Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Jac Phys 1 2E - 01 Page 1 Tuesday, October 21, 2003 1:41 PM AREA OF STUDY 1 Reflecting light Chapter 2 Refracting light Chapter 3 Seeing colours 1 Wave-like properties of light UNIT Chapter 1 Jac Phys 1 2E - 01 Page 2 Tuesday, October 21, 2003 1:41 PM Chapter Reflecting light 1 Remember Before beginning this chapter, you should be able to: • use ray diagrams to show how light is reflected from smooth surfaces. Key ideas After completing this chapter, you should be able to: • use the speed of light to calculate the time light takes to travel a stated distance • appreciate that light travels in straight lines and be able to give evidence for this • use the ray model to explain the reflection of light and the formation of images in plane and curved mirrors • describe images fully, referring to their location, size, orientation and nature • describe various applications of plane and curved mirrors. Figure 1.1 S traight rays of sunlight coming through the forest are seen because the light is scattered by airborne dust particles and light fog into our eyes. Jac Phys 1 2E - 01 Page 3 Tuesday, October 21, 2003 1:41 PM Sight is the sense by which humans and most other mammals get most of their information about the world. This sense responds to light. Questions about light naturally arise. Where does light come from? What can it do? How can its properties be explained? LIGHT AND ITS PROPERTIES Some obvious observations of light are: • Sources of light are needed to see. • Light travels very fast. • Light produces shadows. Sources of light Objects that give off their own light are described as luminous. Luminous objects that produce light as a result of being hot are described as incandescent. When we experience darkness at night or in an enclosed room, we know that a source of light, such as the Sun or a lamp, is needed to light up the darkness. Once a lamp is turned on, we can see features in the room because the light from the lamp shines on them and is then reflected into our eyes. This means that objects can be classified into two groups. Objects seen because they give off their own light are called luminous objects; those seen because they reflect light are called non-luminous objects. The Sun, torches and candles are luminous objects. Tables, chairs, cats and dogs are non-luminous objects. Some luminous objects produce light because they are hot. The Sun is one example. The higher the temperature, the brighter the light, and the colour also changes. These objects are called incandescent. Figure 1.2 The Pleiades open star cluster in the constellation Taurus. All stars are incandescent sources of light. CHAPTER 1 REFLECTING LIGHT 3 Inve Jac Phys 1 2E - 01 Page 4 Tuesday, October 21, 2003 1:41 PM ations stig Inve Investigation 1.1 Luminous or not? ations stig Investigation 1.2 Luminosity and temperature Other objects are cold and produce light in another way. It involves changes in the energy of electrons in the material brought about by either chemical or electrical processes. Speed of light The gap we experience between seeing lightning and hearing thunder shows that sound travels relatively slowly. Light seems to travel so fast that to our experience its speed seems infinite; that is, we seem to observe events at the instant they happen. Galileo Galilei (1564–1642) was not convinced of this. He attempted to determine the speed of light by measuring the time delay between the flash of his lamp to an assistant on a distant mountain and the return flash from his assistant’s lamp (see figure 1.3). No detectable delay was observed and Galileo concluded that the speed of light was very high. A longer distance was needed. Figure 1.3 Web Galileo used this method to measure the speed of light. He attempted to time, with his pulse, the delay between uncovering his lantern and seeing the light from his partner’s lantern, which his partner uncovered at the moment when he saw the light from Galileo’s lantern. s link Speed of light Skill cks che Significant figures (p. 493) 4 Olaus Roemer was a Danish astronomer born two years after Galileo’s death. He observed that the time between eclipses of Jupiter’s moons by Jupiter decreased as the Earth moved closer to Jupiter and increased as the Earth moved away. Roemer reasoned that this was because the distance the light travelled from Jupiter to Earth became greater as the Earth’s orbit took it further from Jupiter (see figure 1.4). Roemer used this time and the known diameter of the Earth’s orbit about the Sun to estimate the speed of light. The value he obtained was 2.7 × 108 m s−1. Eventually, in the nineteenth century, with stronger light sources and more precise timing devices, Galileo’s method could be used, but the assistant was replaced by a mirror. The values obtained then were about 3.0 × 108 m s−1. Early in the twentieth century, the American scientist Albert A. Michelson (1852–1931) used a rapidly rotating eight-sided mirror (see figure 1.5). The light was reflected to a distant mirror about 35 kilometres away then WAVE-LIKE PROPERTIES OF LIGHT Jac Phys 1 2E - 01 Page 5 Tuesday, October 21, 2003 1:41 PM Skill cks che SI units (p. 490) reflected back to the rotating mirror. For some particular rotation rates, this light is reflected by one of the sides of the rotating mirror directly to the observer. The rotation rate can be used to calculate the speed of light. The value Michelson obtained was 2.997 96 × 108 m s−1. He actually measured the distance of 35 km to an accuracy of 2.5 cm. The speed of light is currently measured at 2.997 924 58 × 