Conceptual Physics

1. Conceptual Physics Chapter Thirty Notes: Lenses .

2. Introduction • The Big Idea: Lenses change the path of light! • A light ray bends as it enters glass, and bends again as it leaves. • We will discuss two types of lenses: the double convex and the double concave. • The double convex lens is a converging lens. When light waves parallel to the principal axis from an infinitely far object passes through the lens, it will converge at a focal point F on the principal axis. The distance between the focal point and the lens is the focal length, which is always a positive value for converging lenses. • The double concave lens is a diverging lens. When light waves from an infinitely far object passes through the lens, the light waves will diverge as if it originated from a focal point F on the principal axis. The focal length is always a negative value for diverging lenses. • Can you see where these two types of lenses could be applied?

3. 30.1 Converging and Diverging lenses • If a piece of glass or other transparent material takes on the appropriate shape, it is possible that parallel incident rays would either converge to a point or appear to be diverging from a point. A piece of glass which has such a shape is referred to as a lens. • A lens is merely a carefully ground or molded piece of transparent material which refracts light rays in such as way as to form an image. Lenses can be thought of as a series of tiny refracting prisms, each of which refracts light to produce their own image. When these prisms act together, they produce a bright image focused at a point. Converging Lens Convex Lens Diverging Lens Concave Lens

4. There are a variety of types of lenses. Lenses differ from one another in terms of their shape and the materials from which they are made. Our focus will be upon lenses which are symmetrical across their horizontal axis - known as the principal axis. In this unit, we will categorize lenses as converging lenses and diverging lenses. A converging lens is a lens which converges rays of light which are traveling parallel to its principal axis. Converging lenses can be identified by their shape; they are relatively thick across their middle and thin at their upper and lower edges. A diverging lens is a lens which diverges rays of light which are traveling parallel to its principal axis. Diverging lenses can also be identified by their shape; they are relatively thin across their middle and thick at their upper and lower edges.

5. A double convex lensis symmetrical across both its horizontal and vertical axis. Each of the lens' two faces can be thought of as originally being part of a sphere. The fact that a double convex lens is thicker across its middle is an indicator that it will converge rays of light which travel parallel to its principal axis. A double convex lens is a converging lens. A double concave lensis also symmetrical across both its horizontal and vertical axis. The two faces of a double concave lens can be thought of as originally being part of a sphere. The fact that a double concave lens is thinner across its middle is an indicator that it will diverge rays of light which travel parallel to its principal axis. A double concave lens is a diverging lens. These two types of lenses - a double convex and a double concave lens will be the only types of lenses which will be discussed in this chapter.

6. If you are uncertain of the meaning of the terms, spend some time reviewing them so that their meaning is firmly internalized in your mind. They will be essential as we proceed through the chapter. These terms describe the various parts of a lens and include such words as • If a symmetrical lens is thought of as being a slice of a sphere, then there would be a line passing through the center of the sphere and attaching to the mirror in the exact center of the lens. This imaginary line is known as the principal axis. A lens also has an imaginary vertical axiswhich bisects the symmetrical lens into halves. As mentioned above, light rays incident towards either face of the lens and traveling parallel to the principal axis will either converge or diverge. If the light rays converge (as in a converging lens), then they will converge to a point. This point is known as the focal pointof the converging lens. If the light rays diverge (as in a diverging lens), then the diverging rays can be traced backwards until they intersect at a point. This intersection point is known as the focal point of a diverging lens. The focal point is denoted by the

7. letter F on the diagrams below. Note that each lens has two focal points - one on each side of the lens. Unlike mirrors, lenses can allow light to pass through either face, depending on where the incident rays are coming from. Subsequently, every lens has two possible focal points. The distance from the mirror to the focal point is known as the focal length(abbreviated by f). Technically, a lens does not have a center of curvature (at least not one which has any importance to our discussion). However a lens does have an imaginary point which we refer to as the 2F point. This is the point on the principal axis which is twice as far from the vertical axis as the focal point is. Anatomy of a Convex Lens is identical, except the lens allows light to pass through

8. 30.2 Image Formation by a Lens • The type of image formed by a lens depends upon the shape of the lens and the position of the object. • First lets consider a double convex lens. Suppose that several rays of light approach the lens; and suppose that these rays of light are traveling parallel to the principal axis. Upon reaching the front face of the lens, each ray of light will refract towards the normal to the surface. At this boundary, the light ray is passing from air into a more dense medium (usually plastic or glass). Since the light ray is With unaided vision, the size of an object appears to change depending on your distance to the object. a. A distant object is viewed through a narrow angle. b. When the same object is viewed from a closer distance and thus through a wider angle, more detail is seen.

