Chapter 31 The Law of Refraction

Transcription

1 If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics. Roger Bacon 31.1 Refraction In chapter 30, we saw that an incident ray of light is reflected from a piece of reflecting material such that the reflected ray makes the same angle with the normal as the incident ray. If the reflecting surface is a boundary between two different transparent mediums, such as air and glass, some of the incident light is also transmitted into the glass, as shown in figure However, it is observed Figure 31.1 Reflection and refraction of light. experimentally that this transmitted ray of light is bent as it enters the second medium. The bending of light as it passes from one medium into another is called refraction. Refraction of light occurs because light travels at different speeds in different mediums. Light traveling through a vacuum travels at the speed c = m/s. But when light enters a medium there is a complex interaction between the electromagnetic wave (light) and the atomic configuration of the medium. This interaction causes the electromagnetic wave to slow down in the medium. This slowing down of the wave as it goes from a vacuum into the medium causes it to bend. We will use Huygens principle to show how this is accomplished in section

2 31.2 The Law of Refraction Let us consider a wave front B 1 B 2 of a plane parallel monochromatic wave impinging on the boundary of two different mediums, as shown in figure The incident ray makes an angle of incidence i with the normal N. The incident light Figure 31.2 The law of refraction by Huygens principle. moves at a speed v 1 in medium 1 and v 2 in medium 2, and we assume that v 1 is greater than v 2. The incident wave has just touched the boundary at B 1. In a time t, B 2, the upper portion of the initial wave front, travels a distance v 1 t, and impinges at the boundary of the interface at B 2. In this same time interval t, the wave front at B 1 enters the second medium. By Huygens principle, a secondary wavelet can be drawn emanating from the point B 1. This wave moves a radial distance v 2 t in the second medium in the time interval t, and is shown as the circle of radius v 2 t in the figure. The radial distance v 2 t is less than the distance v 1 t because v 2 is less than v 1. By Huygens principle, the line drawn from B 2 that is tangent to the secondary wavelet is the new wave front. The point of tangency is denoted by B 1 and the new wave front in medium 2 is B 1 B 2.The radius from B 1 to B 1, when extended, becomes the refracted ray B 1 C. The other refracted rays are drawn parallel to B 1 C, as shown in figure The angle that the refracted ray makes with the normal is called the angle of refraction r. We can obtain the relation between the angles i and r from the geometry of figure Since line B 2 B 2 makes an angle i with the dashed normal, angle B 2 B 2 B 1 is equal to (90 0 i), and since the sum of the angles in triangle B 1 B 2 B 2 must equal 180 0, it follows that angle B 2 B 1 B 2 is also equal to the angle i. Using similar reasoning, angle B 1 B 2 B 1 is equal to r. Hence, from the trigonometry of figure 31.2, 31-2

3 and sin i = v 1 t (31.1) B 1 B 2 sin r = v 2 t (31.2) B 1 B 2 Let us divide equation 31.1 by equation 31.2 and obtain and sin i sin r = v 1 t B 1 B 2 v 2 t B 1 B 2 sin i sin r = v 1 v 2 = constant = n 21 (31.3) Equation 31.3 is the law of refraction. It says that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speed of light in medium 1 to the speed of light in medium 2. Because the speed of light in medium 1 v 1 is a constant and the speed of light in medium 2 v 2 is a constant, then their ratio v 1 /v 2, must also be a constant. This constant is called the index of refraction of medium 2 with respect to medium 1 and is denoted by n 21. If medium 1 is a vacuum, then v 1 = c, and the index of refraction of the medium with respect to a vacuum is n = c (31.4) v Since the speed v in any medium is always less than c, the index of refraction, n = c/v, is always greater than 1, except for in a vacuum where it is equal to 1. Indices of refraction for various substances are given in table Notice that the index of refraction of air is so close to the value 1, the index of refraction of a vacuum, that in many practical situations, air is used in place of a vacuum. The law of refraction can be put in a more convenient form by using equation We can write the index of refraction of medium 1 with respect to a vacuum as n 1 = c (31.5) v 1 whereas we can write the index of refraction of medium 2 with respect to a vacuum as n 2 = c (31.6) v

4 Table 31.1 Index of Refraction for Various Materials (λ = nm, the D line of sodium) n Substance Air Benzene Diamond Glass, crown Glass, flint Glass, fused quartz Glycerine Ice Plexiglass Quartz crystal Water Solving for the speeds v 1 and v 2 from equations 31.5 and 31.6, respectively, and substituting them into equation 31.3, gives n 21 = v 1 = c/n 1 = n 2 (31.7) v 2 c/n 2 n 1 Using equation 31.7, we can write the law of refraction, equation 31.3, as or sin i = n 21 = n 2 sin r n 1 n 1 sin i = n 2 sin r (31.8) Equation 31.8 is the form of the law of refraction that we will use in what follows. It is also called Snell s law after its discoverer, Willebrord Snell ( ) a Dutch mathematician who discovered it in 1620, the same year the Pilgrims landed at Plymouth Rock. Note that if a ray lies along the normal, then the angle of incidence i is equal to zero, and hence the angle of refraction r must also be zero, and there is no refraction of this ray. The fact that the speed of light varies from medium to medium has an important effect on the wavelength of light. When an initial wave enters a second medium, its wavelength changes. We can see this from equation 31.3, where n 21 = v 1 v 2 However, the speed of any wave is given by 31-4

5 v = λν (31.9) where λ is the wavelength and ν is the frequency of the wave. Hence, n 21 = v 1 = λ 1 ν = λ 1 (31.10) v 2 λ 2 ν λ 2 The frequency ν of the wave does not change as it goes across the boundary because there are the same number of wave fronts passing from medium 1 into medium 2, per unit time 1. Therefore, λ 2 = λ 1 = λ 1 = n 1 λ 1 (31.11) n 21 n 2 /n 1 n 2 That is, the wavelength of the light in the second medium λ 2 is less than the initial wavelength λ 1 by the factor 1/n 21 = n 1 /n 2. To summarize, the speed of light varies from one medium to another but the frequency of the light remains the same in both mediums. Because of the changing speed of light, the wavelength of the light changes as it goes into the second medium. Example 31.1 The refraction of light. A ray of light of nm wavelength in air, impinges on a piece of crown glass at an angle of incidence of Find (a) the angle of refraction, (b) the speed of light in the glass, and (c) the wavelength of light in the glass. Solution a. We find the angle of refraction from Snell s law, equation 31.8, with the indices of refraction n 1 = n air = 1.00 and n 2 = n glass = 1.52, found in table Therefore, and the angle of refraction is n 1 sin i = n 2 sin r sin r = n 1 sin i n 2 = 1.00 sin = r = sin r = That is, you can t have 5 waves per second incident on the boundary between the two mediums and only 3 waves per second transmitted through the boundary. Where did the other waves go? You must have the same number leaving the boundary as you have entering the boundary. Hence the frequency of the wave must be the same on both sides of the boundary. 31-5

6 b. The speed of light in the glass, found from equation 31.6, is v 2 = c = m/s n = m/s c. The wavelength of light in the glass, found from equation 31.11, is λ 2 = n 1 λ 1 n 2 = (1.00) (500.0 nm) 1.52 = 329 nm To go to this Interactive Example click on this sentence. The law of refraction has been derived by Huygens principle, treating light as a wave phenomenon. We will now simplify the analysis of refraction by using only the ray model of light. Thus, the refraction of waves in figure 31.2 will now be shown as the equivalent refraction of rays in figure The dashed line in figure 31.3(a) is the direction that the incident ray would have followed if it had not Figure 31.3 The law of refraction by a ray diagram. entered the second medium. Note that the incident ray was bent away from its original direction and bent toward the normal. Medium 1 has an index of refraction that is less than medium 2. The medium with the smaller value of the index of refraction is called the rarer medium, whereas the medium with the larger value of n is called the denser medium. Hence, whenever a ray of light goes from a rarer 31-6

7 medium to a denser medium the refracted ray is always bent toward the normal. The larger the value of the index of refraction n 2, the greater the amount of bending. By the principle of reversibility, a light ray that reverses the path in figure 31.3(a), goes from a denser medium to a rarer medium and is bent away from the normal, as seen in figure 31.3(b). Hence, whenever a ray of light goes from a denser medium to a rarer medium, the refracted ray is bent away from the normal Apparent Depth of an Object Immersed in Water An interesting example of the refraction of light is the observation of an object when it is under water. A stick immersed in water appears to be bent, figure 31.4(a), and a fish in water is not where it seems to be. Sometimes in a carnival or a bazaar, a game is played where a glass is placed in water, as in figure 31.4(b), and the (a) (b) Figure 31.4 (a) The stick appears bent because of refraction at the water s surface. (b) The apparent depth of an object immersed in water. patrons try to throw a quarter into the glass, receiving a prize if they are successful. Of course it is much more difficult than it appears because the glass is not where it seems to be. As an example, let the object, the quarter in the glass, be at the bottom of the water at Q, a distance p from the top of the water. A ray from Q makes an angle of incidence i with the normal. Because the ray is going from the more dense medium, water, to the less dense medium, air, the refracted ray is bent away from the normal, as shown. The observer s eye, located at R, sees the ray ROQ, but the person believes that light rays travel in straight lines, and thinks he sees the ray ROI. The person, therefore, assumes that the quarter in the glass is located at I, a distance q below the surface of the water. In a sense, you might say that he is 31-7

8 looking at the image of the glass. The vertical ray QN lies along the normal and hence is not refracted, as shown. These two rays QOR and QN do not intersect anywhere in medium 2, the air, but when produced backward, the ray ROI intersects QN at I, and in this sense, I appears as the image of Q. The glass appears to be much closer to the top of the water than it really is. The distance q from the top of the water to I is called the apparent depth of the object. Let us now solve for the apparent depth. From Snell s law of refraction, equation 31.8, we have n 1 sin i = n 2 sin r We determine the angles i and r from the geometry of figure 31.4 by observing that and tan i = x (31.12) p tan r = x (31.13) q To simplify the solution, let us assume that the angles i and r are small enough (less than 10 0 ) to use the small-angle approximation, that is, sin θ tan θ With this assumption, equations and can be substituted back into equation 31.8 as n 1 tan i = n 2 tan r n 1 x = n 2 x p q Solving for the apparent depth, q = n 2 p (31.14) n 1 Example 31.2 Apparent depth. A glass is placed in the bottom of a tub of water 25.0 cm deep. What is its apparent depth? Solution The apparent depth, found from equation 31.14, is 31-8

