CH 41 - Optics Reflection PDF

Chapter 1 - Reflection and Mirrors (Geometrical)  Objectives: After completing this module, you should be able to: Explain and discuss with diagrams, reflection, absorption, and refraction of light rays. Illustrate graphically the reflection of light from plane, convex, and...

Chapter 1 - Reflection and Mirrors (Geometrical)  Objectives: After completing this module, you should be able to: Explain and discuss with diagrams, reflection, absorption, and refraction of light rays. Illustrate graphically the reflection of light from plane, convex, and concave mirrors. Define and illustrate your understanding of real, virtual, erect, inverted, enlarged, and diminished as applied to images. Use geometrical optics to draw images of an object at various distances from converging and diverging mirrors.  Geometrical Optics In the study of how light behaves, it is useful to use “light rays” and the fact that light travels in straight lines. When light strikes the Air reflection boundary between two media, three things may happen: reflection, absorption Water refraction, or absorption. refraction Reflection, Refraction, and Absorption Air reflection Reflection: A ray from air strikes the water and returns to the air. absorption Water refraction Refraction: A ray bends into Absorption: A ray is absorbed the water toward the normal atomically by the water and does line. not reappear.  The Laws of Reflection 1. The angle of inci- Air N reflection dence qi is equal to the angle of qi qr reflection qr : Water qi = qr All ray angles are measured with respect to normal N. 2. The incident ray, the 3. The rays are reflected ray, and completely the normal N all lie reversible. in the same plane.  The Plane Mirror A mirror is a highly polished surface that forms images by uniformly reflected light. Note: images appear to be equi-distant behind mirror and are right- left reversed.  Definitions Object distance: The straight-line distance p from the surface of a mirror to the object. Image distance: The straight-line distance q from the surface of a mirror to the image. Object Image Object Image distance = distance p=q p q qi = qr  Real and Virtual Real images and objects are formed by actual rays of light. (Real images Light rays No light can be projected on a screen.) Real Virtual object image Virtual images and objects do not really exist, but only seem to be at a Virtual images are on the opposite side of location. the mirror from the incoming rays.  Image of a Point Object Plane mirror q=p p q Virtual Real object image Image appears to be at same distance behind mirror regardless of viewing angle.  Image of an Extended Object Plane mirror q=p p q Virtual image Image of bottom and top of guitar shows forward-back, right-left reversals.  Terms for Spherical Mirrors A spherical mirror is formed by the inside (concave) or outside Concave Mirror (convex) surfaces of a sphere. R Axis V A concave spherical mirror is C shown here with parts identified. Linear aperture Center of Curvature C The axis and linear aperture are Radius of curvature R shown. Vertex V  The Focal Length f of a Mirror Since qi = qr, we find that F is Incident parallel ray mid- way between V and C; we find: qi R qr C F V axis f The focal length f is: Focal point R The focal length, f f  2 The focal length f is equal to half the radius R  The Focus of a Concave Mirror The focal point F for a concave mirror is the point at which all parallel light rays converge. For objects located at infinity, the Incident parallel real image appears at the focal Rays point since rays of light are almost parallel. F C axis R f  Focal point 2  The Focus of a Convex Mirror The focal point for a convex mirror is the point F from which all parallel light rays diverge. Virtual focus; Incident Rays reflected rays diverge. R axis C F R f  2 Reflected Rays  Image Construction: Ray 1: A ray parallel to mirror axis passes through the focal point of a concave mirror or appears to come from the focal point of a convex mirror. Ray 1 Convex Ray 1 mirror C F C F Object Object Concave mirror  Image Construction (Cont.): Ray 2: A ray passing through the focus of a concave mirror or proceeding toward the focus of a convex mirror is reflected parallel to the mirror axis. Ray 1 Convex Ray 1 mirror Ray 2 C F Ray 2 Image C F Image Concave mirror  Image Construction (Cont.): Ray 3: A ray that proceeds along a radius is always reflected back along its original path. Ray 1 Ray 3 Ray 1 Ray 2 Ray 2 C F C F Image Ray 3 Concave Convex mirror mirror  Object at Focal Point When the object is located at the focal point of the mirror, the Ray 1 image is not formed (or it is Ray 3 located at infinity). C F Reflected rays are parallel The parallel reflected rays never cross. Image is located at infinity (not formed).  Object Inside Focal Point 1. The image is erect; i.e., same orientation as the object. C 2. The image is virtual; F Virtual Erect and that is, it seems to be image enlarged located behind mirror. 3. The image is enlarged; Image is located bigger than the object. behind the mirror  Convex Mirror Imaging All images are erect, virtual, and diminished. Images get larger as object Convex Ray 1 approaches. mirror 2 C F Image gets larger as object gets closer  Converging and Diverging Mirrors Concave mirrors and converging parallel rays will Convex mirrors and diverging parallel rays be called converging mirrors from this point will be called diverging mirrors from this onward. point onward. Converging Mirror Diverging Mirror F C C F Concave Convex  Summary (Definitions) Object distance: The straight-line distance p from the surface of a mirror to the object. Image distance: The straight-line distance q from the surface of a mirror to the image. Real image: An image formed by real light rays that can be projected on a screen. Virtual image: An image that appears to be at a location where no light rays reach. Converging and diverging mirrors: Refer to the reflection of parallel rays from surface of mirror.  