108 m s−1. It is rounded off to 300 000 km s−1 for calculation purposes. observer B fixed mirror Io (moon) rotating mirror with eight sides A Sun Jupiter Earth’s orbit source of light Figure 1.4 The time, as seen from the Earth, for Jupiter’s moon, Io, to orbit Jupiter increases as the Earth moves from A to B. (The diagram is not to scale.) SAMPLE PROBLEM 1.1 Solution: Figure 1.5 Light from the source reflects off one of the sides of the rotating mirror towards a mirror 35 kilometres away. The returning beam hits the rotating mirror. If one of the sides of the mirror is in the right position, the light enters the eyepiece and can be seen by the observer. By measuring the speed of rotation when the beam enters the eyepiece, the speed of light can be calculated. How long does light take to travel from the Sun to the Earth? speed of light = 3.00 × 108 m s−1 distance from Sun to Earth = 1.49 × 1011 m distance travelled speed = --------------------------------------------time taken distance travelled ⇒ time taken = --------------------------------------------speed 11 1.49 × 10 m time = ---------------------------------------8 –1 3.00 × 10 m s = 0.497 × 103 s = 497 s = 8 minutes 17 seconds. Skill cks che Scientific notation (p. 492) SAMPLE PROBLEM 1.2 Solution: 35 km How far does light travel in one year (one light-year)? distance travelled = speed × time taken distance = 3.00 × 108 m s−1 × (365.25 × 24 × 60 × 60) s = 9.47 × 1015 m = 9.47 × 1012 km. Shadows The bright Sun produces sharp shadows on the ground. The shape of the shadow is the same shape as the object blocking the light (see figure 1.6). This could happen only if light travels in a straight line. CHAPTER 1 REFLECTING LIGHT 5 Jac Phys 1 2E - 01 Page 6 Tuesday, October 21, 2003 1:41 PM Figure 1.6 The straight rays passing the edge of a skateboarder leave a sharp shadow on the wall. Ray model A ray of light is a very narrow pencil-like beam of light. The need for sources of light, the great speed of light and the existence of sharp shadows can be described by a ray model. The model assumes that light travels in a straight line path called a light ray. A light ray can be considered as an infinitely narrow beam of light and can be represented as a straight line (see figure 1.7). Figure 1.7 Light rays leave a point on this pencil and travel in straight lines in all directions. The pencil is seen because of the ‘bundle’ of rays that enter the eye. PLANE MIRROR REFLECTION When you look at yourself in a plane mirror, some of the light rays from your nose, for example, travel in the direction of the mirror and reflect off in the direction of your eye (see figure 1.8). What is happening at the surface of the mirror to produce such a perfect image? mirror Figure 1.8 Light rays from the tip of the nose reflect off the mirror and enter the eye. 6 WAVE-LIKE PROPERTIES OF LIGHT Jac Phys 1 2E - 01 Page 7 Tuesday, October 21, 2003 1:41 PM The angle of incidence is the angle between the incident ray and the normal. The angle of reflection is the angle between the reflected ray and the normal. To investigate the reflection of light, the angles made by the rays need to be measured. Measurements of these angles show that, like a ball bouncing off a flat wall, the angle of incidence equals the angle of reflection (see figure 1.9). normal incident ray reflected ray angle of angle of incidence reflection mirror Figure 1.9 The ray approaching the mirror is called the incident ray. The ray leaving the mirror is called the reflected ray. The normal is a line at right angles to the mirror. The angles are measured between each ray and the normal. When the path of a light ray is traced, it is found that the angle of incidence always equals the angle of reflection. The normal is a line that is perpendicular to a surface or a boundary between two surfaces. The other seemingly trivial conclusion that can be drawn from the investigation is that the incident ray, the normal and the reflected ray all lie in the same plane (see figure 1.10). normal reflected ray Figure 1.10 The incident ray, the ‘normal’ to the surface of the mirror and the reflected ray all lie in the same plane, which is at right angles to the plane of the mirror. mirror incident ray REGULAR AND DIFFUSE REFLECTION Regular reflection, also referred to as specular reflection, is reflection from a smooth surface. Diffuse reflection is reflection from a rough or irregular surface. Reflection from a smooth surface is called regular or specular reflection. But what happens with an ordinary surface, such as this page? A page is not smooth like a mirror. At the microscopic level, there are ‘hills and valleys’. As the light rays come down into these hills and valleys, they still reflect with the two angles the same but, because the surface is irregular, the reflected rays emerge in all directions (see figure 1.11). This is called diffuse reflection. Light rays from diffuse reflections — from the ground, trees and other objects — enter the eye and enable the brain to make sense of the world. observer A Figure 1.11 This is diffuse reflection. Each of the incoming parallel rays meets the irregular surface at a different angle of incidence. The reflected rays will therefore go off in different directions, enabling observers in all directions to receive light from the surface; in other words, to see the surface. observer B irregular surface CHAPTER 1 REFLECTING LIGHT 7 Jac Phys 1 2E - 01 Page 8 Tuesday, October 21, 2003 1:41 PM FORMING IMAGES WITH A PLANE MIRROR Images are likenesses of objects. Web A virtual image is seen because light appears to be coming