9 q = n 2 p n 1 = 1 (25.0 cm) 1.33 = 18.8 cm The glass appears to be 18.8 cm below the top of the water when in fact it is really 25.0 cm below the top. To go to this Interactive Example click on this sentence. In the solution of the problem of the apparent depth, we made an assumption, just as is often done in physics. The solution is, of course, only as good as the assumption, that is, that the angles i and r are less than Suppose they are not. What effect does this have on the solution? For example, if x in figure 31.4 is 12.5 cm then the angle i, found from equation 31.12, is i = tan 1 x = tan p 25.0 = The angle r, found from Snell s law, is and the angle of refraction is sin r = n 1 sin i n 2 = 1.33 sin r = This is obviously much larger than the assumed limit of The apparent depth, now found from equation 31.13, is q = x = 12.5 cm tan r tan = 16.9 cm If the calculation is repeated with x = 25.0 cm, the angle of incidence i is now equal to , and the angle of refraction r is now equal to and the apparent depth q is 9.05 cm. These values are significantly different from the values obtained with the initial assumptions. This is again a reminder that in trying to represent the physical world in terms of mathematical equations, certain assumptions are made. 31-9

10 When these assumptions are valid, the equations are good. When they are not valid the equations no longer represent the physical world and are essentially useless Refraction through Parallel Faces A ray of light passing through two parallel boundaries is refracted at each of these boundaries, but the final ray is in the same direction as the initial ray, as shown in figure An incident ray makes an angle i 1 with the normal N. Because medium Figure 31.5 Refraction through parallel faces. 2 is more dense than medium 1, the refracted ray is bent toward the normal in medium 2, making an angle of refraction of r 2 with the normal. Snell s law for the first interface becomes n 1 sin i 1 = n 2 sin r 2 (31.15) This refracted ray now becomes the incident ray for the interface between medium 2 and medium 1, and makes an angle i 2 with the normal N.Since normals N and N are parallel, angle r 2 is equal to angle i 2. As the ray goes from the more dense medium 2 to the less dense medium 1, the ray is refracted away from the normal through the angle r 1, as shown. Snell s law for the refraction at this interface is n 2 sin i 2 = n 1 sin r 1 (31.16) Because angle r 2 is equal to angle i 2, the right-hand side of equation is equal to the left-hand side of equation Equating them, we obtain 31-10

11 and n 1 sin i 1 = n 1 sin r 1 i 1 = r 1 Thus, the angle r 1 is equal to the initial angle i 1, and the final refracted ray comes out parallel to the direction of the original ray, but slightly displaced. Example 31.3 Refraction through parallel faces. A ray of light in air makes an angle of incidence of with the normal to a plate of glass of n = 1.50, as in figure Find the angle of refraction of the ray in the glass and the angle of refraction of the final ray as it passes from the glass back into the air. Solution The angle of refraction for the first interface, found from Snell s law, is n 1 sin i 1 = n 2 sin r 2 sin r 2 = n 1 sin i 1 n 2 = 1.00 sin = and the angle of refraction for the first interface becomes r 2 = sin 1 (0.4285) r 2 = For the second interface, the law of refraction is n 2 sin i 2 = n 1 sin r 1 sin r 1 = n 2 sin i 2 = 1.50 sin = n and the angle of refraction for the second interface is r 1 = sin 1 (0.6434) r 1 = Note that the final ray makes the same angle as the initial ray, and they are, therefore, parallel

12 To go to this Interactive Example click on this sentence Total Internal Reflection As shown in figure 31.1, when an incident ray falls on a boundary of a transparent material, part of the ray is reflected back into the first medium, while part of the ray is refracted through the second medium. If the incident ray is in the dense medium, then the refracted ray is bent away from the normal, as shown in figure 31.6(a). If the angle of incidence is increased, the angle of refraction is also increased, as shown in figure 31.6(b). If the angle of incidence is increased still Figure 31.6 Total internal reflection. further, the point is reached where the angle of refraction becomes 90 0, as shown in figure 31.6(c). The angle of incidence that causes the refracted ray to bend through 90 0 is called the critical angle of incidence. When the incident angle becomes greater than the critical angle, no refraction occurs. That is, no light enters the second medium at all; all the light is reflected. This condition is called total internal reflection, because refraction has been eliminated entirely. We obtain the condition for total internal reflection by finding the critical angle of incidence that allows the angle of refraction to be This is done by applying Snell s law of refraction: or n 1 sin i c = n 2 sin 90 0 sin i c = n 2 sin 90 0 = n 2 (31.17) n 1 n 1 Equation gives the condition for total internal reflection. Total internal reflection can only occur when light travels from a denser medium to a rarer medium, because it is only then that the refracted ray is bent 31-12

13 away from the normal. For a light ray going in the opposite direction, the refracted ray bends toward the normal and the angle of refraction could never become Example 31.4 Critical angle for a glass-air interface. What is the critical angle for a ray of light that goes from glass to air? Take n glass = Solution The critical angle, found from equation 31.17, is sin i c = n 2 = n air = 1.00 = n 1 n glass 1.50 i c = sin 1 (0.667) i c = To go to this Interactive Example click on this sentence. Let us now take an ordinary piece of glass, whose index of refraction is 1.50, and cut it into a triangle with angles of , , and , as shown in figure 31.7 An incident ray perpendicular to the first face goes directly into the glass Figure 31.7 Total internal reflection and the prism. without any refraction, because it is along the normal of the first face. This ray makes an angle of incidence of with the normal as it hits the second face. But is greater than the critical angle of just found in example Hence, none of this light crosses the interface into the air, but instead it is totally reflected at this second face. This reflected ray, being normal to the third face, completely 31-13

14 passes into the air at the third face without any refraction there. This ordinary piece of glass, cut into the shape of a triangle, with an angle greater than the critical angle, is called a prism. One of its functions is to completely reflect light. It is even better than a plane mirror for reflection because in a plane mirror some light energy is absorbed by the silver of the mirror, whereas the prism, acting as a device for total internal reflection, reflects everything. No energy is ever transmitted into the other medium at the interface. Prisms are used in binoculars, spectrometers, and reflecting telescopes. Example 31.5 Critical angle for a water-air interface. What is the critical angle between water and air? The index of refraction for water is n w = Solution The critical angle, found from equation 31.17, is sin i c = n 2 = n air = 1.00 n 1 n w 1.33 i c = The critical angle for a water-air interface is thus greater than the critical angle for a glass-air interface. To go to this Interactive Example click on this sentence. A further example of internal reflection can be found in the area of fiber optics. An optical fiber consists of a single flexible glass rod, or an array of them, of high refractive index. Light entering the glass undergoes total internal reflection from the walls of the glass fiber and the light travels down the length of the fiber with little or no absorption. Due to their flexibility, the fibers can be curved into various shapes. But as long as the reflection angle remains greater than the critical angle, the light travels along the length of the fiber Dispersion In all the previous discussions of refraction we assumed that the incident light was monochromatic, that is, it contained light of only one wavelength. If white light, a mixture of all colors and hence numerous wavelengths, impinges on a transparent surface, we obtain the refraction pattern shown in figure The white light is 31-14

15 Figure 31.8 Dispersion of light. refracted such that each color, or wavelength, is refracted by a different angle. Hence, the white light is dispersed into its constituent wavelengths. The separation of white light into its component colors is called dispersion. The band of colors is known as a spectrum. The spectrum of visible light contains the colors red, orange, yellow, green, blue, and violet. Violet light, which has the shortest wavelength, is bent the most, whereas red light with the longest wavelength, is bent the least. Dispersion occurs because the index of refraction is not strictly a constant for a particular material, but rather varies slightly with wavelength. For example, the index of refraction of crown glass for red light (700.0 nm) is about 1.51, whereas for violet light (400.0 nm) it is about 1.53, which is only a slight difference, yet significant enough to cause the phenomenon of dispersion. Example 31.6 Refraction for various wavelengths. White light makes an angle of incidence of as it strikes a piece of glass. Find the angle of refraction for (a) red light (700.0 nm) if n = 1.51 for red light and (b) for violet light (400.0 nm) if n = Solution a. The angle of refraction for red light, found from Snell s law, is n 1 sin i = n 2 sin r r sin r r = n 1 sin i = 1.00 sin = n r r = sin 1 (0.331) r r = b. For violet light, the angle of refraction is n 1 sin i = n 2 sin r v 31-15

16 sin r v = n 1 sin i = 1.00 sin = n r v = sin 1 (0.3267) r v = Notice that the total angular separation between the red and violet light is only 0.2 degrees for this glass. To go to this Interactive Example click on this sentence. Dispersion is most pronounced if a prism is used, as shown in figure Because of the angle between the two faces of the prism, the deviation D of the final refracted ray is greater. Figure 31.9 Dispersion by a prism. Because the index of refraction is a function of the incident wavelength, when defining the index of refraction of a medium, we need to specify the particular wavelength used. When the wavelength is not specified in this book it is assumed that the wavelength is λ = nm, corresponding to the yellow D line of the sodium spectrum Thin Lenses An optical lens is a piece of transparent material, such as glass or plastic. Because of its shape, however, light passing through it either converges or diverges to its principal axes. The entire effect of the lens is due to its shape and the index of refraction of the lens. One such shape of a lens is formed from a piece of plane glass by shaping it into the form of two spherical surfaces with radii of curvature R 1 and R 2, as shown in figure 31.10, where C 1 and C 2 are the centers of the two spherical surfaces. This shaped piece of glass is called a double convex lens, because the shape of its surfaces are convex

17 Figure Formation of a lens. The line going through the center of the lens is called the principal axis and is shown in figure Consider a ray of light AB, parallel and close to the principal Figure Refraction by a lens. axis and impinging on the first surface at the point B. The normal N 1 to the surface at B is a continuation of the radius of curvature of the first surface, which emanates from the center of curvature C 1 of the first surface. The incident ray AB makes an angle of incidence i 1 with the normal N 1. Because the ray is going from the less dense medium, air, to the more dense medium, glass, the refracted ray is bent through an angle r 1 toward the normal, as shown. This refracted ray impinges on the second surface at the point D. The normal at D is the continuation of the radius of curvature of the second surface and is designated N 2. Ray BD makes an angle of incidence i 2 with the normal N 2. Since the ray now goes from glass to air, it is going from a more dense medium to a less dense medium. The refracted ray is, therefore, bent away from the normal N 2 through an angle r 2 as ray DE. The net effect of the 31-17