9. passing from a medium in which it travels fast (less optically dense) into a medium in which it travels relatively slow (more optically dense), it will bend towards the normal line. This is the FST principle of refraction. This is shown for two incident rays on the diagram below. Once the light ray refracts across the boundary and enters the lens, it travels in a straight line until it reaches the back face of the lens. At this boundary, each ray of light will refract away from the normal to the surface. Since the light ray is passing from a medium in which it travels slow (more optically dense) to a medium in which it travels fast (less optically dense), it will bend away from the normal line; this is the SFA principle of refraction.

10. The above diagram shows the behavior of two incident rays approaching parallel to the principal axis. Note that the two rays converge at a point; this point is known as the focal point of the lens. The first generalization which can be made for the refraction of light by a double convex lens is as follows: • Now suppose that the rays of light are traveling through the focal point on the way to the lens. These rays of light will refract when they enter the lens and refract when they leave the lens. As the light rays enter into the more dense lens material, they refract towards the normal; and as they exit into the less dense air, they refract away from the normal. These specific rays will exit the lens traveling parallel to the principal axis.

11. The above diagram shows the behavior of two incident rays traveling through the focal point on the way to the lens. Note that the two rays refract parallel to the principal axis. A second generalization for the refraction of light by a double convex lens can be added to the first generalization. A converging lens can be used as a magnifying glass to produce a virtual image of a nearby object A converging lens forms a real, upside-down image of a more distant object.

12. Diverging lens create virtual images since the refracted rays do not actually converge to a point. In the case of a diverging lens, the image location is located on the object's side of the lens where the refracted rays would intersect if extended backwards. Every observer would be sighting along a line in the direction of this image location in order to see the image of the object. As the observer sights along this line of sight, a refracted ray would come to the observer's eye. This refracted ray originates at the object, and refracts through the lens. The diagram below shows several incident rays emanating from an object - a light bulb. Three of these incident rays correspond to our three strategic and predictable light rays. Each incident ray will refract through the lens and be detected by a different observer (represented by the eyes). The location where the refracted rays are intersecting is the image location. Since refracted light rays do not actually exist at the image location, the image is said to be a virtual image. It would only appear to an observer as though light were coming from this location to the observer's eye.

13. 30.3 Constructing Images Through Ray Diagrams • Now we have three incident rays whose refractive behavior is easily predicted. These three rays lead to our three rules of refraction for converging and diverging lenses. These three rules are summarized below. Refraction Rules for a Converging Lens • Any incident ray traveling parallel to the principal axis of a converging lens will refract through the lens and travel through the focal point on the opposite side of the lens. • Any incident ray traveling through the focal point on the way to the lens will refract through the lens and travel parallel to the principal axis. • An incident ray which passes through the center of the lens will in affect continue in the same direction that it had when it entered the lens. Refraction Rules for a Diverging Lens • Any incident ray traveling parallel to the principal axis of a diverging lens will refract through the lens and travel in line with the focal point (i.e., in a direction such that its extension will pass through the focal point). • Any incident ray traveling towards the focal point on the way to the lens will refract through the lens and travel parallel to the principal axis. • An incident ray which passes through the center of the lens will in affect continue in the same direction that it had when it entered the lens.

14. These three rules of refraction for converging and diverging lenses will be applied through the remainder of this chapter. The rules merely describe the behavior of three specific incident rays. While there are a multitude of light rays being captured and refracted by a lens, only two rays are needed in order to determine the image location. So as we proceed with this lesson, pick your favorite two rules (usually, the ones which are easiest to remember) and apply them to the construction of ray diagrams and the determination of the image location and characteristics.

15. 30.4 Image Formation Summarized Converging Lenses Ray Diagram for Object Located Beyond 2F

16. Diverging Lenses • The diagrams above shows that in each case, the image is • located on the object' side of the lens • a virtual image • an upright image • reduced in size (i.e., smaller than the object) • Unlike converging lenses, diverging lenses always produce images which share these characteristics. The location of the object does not affect the characteristics of the image. As such, the characteristics of the images formed by diverging lenses are easily predictable.