from it. Light doesn’t actually pass through it. Therefore a virtual image cannot be ‘captured’ on a screen. s link The law of reflection can now be used to explain how images are formed in plane mirrors and how to describe images. To describe an image, four aspects need to be determined: 1. its location — where it is in relation to the mirror 2. its size — how big it is compared with the object 3. its orientation — whether it is upright or upside down 4. its nature — whether or not the image can be captured on a screen. Figure 1.12 shows how the image that we see of a cat is formed. All the rays leaving the cat’s ear that reach the mirror are reflected so that they appear to be coming from the same region behind the screen. Some of these rays enter your eye. Someone else standing beside you will receive different rays even though the rays will appear to be coming from the same region. The region from which the rays appear to be coming is the location of the image of the cat’s ear. The image is located as far behind the mirror as the object is in front of the mirror. It is the same size and upright. The light rays entering the eye only appear to come from the location of the image. The image is described as a virtual image. A virtual image cannot be ‘captured’ on a screen. mirror A1 A B1 B Image in a plane mirror applets Figure 1.12 When the rays of light from the cat’s ear reflect off the mirror, they enter a person’s eye as if they had come from a region behind the mirror. This region is the location of the virtual image. It is located as far behind the mirror as the cat’s ear is in front of the mirror. Lateral inversion Lateral inversion is the apparent sideways reversal of an image in a mirror when compared to the object in front of the mirror. When you look into a mirror, you do not see yourself as others see you. Your left and right sides appear reversed. This is called lateral inversion. Some safety vehicles, such as ambulances, have their front labels adjusted for lateral inversion so that drivers looking in their rear vision mirrors will see the letters the right way round (see figures 1.13 and 1.14). mirror A B Figure 1.13 A1 B1 The rays from the letter ‘J’ reflect off the mirror and into the eye. The loop of the ‘J’, from the eye’s point of view, is on the left, whereas in the image the loop is on the right. This situation depends on where the eye is. How would this situation look if the eye moved to the far left of the diagram, behind the ‘J’? 8 WAVE-LIKE PROPERTIES OF LIGHT Inve Jac Phys 1 2E - 01 Page 9 Friday, October 24, 2003 1:19 PM ations stig Figure 1.14 Some vehicles, such as ambulances, have their name laterally inverted on the front of the vehicle. When the word is seen through the rear vision mirror of the car in front, it appears the right way around and triggers a more immediate response on the part of the car’s driver. Inve Investigation 1.3 Lateral inversion ations stig Inve Investigation 1.4 Up periscope! ations stig Investigation 1.5 Multiple images Rotating mirrors Another use of plane mirrors involves the effect on the reflected ray when a mirror is turned. When a mirror rotates through a small angle, the reflected ray is moved through twice that angle. This effect is put to good use in sensitive scientific instruments. Sometimes scientific instruments are used to measure very small values of quantities, such as electric currents. A very small electrical input might turn a mirror slightly. But when a beam of light is shone onto the mirror and reflected onto a wall across the room, a more measurable deflection can be observed (see figure 1.15). mirror g r tiltin y afte d ra flecte re reflected ray before tilting 2θ Web incoming ray Figure 1.15 s link θ One-way mirror The principle of the mirror or spot mirror galvanometer. A ray from a light source, for example, can be tilted reflected across a large distance. A very small rotation of the mirror can move a light spot on a distant wall several centimetres. AS A MATTER OF FACT A normal mirror has a thick coating of silver on the rear side of the glass. The silver reflects in a regular way because the glass surface is optically flat. The outer surface of the silver cannot be used because it does not reflect evenly and it corrodes, so the silver is covered with a protective paint. A two-way mirror, on the other hand, has a thinner coating of silver. Some light can pass through this, but most is reflected. From the front, the mirror produces a normal mirror image. From the rear, sufficient light passes through for the sensitive human eye to see clearly beyond it. However, a ‘secret observer’, behind the mirror, can be detected because light can travel along a light path in both directions. If a light globe is turned on behind the mirror, some light reflecting off the secret observer will pass through the mirror to the front. The viewer in front of the mirror will be able to see the faint appearance of the secret observer. The secret observer cannot be seen if there is no light source behind the mirror. Figure 1.16 CHAPTER 1 REFLECTING LIGHT 9 Jac Phys 1 2E - 01 Page 10 Tuesday, October 21, 2003 1:41 PM CONCAVE MIRRORS A concave mirror is a curved mirror with the reflecting surface on the inside of the curve. When a light source is placed in front of a plane mirror, the reflected rays continue to move apart or diverge. If the mirror is curved into a concave shape, the reflected rays would spread less widely. With just the right shape, the reflected rays form in parallel lines. The shape to achieve this is a parabola (see figure 1.17(c)). The parabolic shape is used in the design of torches, car headlights and searchlights. (a) Plane mirror (b) Circular concave mirror Figure 1.17 (a) Rays from a light source in front of a plane mirror are reflected in all directions. (b) The curved circular mirror brings the reflected rays inwards. (c) When the curved mirror is in the shape of a parabola, the reflected rays become parallel to each other and the light source is said to be at the focus of the curved mirror. The focus of a concave mirror is the point through which a set of rays passes that is parallel to the principal axis and reflected from the mirror. The principal axis of a curved mirror is a line passing through the centre of the mirror and which is perpendicular to the plane of the mirror. The focal length is the distance from the centre of a curved mirror to its focus. 