18 two refractions is to take a ray of light, which is parallel to the principal axis, and bend it such that it crosses the principal axis. The point where this ray crosses the principal axis is called the principal focus, and is designated by the letter F. The distance from the center of the lens to the principal focus is called the focal length f of the lens. It is important to note that for a particular material, it is the shape of the lens that determines the normals and hence the refracted rays and thus the focal length of the particular lens. Since we have already seen that the converging of a parallel ray to the principal axis is caused by the refraction at the two surfaces of the lens, it will not be necessary to go into all this detail every time we consider a lens. Instead, we will say that a ray of light, parallel to the principal axis, is bent in the middle of the lens. It then converges to the principal focus, as shown in figure Figure Bending of a ray of light by a convex lens. All rays parallel and close to the principal axis converge to the principal focus, as seen in figure Such a lens is called a converging lens. If the incident rays are not close to the principal axis, the converging rays do not all Figure A converging lens. intersect at the same point, a result known as spherical aberration. We will assume in this book that all rays are sufficiently close to the principal axis so that spherical aberration can be ignored. If parallel light comes in from the right of the lens it converges at the same distance to the left of the lens. Note also that, by the principle of reversibility, light emanating from the principal focus comes out parallel 31-18

19 to the principal axis. Figure shows some examples of converging lenses. The actual shapes are determined by the radii of curvature, but an easy way to identify such a lens is to note that a converging lens is always thicker at the center of the lens Figure Examples of converging lenses. than it is at the rim. If any of the thin lenses in figure are reversed, they still produce the same effect. A diverging lens is a lens that takes a bundle of parallel light rays and diverges them away from the principal axes, as shown in figure If the Figure A diverging lens. diverging rays are produced backward, they all intersect at the same point F, called the principal focus, or focal point, of the diverging lens. The diverging rays appear to come from the principal focus. The distance from the center of the lens to the focal point is called the focal length f of the diverging lens. Some examples of diverging lenses are shown in figure One characteristic of all diverging lenses is that they are thinner at the center of the lens than they are at the rim. Figure Examples of diverging lenses

20 31.8 Ray Tracing and the Standard Rays To determine the location of an image formed by a lens, we use the technique of ray tracing. An object OP, represented by a brown arrow, is placed a distance p in front of a convex lens in figure A ray, from the bottom of the object at P, travels Figure Ray diagram for a convex lens. along the principal axis. This ray goes through the lens undeviated. Hence the bottom of the image must lie somewhere on the principal axis of the lens. To determine the location of the tip of the image, three rays are drawn from the tip of the object. These three rays are called the standard rays and are shown in figure The first ray (1), in red, is drawn parallel to the principal axis and after refraction it passes through the principal focus. A second ray (2), in green, is drawn from the top of the object and passes through the first focal point. This ray is refracted through the lens so that it comes out parallel to the principal axis. (Recall that by the principle of reversibility, light emanating from the principal focus comes out parallel to the principal axis.) The third of the standard rays (3), in blue, is drawn from the tip of the arrow and goes through the center of the lens. The refraction experienced at the first surface is exactly compensated by the refraction at the second surface, and the ray passes through the lens undeviated. These three rays intersect at the point I and form the real, inverted image IQ, in yellow. If a screen were placed at this point a sharp image would be found there. The distance from the object to the lens is called the object distance p, whereas the distance from the lens to the image is called the image distance q. Thus, we can find any image by drawing a ray diagram. A ray diagram for a diverging lens is shown in figure An object OP is placed a distance p in front of a diverging lens. A ray (1), in red, is drawn from the object parallel to the principal axis. The diverging lens causes the ray to bend away from the principal axis, as if the ray had come from the principal focus F. A second ray (2), in green, is drawn, which would go to the second focal point if there were no 31-20

21 Figure Ray diagram for a concave lens. lens present. This ray is caused to bend by the diverging lens so that it comes out parallel to the principal axis. A third ray (3), in blue, passes through the center of the lens and is not bent. These three rays do not intersect anywhere to the right of the lens and hence cannot form a real image there. However, if we follow each of the three rays backward, they appear to intersect at the point I, and the image is located at that point. The distance from the lens to the image, the image distance, is denoted by the letter q in the diagram. We can find an image of any object for any lens by drawing these standard rays The Lens Equation A mathematical relation between the object distance p, the image distance q, and the focal length f, can be obtained from the geometry of figure In triangle OPC, whereas in triangle QIC, tan θ 1 = OP = h o PC p tan θ 1 = QI = h i QC q Equating the tangent of θ 1 from each equation gives Rearranging terms, this becomes h o = h i (31.18) p q 31-21

22 Figure Determining the lens equation. We will return to this equation shortly. In triangle ACF, whereas in triangle IQF, h i = q (31.19) h o p tan θ 2 = AC = h o CF f tan θ 2 = QI = h i QF q f Equating the tangent of θ 2 from each equation, gives Rearranging terms, this becomes h i = h o q f f h i = q f (31.20) h o f Setting equation equal to equation 31.19, gives or Dividing each term by q, gives Solving for 1/f, we obtain q f = q f p q 1= q f p 1 1 = 1 f q p 31-22

23 1 = (31.21) f p q Equation is the lens equation and gives the relation between the object distance p, the image distance q, and the focal length f of the lens. Notice that it is the same formula as the mirror equation found in chapter 30. The linear magnification M of a lens system is defined as the ratio of the height of the image to the height of the object, that is, Using equation 31.19, we can also write this as M = height of image = h i (31.22) height of object h o M = h i = q (31.23) h o p The magnification tells how much larger the image is than the object. When we know M, we find the height of the image h i from equation 31.23, as h i = Mh o (31.24) The minus sign in equation is placed there by convention. For a single lens, when the magnification is negative the image is real and inverted, when it is positive the image is virtual and erect. Notice that the lens equation, the magnification equation, and the height of the image equation, are the same equations that were obtained for the spherical mirror in chapter 30. Example 31.7 A converging lens. An object 5.00 cm high is placed 35.0 cm in front of a converging lens of 10.0 cm focal length. (a) Where is the image located? (b) What is the magnification? (c) What is the size of the image? Solution a. The image distance, found from the lens equation, equation 31.21, is 1 = f p q 1 = 1 1 = 1 1 = cm 1 q f p 10.0 cm 35.0 cm q = 14.0 cm 31-23

24 b. The magnification, found from equation 31.23, is M = q = 14.0 cm = p 35.0 cm Since M is negative, the image is real and inverted. c. The size of the image, found from equation 31.24, is h i = Mh o = (0.400)(5.00 cm) = 2.00 cm The image is thus smaller than the object. The minus sign on h i simply means that the distance h i is in the negative y direction, and hence, the image is inverted. To go to this Interactive Example click on this sentence. Example 31.8 A diverging lens. An object 5.00 cm high is placed 35.0 cm in front of a diverging lens of 10.0 cm focal length. (a) Where is the image located? (b) What is the magnification? (c) What is the size of the image? Solution a. The image distance, found from the lens equation, equation 31.21, is 1 = f p q 1 = 1 1 = 1 1 q f p 10.0 cm 35.0 cm = cm cm 1 = cm 1 q = 7.78 cm Since the image distance q is negative, the image is virtual and is located 7.78 cm in front of the lens. b. The magnification, found from equation 31.23, is M = q p = 7.78 cm 35.0 cm =

25 Because M is positive the image is virtual and erect. c. The size of the image, found from equation 31.24, is h i = Mh o = (0.222)(5.00 cm) = 1.11 cm To go to this Interactive Example click on this sentence Some Special Cases for the Convex Lens Case 1: The Object Is Located at Infinity When the object is very far away from the lens, we can assume that it is at infinity. Setting p = in the lens equation, gives, for the location of the image, But 1/ = 0, and the equation reduces to 1 = 1 1 q f p = 1 1 f q = f That is, when the object is located at infinity, the image is located at the principal focus. The magnification becomes M = q = q = 0 p which shows that the image is reduced to a point. This case, which was shown in figure 31.13, is used as a simple technique to determine the focal length of a converging lens. A converging lens is held in front of a card and the distance between the card and the lens is varied until a sharp image is obtained on the card of a very distant object ( p equal to infinity). The distance between the lens and the card is the image distance q. But as just seen, if the object is at infinity, then q = f. Hence, the distance from the lens to the card is the focal length of the converging lens. Case 2: The Object Lies between Infinity and 2f, That Is, 2f < p < This is a rather general case and the ray diagram is shown as the continuous lines 31-25

26 in figure The actual location of the image is found from the lens equation when p and f are specified. Figure The object moves closer to the lens. A very interesting result occurs when the object is moved in toward the lens. The ray diagram for this case is shown in dashed lines in figure Notice that as the object moves toward the lens the image moves away from the lens and gets bigger. However, the height of the image h i never gets greater than the height of the object h o, and thus the magnification is always less than 1. (This will be proved in case 3.) Case 3: The Object Is Located at a Distance of Twice the Focal Length from the Lens, That Is, p = 2f For this case the image, found from the lens equation, is or and hence, 1 = = f p q 2f q 1 = 1 1 = 1 q f 2f 2f q = 2f That is, when the object is placed a distance of 2f to the left of the lens, the image is located at the same distance of 2f to the right side of the lens. The magnification for this case becomes M = h i = q = 2f = 1 h o p 2f and h i = h o 31-26

27 Hence, when p = q = 2f, the height of the image is the same size as the height of the object. For any value of p greater than 2f the magnification is always less than 1, and hence the height of the image is less than the height of the object. For any value of p less than 2f, the magnification is greater than 1, and hence the height of the image is always greater than the height of the object. Case 4: The Object Is Located between the Focal Length and Twice the Focal Length, That Is, f < p < 2f It is within this region that the lens acts as a magnifier, because if p is less than 2f, the magnification is greater than 1. The ray diagram is shown in figure It is obvious from the diagram that q is greater than p, and hence, the magnification, M Figure Ray diagram for object located between the focal length and twice the focal length. = q/p, is greater than 1, and the image is enlarged. This is also seen in the diagram, that is, the height of the image is greater than the height of the object. Case 5: The Object Is Located at the Principal Focus of the Lens, That is, p = f When the object is placed at the principal focus, p = f, and the lens equation gives for the location of the image, 1 = 1 1 = 1 1 = 0 q f p f f The only way for 1/q to be equal to zero, is for q to equal infinity, that is, q = 31-27

28 The ray diagram for such a case is shown in figure The standard ray (2) cannot be drawn, but rays (1) and (3) can, and as we see from the diagram, they become parallel on the right-hand side of the lens and hence never intersect to form a real image there at any finite distance from the lens. The parallel rays only intersect at infinity and it is there where the image may be found. Note that if rays (1) and (3) were dashed backward, they would still be parallel, and hence, could not form a virtual image either. Figure Ray diagram for an object at the principle focus. Case 6: The Object Is Placed within the Principal Focus, That Is, p < f A ray diagram for this case is shown in figure Standard ray (1) is drawn Figure Ray diagram for an object placed within the principle focus