Image Construction Summary: Ray 1: A ray parallel to mirror axis passes through the focal point of a concave mirror or appears to come from the focal point of a convex mirror. Ray 2: A ray passing through the focus of a concave mirror or proceeding toward the focus of a convex mirror is reflected parallel to mirror axis. Ray 3: A ray that proceeds along a radius is always reflected back along its original path.  Objectives: After completing this module, you should be able to: Define and illustrate the following terms: real and virtual images, converging and diverging mirrors, focal length, and magnification. Understand and apply the sign conventions that apply to focal lengths, image distances, image heights, and magnification. Predict mathematically the nature, size, and location of images formed by spherical mirrors. Determine mathematically the magnification and/or the focal length of spherical mirrors.  Analytical Optics In this unit, we will discuss analytical relationships to describe mirror images more accurately. But first we will review some graphical principles covered in Module 34a on light reflection.  The Plane Mirror Object Image Object Image distance = distance p=q p q Image is virtual Object distance: The straight-line distance p from the surface of a mirror to the object. Image distance: The straight-line distance q from the surface of a mirror to the image.  Spherical Mirrors A spherical mirror is formed by the inside (concave) or outside Concave Mirror (convex) surfaces of a sphere. R Axis V A concave spherical mirror is C shown here with parts identified. Linear aperture Center of Curvature C The axis and linear aperture are Radius of Curvature R shown. Vertex V  The Focal Length f of a Mirror Since qi = qr, we find that F is Incident parallel ray mid- way between V and C; we find: qi R qr C F V axis f The focal length f is: Focal point R The focal length, f f  2 The focal length f is equal to half the radius R Converging and Diverging Mirrors Concave mirrors and converging parallel rays Convex mirrors and diverging parallel will be called converging mirrors. rays will be called diverging mirrors. Converging Mirror Diverging Mirror F C C F Concave Convex  Definitions Focal length: The straight-line distance f from the surface of a mirror to focus of the mirror. Magnification: The ratio of the size of the image to the size of the object. Real image: An image formed by real light rays that can be projected on a screen. Virtual image: An image that appears to be at a location where no light rays reach. Converging and diverging mirrors: Refer to the reflection of parallel rays from surface of mirror.  Image Construction Summary: Ray 1: A ray parallel to mirror axis passes through the focal point of a concave mirror or appears to come from the focal point of a convex mirror. Ray 2: A ray passing through the focus of a concave mirror or proceeding toward the focus of a convex mirror is reflected parallel to the mirror axis. Ray 3: A ray that proceeds along a radius is always reflected back along its original path.  Mirror Equation The following equations are given without derivation. They apply equally well for both converging and diverging mirrors. p 1 1 1   R p q f y Y’ f R f  q 2  Sign Convention 1. Object distance p is positive for real objects and negative for virtual objects. 2. Image distance q is positive 1 1 1 for real images and negative   for virtual images. p q f 3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.  Magnification of Images The magnification M of an image is the ratio of the image size y’ to the object size y. Magnification: Obj. Img. Obj. Img. y ' q y y’ y y’ M  y p M = +2 M = -1/2 y and y’ are positive when erect; negative inverted. q is positive when real; negative when virtual. M is positive when image erect; negative inverted.  Alternative Solutions It might be useful to solve the mirror equation algebraically for each of the parameters: 1 1 1   p q f qf pf qp p q f  q f p f q p Be careful with substitution of signed numbers!  Summary The following equations apply equally well for both converging and diverging mirrors. p 1 1 1   R p q f y Y’ f R f  q 2  Summary: Sign Convention 1. Object distance p is positive for real objects and negative for virtual objects. 2. Image distance q is positive for real images and negative for virtual images. 1 1 1 3. The focal length f and the radius of   curvature R is positive for converging p q f mirrors and negative for diverging mirrors. 4. The image size y’ and the magnification M of images is positive for erect images and negative for inverted images.  Summary: Magnification The magnification M of an image is the ratio of the image size y’ to the object size y. Magnification: Obj. Img. Obj. Img. y ' q y y’ y y’ M  y p M = +2 M = -1/2 y and y’ are positive when erect; negative inverted. q is positive when real; negative when virtual. M is positive when image erect; negative inverted.  Objectives: After Completing This Module, You Should Be Able To: Determine the focal length of converging and diverging lenses. Apply the lensmaker’s equation to find parameters related to lens construction. Use ray-tracing techniques to construct images formed by converging and diverging lenses. Find the location, nature, and magnification of images formed by converging and diverging lenses.  