10 (c) Parabolic concave mirror Figure 1.18 In a car headlight, the filament of the globe is placed at the focus of the curved reflective surface. This produces a narrow beam of light. Light can travel either way along a light path, so the arrows of the light rays in figure 1.17 can be reversed. This means that all the parallel rays of light that hit the concave mirror will pass through a point. This point therefore has special significance. It is called the focus of the concave mirror. The line through the centre of the mirror and the focus is called the principal axis and the distance from the centre of the mirror to the focus is called the focal length. WAVE-LIKE PROPERTIES OF LIGHT Jac Phys 1 2E - 01 Page 11 Tuesday, October 21, 2003 1:41 PM The pole of a curved mirror is at the centre of its reflecting surface. The centre of curvature of a concave mirror is a point on the principal axis at twice the focal length from the pole of the mirror. A real image is one through which light passes. A real image can be seen on a screen placed at the location of the image. Ray tracing can be used to investiconcave gate further the properties of the mirror principal axis image formed by a concave mirror. Ray tracing is a graphical technique for finding the position and size of images formed by mirrors. centre of focus (F) As shown in figure 1.19, the centre curvature (C) pole of a concave mirror is called the pole. The centre of curvature (see figure 1.20) is a point twice as far from the pole as the focus. This statement is true as long as the curved mirror is Figure 1.19 For small small. For small mirrors, the circular mirrors, the focus is halfway or spherical curved shape is easier to between the centre of curvature manufacture than the ideal parabolic and the pole of the mirror. shape and the difference between the centre of two shapes — spherical and parabolic curvature — would be minor. For specialist mirrors, such as in the Hubble telescope, the parabolic shape is used. The ray model can now be applied to the concave mirror. In figure 1.21, rays of light spread out from the head of the object, hit the mirror and pass through the image. This is called a real image. (The concave mirror has been represented as a straight line because the Figure 1.20 A curved rays are close to the principal axis.) or spherical mirror can be This is called a real image because the imagined as being part of a light passes through this location and complete sphere. The centre of the image can be captured on a screen. curvature is at the point that would Four rays can be drawn to locate the be at the centre of that sphere. image. • Ray 1 leaves the head of the object parallel to the principal axis, reaches the mirror and is reflected back through the focus. • Ray 2 passes through the focus before reaching the mirror. It is then reflected in a direction parallel to the principal axis. • Ray 3 travels towards the pole of the mirror. At this point, the section of the concave mirror is at right angles to the principal axis, so the ray is reflected at an equal angle below the axis. Thus the reflected ray passes the object at the same distance below the axis that the object was above it. • Ray 4 passes through the centre of curvature before reaching the mirror. Because the centre of curvature is the centre of a sphere, this ray is travelling along the radius. The ray therefore meets the spherical mirror at zero angle of incidence and thus is reflected back along the same line. In figure 1.22, light rays drawn from the head of the object are reflected by the concave mirror and all appear to come from another point behind the mirror. This is the image of the head of the object. It is called a virtual image because no light passes through it and it cannot be captured on a screen. The four rays have been used to locate the image, but every single ray from the head of the object that reaches the mirror appears to come from the image. In figures 1.21 and 1.22, the rays that leave the base of the object along the principal axis towards C and F all come back along the axis. This means that the image of the base of the object is also on the axis. CHAPTER 1 REFLECTING LIGHT 11 eMo Jac Phys 1 2E - 01 Page 12 Tuesday, October 21, 2003 1:41 PM 1 ling del object Model of a concave mirror C F 2 image Inve 3 ations stig 1 4 Figure 1.21 Web Investigation 1.6 Concave mirrors — and observation exercise s link Concave mirror applet The magnification of a mirror is the ratio of the height of the image to the height of the object. Light rays from the head of the object are reflected by the concave mirror and all pass through another point, which is the image of the head of the object. This is called a real image because the light passes through this location and the image can