29 parallel to the principal axis and is refracted in the lens such that it passes through the principal focus F, as shown. Of all the infinite rays that emanate from the tip of the arrow, standard ray (2) is a ray that is aligned with the dashed straight line from the principal focus to the tip of the arrow of the object. Therefore, ray (2) appears to come from the first principal focus, and hence, is refracted in the lens such that it comes out parallel to the principal axis. The third standard ray (3) is drawn from the tip of the object and goes through the center of the lens undeviated, as shown. The three standard rays do not converge anywhere on the right-hand side of the lens to form a real image there. However, if these standard rays are produced backward they intersect on the left-hand side of the lens and form a virtual image there. Thus a person looking through the lens from the right would see the enlarged, erect, virtual image of the object on the left-hand side of the lens. When a convex lens is used with the object located within the principal focus, it is called a simple magnifying glass. In this case, the image distance q is always negative. Example 31.9 A simple magnifying glass. An object 5.00 cm high is placed 3.00 cm in front of a convex lens of 5.00-cm focal length. (a) Where is the image located? (b) What is the magnification? (c) How high is the image? Solution a. The image distance, found from the lens equation, is and Note that q is a negative quantity. 1 = 1 1 = 1 1 = cm 1 q f p 5.00 cm 3.00 cm q = 7.50 cm b. The magnification, found from equation 31.23, is M = q p = 7.50 cm 3.00 cm = The magnification is thus positive and indicates that the image is virtual and erect. c. The image height, found from equation 31.24, is h i = Mh o = (2.50)(5.00 cm) = 12.5 cm 31-29

30 Thus the lens used in this way gives an enlarged (magnified 2.5 times), erect, virtual image of the object. The fact that q is negative is an indication that the image is virtual. To go to this Interactive Example click on this sentence. In summary, for a converging lens, with the object distance p positive, if the image distance q is positive the image is real and located on the other side of the lens. If the image distance q is negative, the image is on the same side of the lens as the object and is a virtual image Combinations of Lenses Most optical systems consist of a combination of two or more lenses, prisms, or mirrors. Figure shows a combination of two convex lenses separated by a Figure Combination of two convex lenses

31 distance d. The first lens has a focal length f 1 and the second lens has the focal length f 2, as illustrated. An object of height h o1 is placed a distance p 1 in front of the first lens. Let us find the location and size of the resulting image of the lens combination. We begin the analysis by considering the first lens only, that is, we temporarily ignore the existence of the second lens. We find the image distance q 1 of the first lens by drawing the standard rays of a ray diagram, as in figure This image of the first lens now acts as the object of the second lens. We now completely ignore the existence of the first lens and consider the second lens as though an object has been placed a distance p 2 in front of it. We again find the image from this second lens by drawing the standard rays of a ray diagram, as shown in figure Thus, the image distance q 2 of the second lens and the image height h i2 are easily found. In addition to using the ray diagram, we can also find the final image from the lens equation. The image distance of the first lens is found from whereas the image distance of the second lens is found from 1 = 1 1 (31.25) q 1 f 1 p 1 1 = 1 1 (31.26) q 2 f 2 p 2 However, as we can see from the diagram, the object distance p 2 is p 2 = d q 1 (31.27) In most of the cases that we consider in this book, the image distance q 1 turns out to be less than the separation distance d, and the term (d q 1 ) is positive, giving a positive object distance p 2. There are some cases where the image distance q 1 is greater than the separation distance d. In these cases, the term (d q 1 ) is a negative quantity and hence, the object distance p 2 is also a negative quantity. An object having a negative object distance is called a virtual object. For a virtual object, the rays strike the lens before the object is formed. We will see an example of this later. The magnification of the lens combination is defined as the ratio of the height of the final image to the height of the initial object. That is, But the magnification of the first lens is M = height of image 2 = h i2 (31.28) height of object 1 h o

32 whereas the magnification of the second lens is Thus, from equation 31.30, and from equation 31.29, M 1 = height of image 1 = h i1 (31.29) height of object 1 h o1 M 2 = height of image 2 = h i2 (31.30) height of object 2 h o2 h i2 = h o2 M 2 (31.31) h o1 = h i1 (31.32) M 1 Substituting equations and back into equation 31.28, and noting that the height of image 1 h i1 is equal to the height of object 2 h o2, we get or M = h i2 = h o2 M 2 = h o2 M 2 h o1 h i1 /M 1 h o2 /M 1 M = M 1 M 2 (31.33) Equation says that the magnification of a combined optical system is equal to the product of the magnification of each individual lens. The height of the final image, obtained from equation 31.28, is h i2 = Mh o1 (31.34) The location and nature of the image of a particular optical system depends on the focal lengths f 1 and f 2, the location of the first object p 1, and the separation d between the two lenses. We will consider different optical systems in section Example A combination of lenses. An object 5.00 cm high is placed 15.0 cm in front of a convex lens of 10.0-cm focal length. A second convex lens, also of 10.0-cm focal length, is placed 48.0 cm behind the first lens. Find (a) the image of the lens combination, (b) the magnification of the system, and (c) the height of the final image. Solution a. The image of the first lens, found from equation 31.25, is 31-32

33 1 = 1 1 = 1 1 = cm 1 q 1 f 1 p cm 15.0 cm q 1 = 30.0 cm The object distance for the second lens, found from equation 31.27, is p 2 = d q 1 = 48.0 cm 30.0 cm = 18.0 cm Hence, the final image, found from equation 31.26, is 1 = 1 1 = 1 1 = cm 1 q 2 f 2 p cm 18.0 cm q 2 = 22.5 cm Thus, the final image of the lens combination is found 22.5 cm behind the second lens, or 70.5 cm behind the first lens. b. The magnification of each lens is found as The final magnification of the system is M 1 = q 1 = 30.0 cm = 2.00 p cm M 2 = q 2 = 22.5 cm = 1.25 p cm M = M 1 M 2 = ( 2.00)( 1.25) = 2.50 Note that M 1 is negative indicating that the first lens inverts the first image. The fact that M 2 is also negative means that lens 2 inverted the second image from upside down to erect. The total magnification M is positive indicating that the final image is right side up. In dealing with combinations of lenses we must be careful of the convention adopted for the magnification of a single lens. The convention adopted is that if M is negative, the image is real and inverted, whereas if M is positive, the image is erect and virtual. In this case of the combined lens, M 1 is negative and image 1 is real and inverted, as expected. Also M 2 is negative and image 2 is real and inverted from its original orientation when in front of lens 2, also as expected. The inversion of the inverted image, gives a final image that is erect. Notice however, that the final magnification is positive because it is the product of two negative numbers. But the final image is real not virtual. Hence, the convention that a positive magnification indicates that the image must be virtual cannot be used for a system of multiple lenses

34 c. The final height of the image of the system, found from equation 31.34, is h i2 = Mh o1 = (2.50)(5.00 cm) = 12.5 cm To go to this Interactive Example click on this sentence Thin Lenses in Contact An interesting case results from the lens combination of section when the distance d between the two lenses is made equal to zero, thereby placing the two lenses in contact with each other. The lens equation applied to the second lens is 1 = (31.35) f 2 p 2 q 2 Setting p 2 = d q 1 from equation into equation 31.35, yields 1 = (31.36) f 2 d q 1 q 2 Setting the separation distance d equal to zero in equation yields But the lens equation applied to the first lens is given by 1 = (31.37) f 2 q 1 q 2 and 1 = 1 1 q 1 f 1 p 1 1 = (31.38) q 1 f 1 p 1 Substituting equation into equation 31.37, yields and 1 = f 2 f 1 p 1 q

35 1 + 1 = (31.39) f 1 f 2 p 1 q 2 Let us examine equation and see what it is telling us. Here p 1 is the object distance of the first lens, but it is also the object distance of the lens combination, and we will designate it as p c. Although q 2 is the image distance of the second lens, it is also the image distance of the lens combination, and we can call it q c. Hence, we can rewrite equation as = (31.40) f 1 f 2 p c q c The right-hand side of equation looks like part of the lens equation, so it is only natural to define the focal length for the two lenses in contact as f c, and the lens equation for the combination should then be Combining equations and 31.41, gives 1 = (31.41) f c p c q c 1 = (31.42) f c f 1 f 2 The focal length of the combination is thus obtained from equation Hence, whenever two lenses are in contact, the reciprocal of the focal length of the combination is equal to the sum of the reciprocals of the focal lengths of the individual lenses. Example The focal length of two convex lenses in contact. A convex lens of 10.0-cm focal length is placed in contact with a convex lens of 15.0-cm focal length. What is the focal length of the combination? Solution The focal length of the combination, found from equation 31.42, is and 1 = = = cm 1 f c f 1 f cm 15.0 cm f c = 6.00 cm 31-35

36 To go to this Interactive Example click on this sentence. Example The focal length of a convex and a concave lens in contact. A 10.0-cm convex lens is placed in contact with a 15.0-cm concave lens. What is the focal length of the combination? Solution The focal length of the combination is found from equation 31.42, but since the second lens is concave its focal length is negative, that is, f 2 = 15.0 cm. Therefore and 1 = = 1 1 = cm 1 f c f 1 f cm 15.0 cm f c = 30.0 cm To go to this Interactive Example click on this sentence. Equation 31.42, although derived for only two lenses, is completely general and thus, for any number of lens placed in contact the focal length of the combination is 1 = (31.43) f c f 1 f 2 f 3 f 4 When dealing with many lenses in contact it is convenient to define the dioptric power of a lens as the reciprocal of the focal length in meters. Thus P = 1 (31.44) f The reciprocal of the focal length in meters is called a diopter. Example The power of a lens. What is the power of a 15.0-cm focal length lens? Solution 31-36

37 The focal length of the lens is f = 15.0 cm = m. The power of the lens, found from equation 31.44, is P = 1 = 1 = 6.67 diopters f m To go to this Interactive Example click on this sentence. The advantage of expressing a lens in terms of its dioptric power is evident when the lenses are in contact. For example, if there are four lenses of focal lengths f 1, f 2, f 3, and f 4, then the power of each lens is P 1 = 1 f 1 P 2 = 1 f 2 P 3 = 1 f 3 P 4 = 1 f 4 The power of these lenses in combination, determined from equation 31.43, is and substituting the values of 1/f, we get 1 = f c f 1 f 2 f 3 f 4 The power of the combination is thus 1 = P 1 + P 2 + P 3 + P 4 f c P c = P 1 + P 2 + P 3 + P 4 (31.45) Expressing a lens in terms of dioptric power is convenient for an optician, who places many different lenses in contact. When a patient is given an eye exam, the optician places different lenses in front of the patient s eyes and asks him to read a chart. When the patient finds the sharpest image, the optician immediately gets the power of the combination of lenses used, P c, from equation The reciprocal of P c is the focal length of the eyeglasses needed for the patient and a lens of this focal length can be ground using the lensmaker s formula, which we now discuss

