the waves. To account for the observed direction of the bending, it is necessary to assume that light travels more slowly in water or glass than it does in air. In Fig. 383 is shown the effect of a change in the velocity of waves as they enter a medium where the velocity is less (see also section 323). In general, waves travel in directions perpendicular to themselves. Hence in the case shown in the figure the arrows, drawn perpendicular to the waves, indicate the directions in which the light is traveling. When light travels from one medium into another where its velocity is greater, it will be bent away from the normal (Fig. 384). We thus see that in order to explain refraction the corpuscular theory states that the velocity of light in water is greater than that in air, while the wave theory states that the velocity in water must be less than that in air. In 1850 Foucault announced that he had proved by direct experiment (sect. 567) that the velocity of light |