only when light attempts to go from water into air and when the angle of incidence is greater than about 49 degrees. When light is totally reflected from a surface to the eye, the surface looks like a silvered mirror. For example, if one holds a tumbler of water at about the height of the top of his head and looks through the side of the glass at the underside of the surface of the water, it will look like a silvered surface. In that case it is totally reflected light that comes to the eye.* A more rigorous explanation of total reflection based on the sine law of refraction will now be given. In the case of air and water where the angle i, is the angle in air, and i2 is the angle in water. Whether the angle i, is the angle of incidence or the angle of refraction depends on which way the light is traveling: if the light is incident in the water, traveling out, i1 is the angle of refraction, and i2 the angle of incidence. The sine law may be written Suppose that light is incident inside the water at a known angle; that is, i2 is the angle of incidence and is known. To find the value of sin i1 one must look up the sine of i, and multiply it by four thirds. This works all right provided i2 is less than about 49 degrees. Since the sine of 49 degrees is three fourths (approximately), sini == 1. When the sine of an angle is equal to 1, the angle is equal to 90 degrees. Hence, if light is incident in the water at an angle of 49 degrees, it will be refracted at an angle of 90 degrees, or tangentially to the surface. But when the angle of incidence is greater than 49 degrees, sin i2, will be greater than three fourths, and sin i1, as computed by equation (15), will be greater than 1. But no angle can have a sine greater than unity; hence there can be no angle of refraction, there cannot be refraction, and the light must be totally reflected. For an air-water surface the angle of 49 degrees is the critical angle. In general, if the index of refraction is denoted by n, then * Total reflection can easily be shown by the aid of a glass tank (Fig. 379). When the angle i, is equal to 90 degrees, i2 is equal to the critical angle. For this case (sin i1 = 1), equation (16) becomes 594. Examples of total reflection. In the case of an air-water surface, total reflection takes place only when the incident light is inside the water. This is the case in the luminous jets sometimes seen in fountains. Fig. 392 shows one method of producing a luminous jet. Light enters through a glass window into a chamber through which water is flowing. As shown in the figure the light entering the water jet is totally reflected on the inside of the jet and follows it around a curve. There is enough light scattered or diffusely reflected in all directions from particles in the water to make the jet appear luminous. The totally reflecting prism is used in many kinds of optical instruments, because it gives reflections superior to those from a silvered mirror. As shown in Fig. 393, a section of such a prism is an isosceles right-angled triangle. Light falling perpendicularly to one face enters the prism without any bending of the beam and strikes the inclined face at an angle of incidence of 45 degrees. Since this angle is greater than the critical angle of glass, the light is totally reflected, and passes out through the third face, as shown in the figure. 595. The relative index and the absolute index. The index of refraction between two media, or the relative index, is defined by the following equation: Relative index of two media = speed in first medium speed in second medium Hence the sine law of refraction may be stated thus: = sin i1 where i1 is the angle the ray makes with the normal to the surface in the Accurate values given in tables for the indexes of substances are the absolute ones. However, as the speed of light in air differs so little from that in a vacuum, it is only in very accurate work that one needs to distinguish between them. In designing high-grade lenses it is necessary to know how much the light is bent in passing from one kind of glass to another. In that case one must know the relative index of refraction. But the tables give only the absolute indexes of different substances. How can one use the sine law and make the necessary computations? In answering this question not only will a method of computing the relative index be given but a more general form of the sine law will be stated. To make the discussion more concrete we shall take the case where light is traveling from water to glass. The absolute index of water is, by definition, 0 where Vo is the speed of light in a vacuum, and V1 is the velocity in the water. The absolute index of glass is (19) where V2 is the speed in glass. If equation (19) is divided by equation (18), we have n V 1. (20) But the ratio V1/V2 is the relative index for the two substances designated as 1 and 2; hence the relative index is equal to the inverse ratio of the absolute indexes of the two substances. The sine law of refraction can now be written in terms of the absolute indexes. From equations (14) and (20) we have where the n's are the absolute indexes. This may be written a form very easy to remember on account of the symmetry of the subscripts. Equation (21) may be called the general law of refraction. It is the equation that is used in computing the change in direction of light which is traveling from water to glass or from one kind of glass to another. 596. The prism. The prism is used so much in optics that the student should be certain that he understands why a ray follows the path shown in Fig. 394. The inci dent ray is bent toward the normal when it enters the prism and is bent away from the normal as it leaves. The result of this is that the angle of deviation (D in the figure) is large. 597. Dispersion. Whenever a narrow beam of light from any ordinary source D 12 12 FIG. 394 passes through a prism and then falls on a screen, a colored band, or spectrum, is produced (Fig. 395). To understand this three points must be clear: 1. It was proved by Sir Isaac Newton that sunlight is a mixture of different kinds of light. As will be explained later, these kinds differ in the fact that their wave-lengths, and hence their frequencies, are not the same. 2. Different kinds of light produce different color sensations in the eye. Color depends on the frequency of the light-vibrations. The lowest frequencies, or longest waves, of visible light produce the sensation of red; while the highest frequencies, or shortest waves, produce the violet sensation. The principal colors in the spectrum, in the order of their frequencies, are red, yellow, green, blue, and violet. 3. The velocity of light in glass depends on the frequency of vibration of the waves. Red light travels faster in glass than those kinds which have a higher frequency. Violet light travels slower than blue, blue slower than green, green slower than yellow, and yellow slower than red. FIG. 395 RGB Since the amount that light is bent by glass depends on the velocity in the glass, and since the velocity is different for different frequencies, it follows that when a beam of white light, which is a mixture of different frequencies, passes through a glass prism, the beam will be spread out as indicated in Fig. 395. Since the different frequencies produce different color sensations, the band formed on the screen is colored. The breaking up of a beam of light into its component parts is called dispersion. 598. Newton's experiment with colors. As stated in the last section, it was Newton who showed that sunlight is a mixture of different kinds of light. The method of his experiment was relatively simple. A narrow beam of sunlight, after passing through the prism P, (Fig. 396), was broken up into a spectrum on the screen |