38 31.13 The Lensmaker s Formula We showed in figure that a spherical lens was made from a piece of plane glass by shaping it into the form of two spherical surfaces with radii of curvature R 1 and R 2. We then showed in figure that a ray of light parallel to the principal axis of this lens was refracted and then crossed the principal axis at the focal point F. Then in figure we showed that all rays parallel and close to the principal axis converge to the principal focus, and we defined the distance from the center of the lens to the focal point as the focal length f of the lens. We then showed in section that the focal length of any lens could be easily determined experimentally by placing a converging lens in front of a card and the distance between the card and the lens is varied until a sharp image is obtained on the card of a very distant object ( p equal to infinity). The distance between the lens and the card is the image distance q. But if the object is at infinity, then q = f. Hence, the distance from the lens to the card is the focal length of the converging lens. In this way we could determine the focal length of any lens experimentally. But how do we know how to grind the lens into the spherical shape that will gives us this focal length? Recall from chapter 30 that the focal length of a spherical mirror was found by equation to be f = R/2, where R was the radius of curvature of the mirror. That is, if you want a particular focal length f, just grind the mirror to the radius R = 2f. In this way you can make a mirror of any focal length. In a similar way, we would now like to determine the focal length of the lens in terms of its radii of curvature and the index of refraction of the lens material. The equation that relates the focal length, index of refraction, and radii of curvature of a lens is called the lensmaker s formula. To derive this relation we place an object at the point A, at the object distance p, in a medium of index of refraction n 1, in front of a thick lens as seen in figure We consider the ray of light that leaves the object at A and t = p 2 + q 1 n 1 B n 2 n 1 A I F E p 2 p 1 q 1 q 2 Figure Refraction of a ray of light through a thick lens

39 strikes the spherical surface at the point B. The incident ray AB, upon entering the second medium of index of refraction n 2, is bent toward the principal axis and this refracted ray crosses the principal axis at the point I, and at this point represents the image I of the object at A. The distance from the front of the lens to the point I is the image distance q 1. The ray then continues through the second medium until it arrives at the point E where it is again refracted toward the principal axis and intersects the principal axis at the point F, which will be the final image of the thick lens. The distance from the point I to the right surface of the lens is the object distance for the second spherical surface and is designated as p 2. The distance form the right surface to the final image F is the image distance q 2 of the second spherical surface. Let us now look at what happens at each of the spherical surfaces of the thick lens. The incident ray leaves the object at A and strikes the spherical surface at the point B. The normal at the point B is drawn as the extension of the radius R 1 of the spherical surface whose center of curvature is located at the point C 1, figure The incident ray AB makes an angle of incidence i 1 with the normal N. Upon entering the second medium of index of refraction n 2, the light ray is bent toward the normal making the angle of refraction r 1 as shown in the figure. This refracted ray crosses the principal axis at the point I, and represents the image I of the object at A. The law of refraction applied to the interface at point B gives n 1 sin i 1 = n 2 sin r 1 (31.46) n 1 N B i 1 δ r 1 R 1 θ D R 1 n 2 A α β C γ I p 1 q 1 Figure Refraction of a ray of light at the first spherical surface. We consider only incident rays that are close to the principal axis. Such rays are called paraxial rays. For paraxial rays, the angle α and hence i 1 and r 1 will all be small angles 2 and hence the sine of these angles will be equal to the angle itself. That is, we let sin i 1 = i 1 and sin r 1 = r 1, hence the law of refraction becomes 2 The angles in our diagrams are greatly exaggerated to show the details of the refraction

40 n 1 i 1 = n 2 r 1 (31.47) Recall from geometry that the exterior angle of a triangle is equal to the sum of the two interior angles. Applying this result to triangle ABC we see that and for triangle BIC, Solving for r 1 gives i 1 = α + θ (31.48) θ = r 1 + γ r 1 = θ γ (31.49) Substituting equations and into equation gives n 1 (α + θ) = n 2 (θ γ) (31.50) Now recall that the arc s of a circle is given by s = rθ. Applying this relation to arc BD of the lens we get BD = R 1 θ and hence θ = BD (31.51) R 1 Because the ray AB is a paraxial ray, the arc BD is approximately equal to q 1 γ, that is BD q 1 γ and hence γ = BD (31.52) q 1 Also arc BD is approximately equal to p 1 α, that is and hence BD p 1 α α = BD (31.53) p 1 Replacing equations 31.51, 31.52, and into equation gives n 1 (BD + BD) = n 2 (BD BD) p 1 R 1 R 1 q 1 Canceling out the term BD on both sides of the equation gives 31-40

41 Rearranging terms yields and n 1 + n 1 = n 2 n 2 p 1 R 1 R 1 q 1 n 1 + n 2 = n 2 n 1 p 1 q 1 R 1 R 1 n 1 + n 2 = n 2 n 1 For first spherical surface (31.54) p 1 q 1 R 1 Let us now consider what happens at the second spherical surface. The ray of light from the first surface that converged to the point I as the image of the first surface now becomes the object for the second spherical surface, figure The n 2 n 1 I γ C 2 R 2 β θ R 2 G α i 2 δ r 2 E p 2 q 2 N F Figure Refraction of a ray of light at the second spherical surface. distance from the point I to the right surface of the lens is the object distance for the second spherical surface and is designated as p 2. The distance from the right surface to the final image F is the image distance q 2 of the second spherical surface. Using the same analysis on the right spherical surface as on the left spherical surface we obtain the equivalent relation of equation as n 2 + n 1 = n 1 n 2 For second spherical surface (31.55) p 2 q 2 R 2 The combined effects of both refractions are found by adding equations and as n 1 + n 2 + n 2 + n 1 = (n 2 n 1 ) + (n 1 n 2 ) p 1 q 1 p 2 q 2 R 1 R 2 n 1 + n 1 + n 2 + n 2 = (n 2 n 1 ) (n 2 n 1 ) p 1 q 2 p 2 q 1 R 1 R 2 n 1 p 1 + n 1 q 2 + n 1 2 p2 + q 1 1 = (n 2 n 1 ) 1 1 (31.56) R1 R

42 Let us now analyze the third term in equation n 2 1 p2 + 1 q 1 = n 2 q 1 + p 2 p 2 q 1 = n 2 t p 2 q 1 (31.57) where we have used the fact that the sum q 1 + p 2 = t, the thickness of the lens as seen in figure If we make the lens a thin lens then in the limit we can consider the thickness t of the lens to be small compared to the object and image distances and hence we let t approach zero. That is, as t 0, the term in equation approaches zero and can be dropped from equation Thus equation becomes n 1 p 1 + n 1 q 2 = (n 2 n 1 ) 1 1 R1 R 2 or 1 p q 2 = n 2 n 1 n 1 1 R1 1 R 2 (31.58) The object distance p 1 for the first spherical surface is also the object distance of the entire lens p and so we let p 1 = p, and q 2 the image distance of the second spherical surface is the image distance q of the entire lens and so we now let q 2 = q. Therefore equation becomes 1 p + 1 q = n 2 n 1 1 n 1 1 (31.59) R1 R 2 But we showed in equation 31.21, the lens equation, that 1 p + 1 q = 1 f (31.21) Setting equation equal to equation gives 1 f = n 2 n 1 n 1 1 R1 1 R 2 (31.60) Equation is called the lensmaker s equation because it allows the lens maker to make a lens of a particular focal length f by the appropriate choice of radii of curvature R 1 and R 2 of the spherical lens and the index of refraction n 1 of the medium in which the lens will be used and n 2 the index of refraction of the lens itself. If the lens is to be used in air, then n 1 = 1 and n 2 = n, the index of refraction of the lens, and equation becomes 1 (31.61) f = (n 1) 1 1 R1 R 2 Equation is the lensmaker s formula for a lens used in air, n is the index of refraction of the glass, R 1 is the radius of curvature of the first surface of the lens, R 2 is the radius of curvature of the second surface, and f is the focal length of the lens

43 By convention, if the surface is convex to the incident ray, R is considered positive. If the surface is concave to the incident ray, R is considered negative. It is assumed that the ray of light is incident from the left. Thus, for a double convex lens, R 1 would be positive and R 2 would be negative. Equation says that in order to make a lens of a particular focal length f for a medium of index of refraction n, the medium must be ground to the radii of curvature R 1 and R 2 that satisfies equation Note that in the derivation of equation 31.61, we assumed that the thickness t of the glass lens is negligible compared to the distance to the principal focus and to any object or image distance concerned. Such a lens of negligible thickness is called a thin lens. We have only considered thin lenses in this book. The focal length of any converging or diverging lens is found from the lensmaker s formula, equation However, it is important to remember the convention that R is positive for an incident ray that impinges on a convex surface and negative for one that impinges on a concave surface. A plane surface is a surface whose radius of curvature is infinite, and hence, 1/R = 0. Example Finding the focal length of a converging lens. A double convex lens has radii of curvature of 25.0 cm. The index of refraction is What is the focal length of the lens? Solution By our convention, the radius of curvature of the first convex surface is cm. The inside of the second convex surface is concave to the incident light and its radius of curvature is therefore 25.0 cm. The focal length of the lens, found by the lensmaker s formula, equation 31.61, is 1 f = (n 1) 1 1 R1 R 2 = (1.52 1) 1 25 cm 1 = cm cm f = 24.0 cm To go to this Interactive Example click on this sentence. Example Finding the focal length of a diverging lens. A double concave lens has radii of curvature of 25.0 cm. The index of refraction is What is the focal length of the lens? 31-43