17. 30.5 Some Common Optical Instruments • Optical instruments that use lenses include the camera, the telescope (and binoculars), and the compound microscope. Cameras and Projectors The camera is essentially a dark box (camera oscuro = dark room) into which light is passed through a lens, forming a real image on the back wall where a light sensitive substance (film or photoelectric material) is placed. An example is shown. The camera is “focused” (so that the image forms in the right place) by manipulating the lens, either by moving it back and forth or by changing its focal length. Usually the object is beyond 2f distance, and the image distance is between f and 2f, so the image is reduced. Special close-up cameras can focus on objects at 2f or closer. The projector is a camera in reverse. The object is an illuminated transparency, placed between f and 2f distance. The image is formed on a screen at distance greater than 2f, so it is enlarged (and inverted relative to the object).

18. The Refracting Telescope A telescope is used to give an enlarged image of a distant object. We will assume the object is very far away, so that the rays from any point in it are essentially parallel. This means that the image formed by the first lens (the "objective") is real and located at the focal point of that lens. This real image is then magnified by the second lens (the "ocular" or "eyepiece") acting as a simple magnifier to form a virtual final image. The eye of the viewer is relaxed to see a final image located essentially at infinite distance, so the lenses are adjusted as shown. We assume that the ray from the bottom of the object passes along the axis (the dotted line) and is unaffected by the telescope. The ray from the top of the object is #1 in the diagram. Without the telescope, this ray would make angle α at the viewer’s eye with the ray from the bottom of the object, so the angular size of the object is α.

19. Ray #1 passes through the center of the objective lens undeviated and forms part of the real image at the focal point of that lens. Light from that real image then passes through the eyepiece. Ray #2 is the one that passes through the center of that lens. (It is not a continuation of ray #1; rays do not suddenly bend in empty space.) Since it came from the part of the image representing the top of the object, it is seen by the viewer as coming from that point in the (inverted) final image. The angular size of that image is thus α’ shown. Since the ratio of the sizes of the images on the viewer’s retina is determined by the ratio of their angular sizes, the viewer sees the final image to be larger than the object by the ratio "!/" . This is the angular magnification of the instrument. A pair of these telescopes side by side, each with a pair of prisms, makes up a pair of binoculars like those shown here. The prisms flip the image right-side up.

20. The Microscope A microscope also uses two lenses, but since the object can be placed as close as desired, one can get enlarged images from both lenses and thus higher magnification. In this case the objective has a short focal length and the object is placed a bit beyond its focal point. This gives an enlarged real image inside the microscope tube. The eyepiece is again used as a simple magnifier, this time producing a virtual image at the viewer's near point. The situation is as shown.

21. 30.6 The Eye The Anatomy of the Eye • The human eye is a complex anatomical device that remarkably demonstrates the architectural wonders of the human body. Like a camera, the eye is able to refract light and produce a focused image that can stimulate neural responses and enable the ability to see. In Lesson 6, we will focus on the physics of sight. We will use our understanding of refraction and image formation to understand the means by which the human eye produces images of distant and nearby objects. Additionally, we will investigate some of the common vision problems which plague humans and the customary solutions to those problems. As we proceed through the chapter, we will apply our understanding of refraction and lenses to the physics of sight. • The eye is essentially an opaque eyeball filled with a water-like fluid. In the front of the eyeball is a transparent opening known as the cornea. The cornea is a thin membrane which has an index of refraction of approximately 1.38. The cornea has the dual purpose of protecting the eye and refracting light as it enters the eye. After light passes through the cornea, a portion of it passes through an opening known as the pupil. Rather than being an actual part of the eye's anatomy, the pupil is merely an opening. The pupil is the black portion in the middle of the eyeball. It's black appearance is attributed to the fact that the light which the pupil allows to enter the eye is absorbed on the retina (and elsewhere) and does not exit the eye. Thus, as you sight at another

22. person's pupil opening, no light is exiting their pupil and coming to your eye; subsequently, the pupil appears black. • Like the aperture of a camera, the size of the pupil opening can be adjusted by the dilation of the iris. The iris is the colored part of the eye - being blue for some people and brown for others (and so forth); it is a diaphragm which is capable of stretching and reducing the size of the opening. In bright-light situations, the iris adjusts its size to reduce the pupil opening and limit the amount of light which enters the eye. And in dim-light situations, the iris adjusts so as to maximize the size of the pupil opening and increase the amount of light which enters the eye.