be captured on a screen. The diagram includes only four rays for the purpose of locating the image, but every single ray from the head of the object that hits the mirror passes through the head of the image. Note that the shaded section of figures 1.21 and 1.22 indicates that all the rays from the head of the object that reach the mirror, not only the four rays described on page 11, will pass through the head of the image. The image can now be described as follows. • Location: The scale of the diagram can be used to work out the dis2 tance of the image from the mirror. • Size: Similarly, the vertical scale — or the comparison of the size of the image with the size of the object — object image can be used to calculate the height of the image. The magnification is C F the height of the image divided by 4 the height of the object. • Orientation: The appearance of the 1 image immediately tells you whether 3 it is the same way up as the object or upside down. If it is on the same side of the principal axis, then the Figure 1.22 All four rays appear to come from the same image is upright. • Nature: The image is real if the rays point. This is where the image of actually pass through its location. the head of the object is located This means that a screen can be put and it is a virtual image. Note that at this point and the image will we could have located the image appear on it. If the rays only look as with any two of the rays, but either if they are coming from the image’s or both of the others could be used location, as in a plane mirror, then to confirm the original location of the image. the image is a virtual image. Looking at stars The focal plane is at right angles to the principal axis and passes through the focus. 12 Parallel rays reaching a concave mirror could be coming from a very distant object, such as a star, so all the light hitting the mirror passes through the focus. This same design can be applied to a telescope in which a larger mirror can collect more light and enable astronomers to see fainter stars. The light from stars at an angle to the principal axis is still parallel and is brought to a focus at a point above or below the focus, but in the same focal plane (see figure 1.23). WAVE-LIKE PROPERTIES OF LIGHT Jac Phys 1 2E - 01 Page 13 Tuesday, October 21, 2003 1:41 PM Figure 1.23 parallel light from star off principal axis principal axis • F concave focal mirror plane If a concave mirror is oriented so that its principal axis is pointing directly at a star, then light from that star will be collected at the focus. For stars off to the side, at a slight angle to the axis, their light is collected at a point to the side of the focus, but still in the same plane at right angles to the axis. This plane is called the focal plane. Figure 1.24 Inve Radio telescopes use parabolic concave mirrors to collect radio waves from distant stars. ations stig Investigation 1.7 Locating images in concave mirrors However, if the light comes from a light source close to the concave mirror so that the incident rays are not parallel, the paths of the reflected rays are not so easy to predict. Using a formula to model images in concave mirrors Ray tracing diagrams can be used to develop a mathematical relationship between the positions of the object and the image relative to the mirror. As shown in figure 1.25: u = distance from object to mirror v = distance of image from mirror f = focal length u Ho = height of object B D Hi = height of image. object Ho Figure 1.25 C A The description of an image can be determined by use of a formula. G F O Hi image E v positive distances negative distances CHAPTER 1 REFLECTING LIGHT 13 Jac Phys 1 2E - 01 Page 14 Friday, October 24, 2003 9:09 AM In figure 1.25 the shaded triangles BAO and EGO are similar. Hi v Therefore: ------- = --u Ho where H -------i is equal to the magnification, M, of the mirror. Ho [1] The triangles DOF and EGF are also similar. Hi v–f Therefore: ------- = ---------Ho f ⇒ ⇒ ⇒ ⇒ v v–f --- = ---------- (from [1]) f u vf = uv − uf vf + uf = uv 1 1 1 --- + --- = --- (dividing by uvf ) u v f [2] Because the image can be either in front of the mirror and therefore a real image or behind the mirror and therefore a virtual image, a sign convention needs to be employed so as to distinguish these two possibilities. For real images that are in front of the mirror, the image distance is positive. For virtual images that are behind the mirror, the image distance is negative. Virtual images have a negative image distance, giving a negative value v to the ratio ---u- . However, the magnification is the ratio of two heights and therefore cannot be negative. In such cases, only the size of the numerical answer is taken. SAMPLE PROBLEM 1.3 Locate and describe the image formed by a 3.0 cm high object 25 cm in front of a concave mirror with a focal length of 10 cm. Ho = 3.0 cm; Hi = ?; u = 25 cm; v = ?; f = 10 cm Solution: 1 1 1 Using --- + --- = --u v f 1 1 1 ------ + --- = -----25 v 10 1 1 1 --- = ------ − -----⇒ v 10 25 (5 – 2) = ----------------50 ⇒ ⇒ 14 WAVE-LIKE PROPERTIES OF LIGHT 3 = -----50 v 50 --- = -----1 3 50 v = -----3 = 16.67 = 17 (substituting data into mirror formula) (combining fractions using lowest common denominator) (inverting each side of the equation) (rounding the answer off to two significant figures) Jac Phys 1 2E - 01 Page 15 Tuesday, October 21, 2003 1:41 PM Skill cks che Significant figures (p. 493) SAMPLE PROBLEM 1.4 The image is 17.0 cm in front of the mirror. It is real and inverted. Hi v Magnification = ------- = --Ho u H 17 -------i = -----3.0 25 2 = --3 2 ⇒ Hi = --- × 3.0 3 = 2.0 The image is 2.0 cm high. Locate and describe the image formed by a 5.0 cm high object 10 cm in front of a concave mirror with a focal length of 15 cm. Solution: Ho = 5.0 cm; Hi = ?; u = 10 cm; v = ?; f = 15 cm Inve Using ations stig ⇒ eMo Investigation 1.8 Soup spoon ling del Exploring a concave mirror with a spreadsheet 1 1 1 --- + --- = --u v f 1 1 1 ------ + --- = -----10 v 15 1 1 1 --- = ------ − -----v 15 10 (2 – 3) = ----------------30 (substituting data into mirror formula) (combining fractions using lowest common denominator) –1 = -----30 v 30 --- = -----⇒ (inverting each side of the equation) 1 –1 ⇒ v = −30 The image is 30 cm behind the mirror. It is virtual and upright. Hi v Magnification = ------- = --Ho u H 30 -------i = ------ = 3 5.0 10 Hi = 3 × 5.0 = 15 The image is 15 cm high. CONVEX MIRRORS A convex mirror is a curved mirror with the reflecting surface on the outside of the curve. Convex mirrors curve outwards rather than inwards. If you look through a convex mirror, you have a wider angle of view. Convex mirrors are used as security mirrors in supermarkets and for safety purposes to assist drivers leaving driveways with restricted views of the traffic (see figures 1.26 and 1.27). Like the concave mirror, the convex mirror has a focus, but it is a virtual focus because it is behind the mirror (see figure 1.28). Incident light rays parallel to the principal axis are reflected away as if they had come from the focus behind the mirror. However, the rays used in the ray tracing for the concave mirror can still be used if you keep in mind that the focus and the centre of curvature are on the other side of the mirror. CHAPTER 1 REFLECTING LIGHT 15 Jac Phys 1 2E - 01 Page 16 Tuesday, October 21, 2003 1:41 PM The rays from the object spread out or diverge after hitting the mirror, so it is impossible to form a real image with a convex mirror. Figure 1.26 Web Convex mirrors are used when a wide view of an area is required. eye s link 3 object 1 Convex mirror applet image F 2 convex mirror convex mirror Figure 1.28 Figure 1.27 Inve A convex mirror is curved outwards. A person standing in front of one will receive light coming from a very wide angle. Compare this with figure 1.17. ations stig Investigation 1.9 Convex mirrors SAMPLE PROBLEM Ray 1 heading towards the focus is reflected parallel to the principal axis. Ray 2 going towards the pole is reflected at an equal angle below the axis. Ray 3 parallel to the principal axis is reflected up as if it came from the focus. 1.5 Solution: The same formula as was used for concave mirrors (see page 14) also works for convex mirrors, but the focal length is represented by a negative number because the focus is a virtual one. For this reason, convex mirrors are sometimes called negative mirrors. Locate and describe the image formed by a 5.0 cm high object 10 cm in front of a convex mirror with a focal length of 15 cm. Ho = 5.0 cm; Hi = ?; u = 10 cm; v = ?; f = −15 cm 1 1 1 Using --- + --- = --u v f 1 1 1 ------ + --- = -----10 v – 5 1 1 1 --- = − ------ − -----⇒ v 15 10 ( −2 + −3 ) = ------------------------ (combining fractions using lowest common 30 denominator) –5 = -----30 v 30 --- = -----⇒ (inverting each side of the equation) 1 –5 ⇒ v = −6.0 The image is 6.0 cm behind the mirror. It is virtual and upright. 16 WAVE-LIKE PROPERTIES OF LIGHT Jac Phys 1 2E - 01 Page 17 Tuesday, October 21, 2003 1:41 PM eMo Hi v Magnification = ------- = --Ho u ling del Exploring a convex mirror with a spreadsheet H 6.0 -------i = ------5.0 10 = 0.6 ⇒ Hi = 0.6 × 5.0 = 3.0 The image is 3.0 cm high. USING CONCAVE MIRRORS IN THE REFLECTING TELESCOPE The reflecting telescope (see figure 1.29) was designed by Isaac Newton (1642–1727) to overcome problems with telescopes that used only lenses. Lenses can produce distortion of the image in that the light can spread into colours when it passes through them (see chapter 3) and large lenses sometimes sag when supported at their edges. Also, large mirrors with one curved surface are easier to make than large lenses with two curved surfaces. As shown in figure 1.30, light from a distant star arrives as parallel rays, so the image of the star is formed in a plane at the focus (F) of the concave mirror. A small, tilted plane mirror is placed just in front of the focus and reflects light out of the side of the tube into a lens which reproduces the original parallel rays. This means the image is magnified. The actual magnification is the ratio of the two focal lengths. focal length of concave mirror Magnification = -------------------------------------------------------------------------------------------------------------focal length of lens Figure 1.29 The Newtonian telescope, with a tube 11.43 cm in diameter concave mirror plane mirror parallel light from distant star F Figure 1.30 convex lens The telescope designed by Isaac Newton uses a concave mirror to bring the light from a distant star to a focus but, before the light reaches the focus, it is reflected out to the side by a plane mirror into a convex lens. CHAPTER 1 REFLECTING LIGHT 17 Jac Phys 1 2E - 01 Page 18 Tuesday, October 21, 2003 1:41 PM As telescopes became more concave mirror powerful, that is, able to collect more light and hence see more stars and convex more detail, the Newtonian design mirror became cumbersome. The French optician N. Cassegrain improved the design by cutting a hole in the middle Figure 1.31 Diagram of the concave mirror and reflecting of a Cassegrain telescope the light through it with a very small convex mirror placed in front of the large concave mirror (see figure 1.31). AS A MATTER OF FACT T Web he eyes of most animals contain transparent lenses. The scallop, however, uses a concave mirror to focus light onto its retina. The retina is the surface of receptor cells that send signals to the brain. s link Scallop eyes concave mirror retina Figure 1.32 A scallop’s eye. The scallop uses a set of reflective cells in the shape of a concave mirror to produce a real image on its ‘screen’ or retina. PROBLEMS WITH CURVED MIRRORS Spherical aberration is the distortion of an image produced by a concave mirror that is spherical rather than parabolic. In small mirrors spherical aberration is less noticeable than in larger mirrors. 