44 Solution By our convention, the radius of curvature of the first concave surface is 25.0 cm. The inside of the second concave surface is convex to the incident ray, and hence, the radius of curvature is cm. The focal length of the diverging lens, found from equation 31.61, is 1 f = (n 1) 1 1 R1 R 2 = (1.52 1) = cm f = 24.0 cm Note that converging lenses have positive focal lengths while diverging lenses have negative focal lengths. To go to this Interactive Example click on this sentence Optical Instruments Most optical devices consist of an arrangement or combination of lenses. The focal lengths of the lenses and their relative positions determines the function of the optical system. A very brief description of these optical instruments is given here as an example of lens applications. Most optical instruments consist of many lens combinations to eliminate various lens defects. We will look at only combinations of single lenses in order to demonstrate the principle of the device. More detailed discussions can be found in books on optics. The first optical instrument considered is the camera. The Camera The basic elements of a camera, figure 31.28(a), are a converging lens, contained in a bellows or tube type mechanism; a roll of photographic film that can be rolled across a focal plane; an aperture that determines the amount of light that enters the camera; and a shutter mechanism that determines the time that the lens is open to light. A schematic diagram of a camera is shown in figure 31.28(b). A ray diagram shows the location of the image. Because the focal length of the lens is a constant, increasing or decreasing the object distance p changes the image distance q, as can easily be determined by the lens equation. Therefore, in order to get a sharp image of any object, we need to be able to change the distance from the lens to the film plane. On studio type cameras, this is accomplished by moving the bellows inward or outward. On small hand cameras, such as a 35 mm camera, there are two concentric tubes containing the lenses. Rotating the tubes clockwise or 31-44

45 counterclockwise causes the lens to move in or out. The total variation of q is relatively small for a 35 mm camera. Figure A simple camera. Example The change in the image distance for a 35 mm camera. A 35 mm camera has a focal length of 50.0 mm. By how much will the image distance change when photographing one object at infinity and another 1.00 m away? Solution When the object is at infinity, the image distance q is equal to the focal length f, as seen in section 31.10, case 1. Therefore, q 1 = f = 50.0 mm = 5.00 cm When p 2 = 1.00 m = 100 cm, the image distance q 2 found from the lens equation, is 1 = 1 1 = 1 1 q 2 f p cm 100 cm q 2 = 5.26 cm The total variation in the image distance q becomes q = q 2 q 1 = 5.26 cm 5.00 cm = 0.26 cm 31-45

46 Thus, the lens does not have to move very far at all to get the two different pictures. One of the reasons for this is, of course, the relatively small focal length of the lens. If the focal length of the camera had been 20.0 cm, then the same calculation as above would have given a variation of the image distance q of 5.00 cm. Therefore, to obtain a small variation of q a small focal length is used. To go to this Interactive Example click on this sentence. Example Getting the entire picture into the camera. At what distance from the 35 mm camera of example should a man, 2.00 m tall, stand in order for his entire length to be in the picture? Solution For 35 mm film, the maximum height that the image h i can be is 35.0 mm. Therefore, the magnification of the camera is M = h i = 3.50 cm = h o 200 cm Since M = q/p, we can solve for the object distance, p, as p = q M Now, as seen above, the image distance q is quite close to the focal length f, so we will assume that they are equal, that is, q = f Hence, p = q = f = 5.00 cm = 286 cm = 2.86 m M M That is, for a 2.00-m tall man to have his entire body in the picture he must stand 2.86 m from the camera. Note that the magnification of a camera is quite small; this is necessary because we usually want to put a large scene onto a small piece of film. The images must be greatly reduced if they are to fit on the film. To go to this Interactive Example click on this sentence

47 The amount of light impinging on the film depends on the speed of the shutter, that is, 1/50, 1/100, 1/200, 1/500 of a second and so forth, and the size of the opening at the lens. The size of the opening is determined by an iris diaphragm and is calibrated in terms of the f-number. The f-number is defined as f# = focal length = f (31.62) diameter of aperture d The standard f-numbers of a camera are: f/2.8, f/4, f/5.6, f/8, f/11, and f/16. Thus, an f-number of f/4 means that the diameter d of the aperture is 1/4 of the value of the focal length. If a 50.0-mm focal length is being used, then an f-number of f/4 implies that the lens opening is d = f = f = 50.0 mm = 12.5 mm f# 4 4 The opening for f/16 is d = f = f = 50.0 mm = 3.13 mm f# Hence, the larger the f-number of a camera, the smaller the lens opening. Thus, a picture taken in bright sunshine would use a larger f-number than one taken in subdued lighting, for the same shutter speed. The Simple Microscope The simple microscope is the same as the magnifying glass that was treated in section 31.10, case 6. The object is placed within the principal focus of a converging lens and an enlarged, erect, and virtual image is formed. The Compound Microscope The compound microscope was invented by Galileo in It consists of two converging lens. The first lens is called the objective lens and is of short focal length. The object to be magnified is placed just outside the focal length f 1 of the objective, so that a real, enlarged image is formed at q 1, figure 31.29(a). The second lens is called the eyepiece, and the image of the objective lens becomes the object for the eyepiece lens. The eyepiece is situated such that the object p 2 always lies just within the principal focus of the eyepiece f 2. Thus the eyepiece acts as a simple magnifying glass, giving an enlarged virtual image at q 2. The lens equation is used to find the location of the final image. Thus, q 1 is found from 1 = 1 1 q 1 f 1 p 1 Once q 1 is determined, the object distance p 2 is found from 31-47

48 Figure The compound microscope. The final image is found from p 2 = d q 1 1 = 1 1 q 2 f 2 p 2 The magnification of the entire system is where M = M 1 M 2 M 1 = q 1 and M 2 = q 2 p 1 p 2 A typical student microscope is shown in figure 31.29(b). Example The microscope. A bug 2.00 mm in diameter is placed 1.50 cm in front of a 1.00-cm focal length objective lens of a compound microscope. The eyepiece lens has a focal length of 23.0 cm and the lenses are separated by a distance of 25.0 cm. Find the size of the resulting image

49 Solution We treat the microscope as a system of two lenses in combination. The image of the first lens is the object for the second lens. The image of the first lens, found from the lens equation, equation 31.25, is 1 = 1 1 q 1 f 1 p 1 = cm 1.50 cm q 1 = 3.00 cm This image serves as the object of the second lens. The object distance for the second lens, found from equation 31.27, is p 2 = d q 1 = 25.0 cm 3.00 cm = 22.0 cm The image of the second lens, found from the lens equation, equation 31.26, is 1 = 1 1 q 2 f 2 p 2 = cm 22.0 cm q 2 = 506 cm The magnification of the first lens is found as M 1 = q 1 = 3.00 cm = 2.00 p cm whereas the magnification of the second lens is found as M 2 = q 2 p 2 = 506 cm 22.0 cm = The total magnification of the system, found from equation 31.33, is M = M 1 M 2 = (2.00)(23.0) = 46.0 The size of the final image, found from equation 31.34, is h i2 = Mh o1 = (46.0)(2.00 mm) = 92.0 mm = 9.20 cm 31-49

50 Notice that because M 1 is negative, the first image is real, whereas M 2 is positive, indicating that the second image is virtual. This is as expected, since the object p 2 lies within the principal focus f 2 of the second lens. The final result of the optical system is to take the initial object, 2.00 mm in size, and make it observable as an image 92.0 mm in size. To go to this Interactive Example click on this sentence. The Astronomical Telescope The astronomical telescope is another example of a combination of two converging lenses. However, this time the objective is a large converging lens of a long focal length. The astronomical telescope is shown in figure 31.30(a). Because the object is very far away, the image of the objective is, for all intents and purposes, located at the principal focus of the objective lens f o. (Recall that when p =, q = f.) It is shown slightly away from there in figure so that the ray diagram is easier to see. The eyepiece is a converging lens of small focal length, f e, and it is placed such that the image of the first lens falls just within the principal focus of the second lens, thereby forming an enlarged, virtual image. That is, the eyepiece acts as a simple magnifying glass. Since the first image falls approximately at the focal distance of the first lens, and the second lens is placed so that the image of the first lens falls just within the focal length of the second lens, the distance separating the lenses is approximately the sum of the two focal lengths. Hence, the overall length of the telescope is L = f o + f e (31.63) The final image of the astronomical telescope is, of course, inverted but this is not serious for analyzing the heavenly bodies. However, the astronomical telescope is not very good for viewing distant objects on the surface of the earth, because they are all inverted. (The astronomical telescope can be converted to a terrestrial telescope by placing a third converging lens between the objective and the eyepiece.) The magnification of a telescope is usually expressed in terms of the angular magnification. The angular magnification M A of the lens is defined as the ratio of the angle subtended at the eye by the object when the lens is used, to the angle subtended by the unaided eye. We state without proof that the total angular magnification of an astronomical telescope when the object is at infinity is given by M A = focal length of objective = f o (31.64) focal length of eyepiece f e 31-50

51 (a) (b) Figure The astronomical telescope. Example An astronomical telescope. Find the angular magnification of an astronomical telescope whose objective has a focal length of 100 cm, and its eyepiece has a focal length of 5.00 cm. Solution The magnification, found from equation is, 31-51

52 M = f o = 100 cm = 20.0 f e 5.00 cm To go to this Interactive Example click on this sentence. The Galilean Telescope The Galilean telescope consists of a converging lens of large focal length for the objective and a diverging lens of short focal length for the eyepiece. The advantage of the Galilean telescope is that it gives an erect image. A ray diagram for a Galilean telescope is shown in figure Only two of the standard rays are Figure Ray diagram for a Galilean telescope. shown to simplify the diagram. The image of the first lens is located at q 1. But note that the second lens is placed before the location of the first image. This is an example of what is called a virtual object, and the object distance p must be written as a negative quantity in the lens equation. The effect of the diverging lens for the eyepiece is to diverge the rays away from the position where they should focus by the objective lens. For example, when ray (2) hits the diverging lens it is parallel to the principal axis. However, a ray parallel to the principal axis is diverged away from the principal axis as though it came from the principal focus of the diverging lens. This ray is now shown as diverging away from the principal axis. In this particular example, the second lens is located at the principal focus of the first lens, hence ray (1) passes through the geometrical center of lens two and is not deviated. We can see from the figure that these two rays diverge on the right-hand side of the second lens and therefore can never form a real image there. However if these rays are produced backward they intersect on the left-hand side of the lens to form an enlarged, erect, virtual image. Thus the Galilean telescope, also called a terrestrial telescope, produces enlarged images that are right side up