23. THE BLIND SPOT: • Many people don't realize they have a blind spot. Actually they have two of them, one in each eye. The point at the back of the eyeball where the optic nerve bundle enters is deficient in light receptors, both rods and cones, and therefore is a blind spot surrounded by the retina. It is displaced about 3mm from the optic axis. You can convince yourself of this by looking at the following picture with one eye covered. • Cover your right eye and look directly at the X with your left eye. Move closer or farther from the page until the O disappears. What has happened is that the image of the O falls directly on the blind spot. This indicates that the image of the O must be on the nasal

24. side of the retina of your left eye. If this is so, and your body is reasonably symmetric, you should be able to find the blind spot in your right eye. Look at the O with your right eye, covering your left eye. Move until the X disappears. • Notice how complete the disappearance is, even though the O and X are quite large. Estimate the angular diameter of the blind spot. This exercise may be done with a ruler held horizontally. Look at one of the numbers marking inches, and see which other number disappears. • The Camera and the Eye: In both the camera and the eye, the image is upside down, and this is compensated for in both cases. You simply turn the camera film around to look at it. Your brain has learned to turn around images it receives from the retina! Your cornea focuses the image on the retina by changing the thickness and shape of the lens. This is called accommodation and brought about by the action of the ciliary muscle, which surrounds the lens.

25. 30.7 Some Defects in Vision Three common vision problems are farsightedness, nearsightedness, and astigmatism. • Farsightedness or hyperopia is the inability of the eye to focus on nearby objects. The farsighted eye has no difficulty viewing distant objects. But the ability to view nearby objects requires a different lens shape - a shape which the farsighted eye is unable to assume. Subsequently, the farsighted eye is unable to focus on nearby objects. The problem most frequently arises during latter stages in life, as a result of the weakening of the ciliary muscles and/or the decreased flexibility of the lens. These two potential causes leads to the result that the lens of the eye can no longer assume the high curvature which is required to view nearby objects. The lens' power to refract light has diminished and the images of nearby objects are focused at a location behind the retina. On the retinal surface, where the light-detecting nerve cells are located, the image is not focused. These nerve cells thus detect a blurry image of nearby objects.

26. Nearsightedness or myopiais the inability of the eye to focus on distant objects. The nearsighted eye has no difficulty viewing nearby objects. But the ability to view distant objects requires that the light be refracted less. Nearsightedness will result if the light from distant objects is refracted more than is necessary. The problem is most common as a youth, and is usually the result of a bulging cornea or an elongated eyeball. If the cornea bulges more than its customary curvature, then it tends to refract light more than usual. This tends to cause the images of distant objects to form at locations in front of the retina. If the eyeball is elongated in the horizontal direction, then the retina is placed at a further distance from the cornea-lens system. Subsequently the images of distant objects form in front of the retina. On the retinal surface, where the light-detecting nerve cells are located, the image is not focused. These nerve cells thus detect a blurry image of distant objects.

27. Astigmatism of the eye is a defect that results when the cornea is curved more in one direction than the other, somewhat like the side of a barrel. Because of this defect, the eye dies not form sharp images. The remedy is cylindrical corrective lenses that have more curvature in one direction than in another. No lens gives a perfect image. The distortions in an image are called aberrations. Two types of aberrations are spherical aberration and chromatic aberration. 30.8 Some Defects of Lenses

28. Spherical aberration The curved surface of most lenses is a small section of a sphere. This is not the ideal shape and the resulting problem is called spherical aberration. After refraction by the lens the rays do not all pass through the same point. This is especially noticeable with the plane/convex lenses used in medium priced optical instruments. The effect is exaggerated in the diagram below. The effect can be reduced by sharing the refraction approximately equally between the two lens surfaces. If the object distance for the lens is large (>25m) then the curved face should be towards the object.

29. If the object distance is small, the lens is used the other way round. An alternative method of reducing the spherical aberration of a lens is simply to reduce the aperture using a diaphragm (also called an iris or a stop). Although this reduces the spherical aberration it also reduces the brightness of the image. Chromatic Aberration The refractive index of the material of which a lens is made is different for different wavelengths (colors) of light. Some dispersion of the light occurs, as with a prism. The resulting false coloring of the image is called chromatic aberration.

30. This effect can be reduced by having a combination of a convex and a concave lens made of glasses having different refractive indices. The dispersion caused by the convex lens is just cancelled by the dispersion caused by the concave lens.