18 As discussed on page 11, most small concave mirrors are spherical rather than parabolic. In small mirrors, differences between the circle and the parabola are not great. For larger mirrors or when high quality images are required, the distortion or spreading of the image produced by a spherical mirror is a problem. This distortion is called spherical aberration. It occurs because a spherical mirror does not reflect parallel rays to a single point. The focus is ‘spread out’, as shown in figure 1.33. WAVE-LIKE PROPERTIES OF LIGHT Jac Phys 1 2E - 01 Page 19 Tuesday, October 21, 2003 1:41 PM (a) Large spherical mirror parallel rays (b) Parabolic mirror parallel rays Figure 1.33 Inve (a) A large spherical mirror reflects the outer parallel rays to a different point from the inner rays. This produces a distorted image, referred to as a spherical aberration. (b) A parabolic mirror brings all the rays to a single point. ations stig An example of spherical aberration is the pattern of light produced by a ceiling light reflecting from the inside of the coffee cup onto the surface of the coffee. The full circular shape of the mug spreads the point focus out into a curve, called a caustic curve. Investigation 1.10 Caustic curve CHAPTER 1 REFLECTING LIGHT 19 Jac Phys 1 2E - 01 Page 20 Tuesday, October 21, 2003 1:41 PM • Light sources are called luminous objects. Some luminous objects give off light because they are hot; these are called incandescent objects. Radius of Earth’s orbit about the Sun = 1.49 × 1011 m Radius of Mars’s orbit about the Sun = 2.28 × 1011 m Radius of Neptune’s orbit about the Sun = 4.50 × 1012 m • Light travels in straight lines through air at a speed of 3.0 × 108 m s–1. Shadows provide evidence that light travels in straight lines. 5. Copy the following figure and draw the incident and reflected rays from the two ends of the object to the eye. Locate the image. SUMMARY • Modelling light as a pencil-like ray helps describe the reflection of light. object plane mirror • When light meets a surface the angle of incidence equals the angle of reflection. The incident ray, the normal to the surface and the reflected ray all lie in the same plane. • If the surface is smooth like that of a mirror, the regular reflection enables images to be seen in the mirror. 6. Calculate the angles, a, b and c in the following figure. • The image in a plane mirror is as far behind the mirror as the object is in front of the mirror. • If a mirror is curved, the properties of the images formed by the mirror are different from the properties of the images in plane mirrors. CHAPTER REVIEW • The ray model explains the formation and properties of images in plane and curved mirrors. • The formation of images in concave and convex mirrors can be mathematically modelled with 50° c a v f Ho A A u QUESTIONS Understanding 1. Draw a Venn diagram for the following words which describe the different objects you can see: luminous, non-luminous, and incandescent. 2. Describe the light path from a light source to your eye in seeing an object. 3. Use the ray model and the sources of light to rephrase the statements (a) ‘I looked at a flower through the window’ and (b) ‘I watched the TV’. 4. Calculate the longest and shortest time for a radio signal travelling at the speed of light to go from the Earth to a space probe when the space probe is (a) near Mars and (b) near Neptune. 20 WAVE-LIKE PROPERTIES OF LIGHT b 7. The two arrowed lines in the figures below represent reflected rays. The line AB represents the plane mirror. Locate the image and the light source in each of the two figures. Hi v - = --- . the equations --1- + 1--- = 1--- and M = ------u mirror B B 8. What type of car does ATOYOT Car Sales sell? 9. A student argues that you cannot photograph a virtual image because light rays do not pass through the space where the image is formed. How would you argue against this statement? 10. Sketch the path of each of the rays entering each of the pair of joined mirrors in the following figure. Jac Phys 1 2E - 01 Page 21 Tuesday, October 21, 2003 1:41 PM 19. (a) You are standing 2.0 m in front of a plane mirror and you wish to take a sharp photograph of yourself in the mirror. At what distance do you set the camera lens? (b) Your friend is standing beside you, 1.0 m away. At what distance do you set the camera lens for a sharp photograph of your friend? 12. You are walking towards a plane mirror at a speed of 1.0 m s−1. How fast is your image walking? How quickly are you and your image approaching each other? 20. Use two or more plane mirrors to produce an image of you that is not laterally inverted; that is, you see yourself as others see you. Use two point sources, marked L and R, at the end of a short line. 13. Use your understanding of the reflection of light to explain how full gloss and matt paints differ. 