53 Example A Galilean telescope. A Galilean telescope is made of a cm objective lens and a 4.50-cm eyepiece lens. The two lenses are separated by a distance d of 15.0 cm. For an object situated at 1000 cm, find (a) the image distance of the first lens, (b) the object distance for the second lens, (c) the image distance for the second lens. Solution a. The image distance of the first lens, found from the lens equation, equation 31.25, is 1 = 1 1 = 1 1 q 1 f 1 p cm 1000 cm q 1 = 20.4 cm b. The object distance for the second lens, found from equation 31.27, is p 2 = d q 1 = 15.0 cm 20.4 cm = 5.4 cm Note that this is a virtual object since p 2 is negative. c. The image distance for the second lens, found from equation 31.26, is 1 = 1 1 = 1 1 q 2 f 2 p 2 ( 4.50 cm) ( 5.4 cm) q 2 = 27 cm Because q 2 is negative the final image is virtual. To go to this Interactive Example click on this sentence. Have you ever wondered? An Essay on the Application of Physics Nature s Camera -- the Human Eye Have you ever wondered how the human eye works? It is one of the greatest marvels of nature, but one that is usually taken for granted. The human eye is very much like a camera. It contains a lens, an iris diaphragm to let the light in, and its film is the retina at the back of the eye, figure 1. The lens focuses incident light onto 31-53

54 the retina so a sharp image is observed. The iris diaphragm opens wide in subdued lighting and closes down in very bright light. In the eye, the distance from the lens to the retina q, does not change as in a camera, instead the shape of the lens is changed by the muscles and ligaments of Figure 1 The human eye. the eye. Hence, the eye changes its focal length in order to bring objects at different object distances to a focus at the same image distance q, the distance from the lens to the retina. This changing of the focal length of the eye is called accommodation. A schematic of the normal eye is shown in figure 2(a). The greatest amount of bending Figure 2 Defects of the human eye. of light in the eye occurs at the cornea, because it is here where the greatest difference in the index of refraction occurs. The index of refraction of air is of course equal to one, whereas the index of refraction of the cornea is about The index of refraction of the lens is about 1.44; that of the aqueous and vitreous humor is about

55 Some eyes do not have the ability to change the shape of the lens as necessary and this is referred to as a defect in vision. One such defect is called hyperopia, or farsightedness, and is shown in figure 2(b). A farsighted person can see distant objects clearly, but objects a short distance away are blurred. The lens is not capable of bending the incoming light of near objects to a focus on the retina. Instead the light is focused behind the retina, leaving blurred vision. The defect is easily remedied by placing a converging lens in front of the eye, as shown in figure 2(c). The converging lens brings these incoming rays to a focus at the retina. Farsightedness is a defect that occurs to most people as they age. The lens hardens with age and the muscles weaken and are no longer capable of shaping the lens as needed. Another defect is myopia, or nearsightedness, as illustrated in figure 2(d). A nearsighted person can see nearby objects clearly, but distant objects are blurred. The distant objects come to a focus in front of the retina leaving a blurred image on the retina. This defect can be eliminated by placing a diverging lens in front of the eye. The diverging lens causes the rays to converge more slowly and hence the image is formed farther back at the retina, as shown in figure 2(e). The minimum distance from the eye at which an object can be seen distinctly is called the near point of the eye. For the average person, the near point is about 25.0 cm. That is, an object at a distance p of 25.0 cm is the minimum distance that an object can be placed to give a distinct image on the retina. If the distance is decreased below this value, the image will be blurred for most people. Remember, this is only an average value, as we get older, the minimum distance of distinct vision increases. For example, the near point can be as close as 6.00 cm for a child and can increase to over a 100 cm for an elderly person. Example 31H.1 Range of focal length for the eye. The distance from the cornea to the retina in the human eye is about 20.0 mm. What should the focal length of the human eye be in order to see clearly an object (a) at infinity and (b) at the near point of most distinct vision, 25 cm? Solution a. For the object at infinity, the image is located at the principal focus. Hence f = q = 20.0 mm = 2.00 cm is the focal length of the eye when it views an object at infinity. b. For the object at p = 25.0 cm, the focal length would have to be 31-55

56 1 = = = cm 1 f p q 25.0 cm 2.00 cm f = 1.85 cm Thus, the eye only has to change its focal length by (2.00 cm 1.85 cm = 0.15 cm) 15 mm to see its entire range of vision. Another common defect of the eye is called astigmatism. Astigmatism occurs in some people, because their eye is not completely spherical. That is, the radius of curvature of the eye in the vertical direction is not the same as the radius of curvature in the horizontal direction. Hence, vertical rays do not converge to the same position as horizontal rays. This defect is usually corrected with a lens of cylindrical curvature. Although the magnification of a lens was defined in equation as M = h i /h o, when viewing an object with the eye it is sometimes more desirable to talk about the angular magnification of the lens. Figure 3(a) shows an object at three distances from the unaided eye. The size of the object, as determined by the unaided Figure 3 Angular magnification and the simple microscope. eye, depends on the angle subtended by the object at the eye. Thus, when the object is at position 1, it subtends an angle θ 1 ; when it is at position 2, it subtends the angle θ 2 ; and when it is at position 3, it subtends the angle θ 3. As seen from the diagram, θ 1 < θ 2 < θ 3. Thus, the closer the object is to the eye, the greater the angle 31-56

57 subtended and the easier it is for the eye to see the object. However, the object cannot be moved closer to the eye than the near point, because that is the closest point that the eye can still see the object distinctly. In figure 3(b), the object is placed at the object distance p 1 = 25.0 cm in front of a normal eye. The height of the object h o subtends an angle α at the lens of the eye. The image of the object is focused on the retina of the eye. This is the largest image that the unaided eye can make of the object. If the object is placed within the focal point of a converging lens, the object can be brought even closer to the eye and hence will subtend a larger angle β, figure 3(c), than with the unaided eye. Therefore, the object is placed within the focal point, at the position p 2, such that the image distance q 2 is equal to the near point of the eye, the closest position of distinct vision. We assume that the value of the near point q 2 = 25.0 cm. The image is virtual, erect, and magnified as seen earlier. The angular magnification M A of the lens is defined as the ratio of the angle subtended at the eye by the object when the lens is used, to the angle subtended by the unaided eye. That is, M A = β (31H.1) α From figure 3(c), we see that tan β = h o p 2 whereas from figure 3(b), we see that Let us take the ratio tan α = h o = h o 25.0 cm p 1 tan β = h o /p 2 tan α h o /25.0 cm = 25.0 cm p 2 Because the angles considered are small, we can use the small-angle approximation, setting the tangent of an angle equal to the angle itself. Hence, β = 25.0 cm α p 2 But this ratio is the definition of the angular magnification, equation 31H.1, hence M A = 25.0 cm p 2 (31H.2) The value of p 2, found from the lens equation, is 31-57

58 1 = 1 1 p 2 f q 2 1 = 1 1 = 25.0 cm + f (31H.3) p 2 f 25.0 cm (25.0 cm)f And the object distance for the image to be at the near point is p 2 = (25.0 cm)f 25.0 cm + f (31H.4) Substituting equation 31H.4 into equation 31H.2, gives M A = β = 25.0 cm = 25.0 cm α p 2 (25.0 cm)f /(25.0 cm + f) M A = 25.0 cm + 1 f (31H.5) Equation 31H.5 gives the angular magnification of a converging lens when the image is located at the near point of the eye, which is assumed to be 25.0 cm for the normal eye. Example 31H.2 The angular magnification of a lens when the image is at the near point of the eye. A converging lens of 10.0-cm focal length is used as a magnifying glass. If the image is to be at the near point of the eye, find the angular magnification of the lens. Solution The angular magnification, found from equation 31H.5, is M A = 25.0 cm + 1 f = 25.0 cm cm = 3.5 The eye is more relaxed when viewing an object at infinity than at the near point. It is, therefore, sometimes more desirable to view an image at infinity than at the near point. For the image to be at infinity, the object must be placed at the principal focus. The lens equation, equation 31H.3, shows that if p is at infinity, 31-58

59 then the object distance p 2 is equal to the focal length f. Hence substituting f for p 2 in equation 31H.2, gives M A = 25.0 cm (31H.6) f Equation 31H.6 gives the angular magnification of a simple microscope when the image is set for viewing at infinity. Example 31H.3 The angular magnification of a lens when the image is at infinity. A magnifying glass of 10.0-cm focal length is used to view an object, such that the image is at infinity. Find the angular magnification of the lens. Solution The angular magnification, found from equation 31H.6, is M A = 25.0 cm f = 25.0 cm = cm Notice that when the image is viewed at the near point the angular magnification is 3.5, whereas when viewed at infinity, it is 2.5. If viewing is for a long time, it might be desirable to view the image at infinity because it is easier on the eyes. If the magnification is the most important consideration, then the image should be viewed at the near point of the eye. The Language of Physics Refraction The bending of light as it travels from one medium into another. It occurs because of the difference in the speed of light in the different mediums. Whenever a ray of light goes from a rarer medium to a denser medium the refracted ray is always bent toward the normal. Whenever a ray of light goes from a denser medium to a rarer medium, the refracted ray is bent away from the normal (p. )

60 Law of refraction The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant. The constant is called the relative index of refraction and it is equal to the ratio of the speed of light in the first medium to the speed of light in the second medium. Because of the changing speed of light, the wavelength of light changes as the light passes into the second medium (p. ). The critical angle of incidence The angle of incidence that causes the refracted ray to bend through When the incident angle exceeds the critical angle no refraction occurs. In that case, it is called total internal reflection because all the light that strikes the interface is reflected (p. ). Prism A triangular piece of transparent material whose angle exceeds the critical angle. A ray of light falling on one of the smaller sides of the prism enters the prism and is totally reflected from the longer side of the prism. Prisms are also used for analyzing the dispersion of white light into its component colors (p. ). Fiber optics A flexible glass rod of high refractive index. Light entering the glass undergoes total internal reflection from the walls of the glass fiber and the light travels down the length of the fiber with little or no absorption of light (p. ). Dispersion The separation of white light into its component colors. It occurs because the index of refraction of a medium varies slightly with the wavelength of light (p. ). Lens A piece of transparent material, such as glass or plastic, that causes light passing through it to either converge or diverge depending on the shape of the material (p. ). Lensmaker s formula An equation that relates the focal length, index of refraction of the medium and the lens, and the radii of curvature of the lens (p. ). Thin lens A lens whose thickness is negligible compared to the distance to the principal focus and to any object or image distance (p. ). Converging lens A lens that causes light, parallel to its principal axis, to converge to the principal axis. Converging lenses have positive focal lengths (p. )

61 Diverging lens A lens that causes light, parallel to its principal axis to diverge away from the principal axis. Diverging lenses have negative focal lengths (p. ). The lens equation An equation that relates the image distance, the object distance, and the focal length of a lens. It has the same form as the mirror equation (p. ). Dioptric power of a lens The reciprocal of the focal length of a lens. The focal length must be expressed in meters. For a combination of any number of lenses in contact, the power of the combination is equal to the sum of the powers of each individual lens (p. ). Accommodation The changing of the focal length of the eye in order to focus an image on the retina of the eye (p. ). Hyperopia, or farsightedness A defect of the eye that causes objects far away to be seen clearly while close objects are blurred. The condition is remedied by placing a converging lens in front of the eye (p. ). Myopia, or nearsightedness A defect of the eye that causes objects close to the eye to be seen clearly, while objects far away are blurred. The condition is remedied by placing a diverging lens in front of the eye (p. ). Near point of the eye The minimum distance from the eye at which an object can be seen distinctly. For the average person the near point is about 25 cm (p. ). Angular magnification The ratio of the angle subtended at the eye by an object, when a lens is used, to the angle subtended by the unaided eye (p. ). Summary of Important Equations The law of refraction sin i = v 1 = constant = n 21 (31.3) sin r v 2 n 1 sin i = n 2 sin r (31.8) The index of refraction n = c (31.4) v 31-61

62 Speed of a wave in terms of wavelength and frequency v = λν (31.9) Index of refraction in terms of wavelengths in two mediums n 21 = λ 1 (31.10) λ 2 Apparent depth for small angles q = n 2 p (31.14) n 1 Critical angle sin i c = n 2 (31.17) n 1 Lens equation 1 = (31.21) f p q Magnification M = h i = q (31.23) h o p Height of image h i = Mh o (31.24) Combination of lenses Final image 1 = 1 1 (31.26) q 2 f 2 p 2 where p 2 = d q 1 (31.27) and 1 = 1 1 (31.25) q 1 f 1 p 1 Magnification M = M 1 M 2 (31.33) Thin lenses in contact 1 = (31.43) f c f 1 f 2 f 3 f 4 Power of a lens P = 1 (31.44) f (in meters) Power of combinations of lenses P c = P 1 + P 2 + P 3 + P 4 + (31.45) Lensmaker s formula, lens in any medium 1 (31.60) f = n 2 n 1 1 n 1 1 R1 R 2 Lensmaker s formula, lens in air 1 (31.61) f = (n 1) 1 1 R1 R 2 f-number of a lens f# = f (31.62) d Angular magnification, astronomical telescope M A = f o (31.64) f e Angular magnification, viewing at near point M A = 25.0 cm + 1 (31H.5) f Angular magnification, viewing at infinity M A = 25.0 cm (31H.6) f 31-62

63 Questions for Chapter 31 *1. Why does a diamond sparkle? 2. When the angle of incidence is equal to zero, the angle of refraction is also zero. Does the wavelength of the light change when going from one medium to another under these circumstances? 3. If you are at the bottom of a pool of water and you look upward into the air at an angle greater than 50 0 can you see anything in the air? 4. Is it possible to have a ray of light refracted into a medium such that the new wavelength decreases to the point where it is no longer in the visible spectrum? What would you see? Does the eye interpret a wavelength or a frequency? 5. What does a wide-angle lens and a telephoto lens do when each is attached to a camera? 6. Describe the optical system used for (a) a slide projector, (b) a movie camera, and (c) an overhead projector. 7. A swimmer forgets to take off her glasses as she enters the pool. When she is under water, wearing the glasses, does the water have any effect on what she can see with the glasses? 8. When you see yourself in a mirror, your right and left hand are interchanged. Yet, if you see a picture of yourself made with a camera, containing lenses, your right and left are not changed. Why? *9. What are bifocals and why are they used? *10. What is a mirage and how is it explained by the index of refraction? *11. How is a rainbow formed? 12. How is a convex lens used, with the help of the sun, to start a fire? Problems for Chapter The Law of Refraction 1. A ray of light impinges on a piece of glass (n g = 1.52) at an angle of incidence of Find the angle of refraction. 2. A ray of light passes from water to glass at an angle of incidence of Find the angle of refraction. 3. A ray of light is refracted by an angle of as it enters water from glass. Find the angle of incidence. 4. A ray of light in air makes an angle of incidence of as it enters an unknown liquid. The refracted ray in the fluid is measured to be Find the index of refraction of the unknown liquid. What substance might it be? 31-63

64 Diagram for problem Determine the speed of light in (a) water, (b) glycerine, and (c) diamond. 6. If a ray of light of nm wavelength in air enters into water, what is the wavelength of the light in the water? Is this light visible? 7. A ray of light of nm wavelength in water, enters into the air. What is the wavelength of the light in the air? 8. A ray of light of nm wavelength in water enters into the air. What is the wavelength of the light in the air? Will the light be seen in the air? 9. A ray of light of nm wavelength in air impinges on a piece of flint glass at an angle of with the vertical. The index of refraction of this flint glass is Find (a) the angle of refraction, (b) the speed of light in the glass, and (c) the wavelength of light in the glass Apparent Depth of an Object Immersed in Water 10. A rock sits at the bottom of a 3.50-m-deep pool. What is its apparent depth? 11. A fish appears to be at a depth of 120 cm in water. What is its actual depth? Diagram for problem Refraction through Parallel Faces 12. A ray of light in air makes an angle of incidence of with a sheet of glass cm thick. Find the distance d that the final ray is displaced from its original direction

65 Diagram for problem Total Internal Reflection 13. What is the critical angle of refraction for a light ray going from water to glass? 14. Find the critical angle of refraction for a ray of light passing from glycerine (n = 1.47) into air. 15. The critical angle of a ray of light is measured as as it goes from an unknown liquid into air. Find the index of refraction of the liquid Thin Lenses 16. A double convex glass lens of index of refraction 1.52, has radii of curvature R 1 = cm and R 2 = 25.0 cm. Find its focal length. 17. A glass lens of index of refraction 1.52, has radii of curvature of R 1 = cm and R 2 = 15.0 cm. What is its focal length and is it a converging or a diverging lens? 18. A glass lens of index of refraction 1.52, has radii of curvature of R 1 = 40.0 cm and R 2 = cm. What is its focal length and is it a converging or a diverging lens? 19. A glass lens of index of refraction 1.52, has radii of curvature of R 1 = 40.0 cm and R 2 = 15.0 cm. What is its focal length and is it a converging or a diverging lens? 20. A lens is made of transparent material, with radii of curvature R 1 = 15.0 cm and R 2 = 90.0 cm. If the focal length of the lens is 5.00 cm, find the index of refraction of the material The Lens Equation, and Section Some Special Cases for the Convex Lens *21. An object 3.00 cm high is placed 20.0 cm in front of a converging lens of 15.0 cm focal length. Draw a ray diagram. Find (a) the image distance, (b) the 31-65

66 magnification, and (c) the final size of the image. (d) Is the image real or virtual? (e) Is the image erect or inverted? *22. An object 3.00 cm high is placed 20.0 cm in front of a diverging lens of 15.0 cm focal length. Draw a ray diagram. Find (a) the image distance q, (b) the magnification M, and (c) the height of the image h i. (d) Is the image real or virtual? (e) Is the image erect or inverted? 23. Where should an object be placed in front of a 20.0-cm lens in order for the image to be the same size as the object? 24. An object 7.00 cm high is placed 5.00 cm in front of a convex lens of 10.0 cm focal length. (a) Draw a ray diagram. Find (b) the image distance, (c) the magnification, and (d) the height of the image. 25. How far in front of a 20.0-cm converging lens should an object be placed in order to produce an image 25.0 cm from the lens on the same side of the lens as the object? 26. An object is placed 15.0 cm in front of a diverging lens of 5.00 cm focal length. Where is the image located and what is its magnification? 27. Where should an object be placed in front of a diverging lens of 10.0 cm focal length in order to give an image a magnification of 0.500? 28. An object 5.00 cm high is placed 20.0 cm in front of a converging lens. The image is measured to be 7.00 cm high. Where is the image located? 29. An object 5.00 cm high is placed 30.0 cm in front of a converging lens of 7.50 cm focal length. What is the size of the image? Combination of Lenses *30. An object 5.00 cm high is placed 20.0 cm in front of a converging lens of 10.0 cm focal length. A second converging lens of 20.0 cm focal length is placed 30.0 cm behind the first lens. Find (a) the image distance for the first lens, (b) the object distance for the second lens, (c) the image distance for the second lens, and (d) the total magnification of the combination. Draw a ray diagram. *31. An object 5.00 cm high is placed 10.0 cm in front of a 20.0-cm converging lens. A second converging lens, also of 20.0 cm focal length, is placed 20.0 cm behind the first lens. Find (a) the location of the final image and (b) its size. Draw a ray diagram. *32. A converging lens of cm is separated by a distance of 20.0 cm from a diverging lens of 5.00 cm. An object is located 30.0 cm in front of the first lens. Find (a) the image distance of the first lens, (b) the object distance for the second lens, (c) the image distance for the second lens, and (d) the magnification of the system Thin Lenses in Contact 33. A 20.0-cm convex lens is placed in contact with a diverging lens of unknown focal length. The lens combination has a focal length of 30.0 cm. Find the focal length of the diverging lens

67 Diagram for problem What is the power of (a) a 50.0-cm converging lens, (b) a 20.0-cm diverging lens, (c) the converging and diverging lenses in contact, and (d) the focal length of the combination when they are in contact? 35. Ten identical converging thin lenses, each of focal length 6.50 cm, are in contact. Find the focal length of the composite lens. Additional Problems 36. A ray of light of nm wavelength in glycerine impinges on a piece of flint glass at an angle of with the vertical. The index of refraction of glycerine is 1.47 and for the flint glass used it is Find (a) the angle of refraction, (b) the speed of light in the glycerine, (c) the speed of light in the glass, and (d) the wavelength of light in the glass. 37. Light travels in medium 1 at m/s. The light is incident at an angle of incidence of when entering medium 2, where the angle of refraction is Find the speed of light in medium A ray of light of nm wavelength makes an angle of incidence of on to the cornea of the eye. The index of refraction of the cornea is Find (a) the angle of refraction, (b) the speed of light in the cornea, and (c) the wavelength of light in the cornea. 39. Light travels from glass of index of refraction 1.50 into water of index If there exists a refracted ray, find the angle of refraction when (a) the angle of incidence is and (b) the angle of incidence is *40. If an object is placed a distance d 1 in front of the principal focus of a lens and the image is located a distance d 2 beyond the other principal focus, show that the focal length of the lens is given by f = d 1 d An object is mounted 60.0 cm in front of a screen. At what two positions will a 12.0 cm focal length lens yield a distinct image on the screen? 42. An optical system creates a virtual object (an object behind the lens) for a diverging lens of focal length f = 8.00 cm. Find the position of the image and describe it for the following two object distances: (a) p = 4.00 cm and (b) p = 10.0 cm