14. Produce a table summarising all the properties of the image produced by a concave mirror for the following object positions: beyond the centre of curvature (C); at C; between C and the focus (F); at F; inside F. 15. Use either a scaled drawing or the concave mirror formula to determine the full description of the image (that is, size, location, orientation and nature) for the following situations: (a) a 2.0 cm high object 12 cm from a concave mirror with a focal length of 8.0 cm (b) a 5.0 cm high object 1.0 m from a concave mirror with a focal length of 10 cm (c) a 4.0 cm high object 12 cm from a concave mirror with a focal length of 20 cm. 16. The formula for the concave mirror can also be used for convex mirrors with only one change: making the focal length a negative quantity. Determine the nature of the image of a 4.0 cm object 15 cm in front of a convex mirror with a focal length of 10 cm. Application 17. Explain how early astronomers knew the Moon must have a rough surface. 18. Assume that your eyes are located at the top of your 1.60 m tall body. (a) What size plane mirror would you need to see your whole body if you were standing 1 m from it? (b) How far from the floor should the bottom edge of the mirror be placed? (c) How would your answers to (a) and (b) change if you assumed your eyes were 10 cm from the top of your head? (d) How would your answers to (a) and (b) change if you were 2 m from the mirror? Use diagrams and words to explain your answers. 21. Design a system of plane mirrors to enable an immobile, bedridden patient to read a book placed on the patient’s chest. 22. Draw three ray diagrams to investigate how the size of an image in a plane mirror depends on the distance of the object from the mirror. 23. A plane mirror reflects a ray of light and changes its direction. The angle of deviation is the angle between the reflected ray and the continuation of the incident ray. How is the angle of deviation related to the glancing angle (that is, the angle between the incident ray and the mirror)? 24. The figure below shows an incident ray (R1) reflected off a mirror (m1). The mirror is rotated (m2) and the reflected ray moves (R2). Use the angles in the diagram to prove that the angle of deflection of the reflected ray equals twice the angle of rotation of the mirror. I N2 N1 m1 m2 a R2 R1 a m2 mirror m1 position 25. Prove that the distance that light travels from a source, P, to another point, Q, via a plane mirror is the shortest when the angle of incidence equals the angle of reflection. (Hint: Draw a straight line from P to the image of Q.) 26. What is the focal length of a plane mirror? 27. A make-up or shaving mirror is a concave mirror designed to produce a magnified image. Your face is inside the focus. A comfortable distance from your face to the mirror would be about 30 cm. (a) If you want a magnification of 2.0, what should be the focal length of the mirror? (b) Where is your image located? CHAPTER 1 REFLECTING LIGHT 21 CHAPTER REVIEW 11. Sketch the path of a ray emitted from a point between two parallel mirrors (see the following figure). Jac Phys 1 2E - 01 Page 22 Tuesday, October 21, 2003 1:41 PM (c) If the height of your face is 16 cm, what diameter mirror would you need to see your whole face? 28. A dentist asks you to design a concave mirror to produce a magnified upright image of a tooth. The magnification should be about 3 with a space of 2 cm between the tooth and the mirror. (a) What focal length mirror will you design? (b) The dentist tests out your mirror but rejects it. She says that it produced a distorted image of the tooth because teeth are three-dimensional objects. Why was the image distorted? 29. A concave mirror with a focal length of 20 cm produces an image located an infinite distance away of an object placed in front of the mirror. Where is the object located? 30. Find the size of the image of the Moon formed by a concave mirror with a focal length of 20 cm if it is 3.8 × 105 km away and its radius is 1740 km. How could the image be made brighter? How could it be made bigger? 31. A real image is formed by a concave mirror. How will the image change if the top half of the mirror is removed? CHAPTER REVIEW More of a challenge 32. When you look into a plane mirror, your left and right sides appear reversed. This is called ‘lateral inversion’. How could you place the mirror so that your image is upside down? 22 WAVE-LIKE PROPERTIES OF LIGHT 33. How can you see raindrops if water is transparent? 34. A laser beam is sent from the surface of the Earth to a corner reflector which was left on the Moon by the Apollo astronauts. The reflector sends the beam back to Earth in the same direction. The time of the round trip for the beam of light is about 2.479 seconds. (a) Calculate the distance of the centre of the Moon from the centre of the Earth, given the following data: speed of light = 2.998 × 108 m s−1 radius of Earth = 6380 km radius of Moon = 1740 km. (b) The Earth was continuing to spin while the laser beam was in transit. How far did the receiver on the Earth’s surface move in that time? (c) What are the implications of these results for the design of this exercise? 35. A corner reflector is made of three plane mirrors each at right angles to each other, like the corner of a room. How many images would you see? 36. The smooth surface of a very large lake is slightly curved because the Earth is round. In fact, the surface is a convex mirror with a focal length equal to half the Earth’s radius. If the radius of the Earth is 6380 km and the moon, which has a radius of 1740 km, is 3.8 × 105 km away, calculate the location and size of the Moon’s image formed by the lake’s surface.