Refraction in Prisms If we apply the laws of refraction to two prisms, the rays bend toward the base, converging light. Parallel rays, however, do not converge to a focus leaving images distorted Two prisms and unclear. base to base  Refraction in Prisms (Cont.) Similarly, inverted prisms cause parallel light rays to bend toward the base (away from the center). Again there is no clear virtual focus, and once again, images are Two prisms distorted and unclear. apex to apex  Converging and Diverging Lens If a smooth surface replaces the prisms, a well-defined focus produces clear images. Converging Lens Diverging Lens Real focus Virtual focus Double-convex Double-concave  The Focal Length of lenses Converging Lens Focal length f Diverging Lens f - F F + f The focal length f is positive for a real focus (converging) and negative for a virtual focus(Diverging).  The Principal Focus Since light can pass through a lens in either direction, there are two focal points for each lens. Left to right The principal focal point F is shown here. Yellow F is the F F F other one. F Now suppose light moves from Right to left F right to left instead... F F F  Types of Converging Lenses In order for a lens to converge light it must be thicker near the midpoint to allow more bending. Double-convex lens Plano-convex lens Converging meniscus lens  Types of Diverging Lenses In order for a lens to diverge light it must be thinner near the midpoint to allow more bending. Double-concave lens Plano-concave lens diverging meniscus lens  Lensmaker’s Equation The focal length 1 1 1   (n  1)    f for a lens. f  R1 R2  The Lensmaker’s Equation: Positive (Convex) R1 R2 Negative (Concave) R Sign Surfaces of different radius convention  Signs for Lensmaker’s Equation R1 and R2 are R1, R2 = Radii interchangeable + n= index of glass R1 R2 f = focal length - 1 1 1   (n  1)    f  R1 R2  1. R1 and R2 are positive for convex outward surface and negative for concave surface. 2. Focal length f is positive for converging and negative for diverging lenses. Terms for Image Construction The near focal point is the focus F on the same side of the lens as the incident light. The far focal point is the focus F on the opposite side to the incident light. Converging Lens Diverging Lens Far Far focus focus F F F F Near Near focus focus  Image Construction: Ray 1: A ray parallel to lens axis passes through the far focus of a converging lens or appears to come from the near focus of a diverging lens. Converging Lens Diverging Lens Ray 1 Ray 1 F F  Image Construction: Ray 2: A ray passing through the near focal point of a converging lens or proceeding toward the far focal point of a diverging lens is refracted parallel to the lens axis. Converging Lens Diverging Lens Ray 1 Ray 1 Ray 2 F F Ray 2  Image Construction: Ray 3: A ray passing through the center of any lens continues in a straight line. The refraction at the first surface is balanced by the refraction at the second surface. Converging Lens Diverging Lens Ray 1 Ray 3 Ray 1 Ray 2 F F Ray 2 Ray 3  Images Tracing Points Draw an arrow to represent the location of an object, then draw any two of the rays from the tip of the arrow. The image is where lines cross. 1. Is the image erect or inverted? 2. Is the image real or virtual? Real images are always on the opposite side of the lens. Virtual images are on the same side. 3. Is it enlarged, diminished, or same size?  Object Outside 2F F 2F Real; inverted; 2F F diminished 1. The image is inverted, 2. The image is real, i.e., i.e., opposite to the formed by actual light object orientation. on the opposite side of the lens. 3. The image is diminished in size, i.e., smaller Image is located than the object. between F and 2F  Object at 2F F 2F Real; inverted; 2F F same size 1. The image is inverted, i.e., opposite to 2. The image is real, i.e., formed by actual the object orientation. light on the opposite side of lens. 3. The image is the same size as the object. Image is located at 2F on other side  Object Between 2F and F F 2F Real; inverted; 2F F enlarged 1. The image is inverted, i.e., opposite to 2. The image is real; formed by actual the object orientation. light rays on opposite side 3. The image is enlarged in size, i.e., larger than the object. Image is located beyond 2F  Object at Focal Length F F 2F Parallel rays; 2F F no image formed When the object is located at the focal length, the rays of light are parallel. The lines never cross, and no image is formed.  Object Inside F F 2F Virtual; erect; 2F F enlarged 1. The image is erect, i.e., same orientation 2. The image is virtual, i.e., formed where as the object. light does NOT go. 3. The image is enlarged in size, i.e., larger than the object. Image is located on near side of lens  Review of Image Formations F FF 2F 2F 2F Real; Real; rays; Virtual; Parallel inverted; inverted; erect; no image 2F 2F 2F F FF diminished same enlarged size enlarged formed Object Outside 2F Region  Diverging Lens Imaging All images formed by diverging lenses are erect, virtual, and diminished. Images get larger as object approaches. Diverging Lens Diverging Lens F F Analytical Approach to Imaging y F 2F 2F F -y’ f p q Lens Equation: Magnification: 1 1 1 y ' q   M  p q f y p  Same Sign Convention as For Mirrors 1. Object p and image q distances 1 1 1 are positive for real and images   negative for virtual images. p q f 2. Image height y’ and magnifi- y ' q cation M are positive for erect M  negative for inverted images y p 3. The focal length f and the radius of curvature R is positive for converging lens or mirrors and negative for diverging lens or mirrors.  Alternative Solutions It might be useful to solve the lens equation algebraically for each of the parameters: 1 1 1   p q f qf pf qp p q f  q f p f q p Be careful with substitution of signed numbers!  Summary A Converging lens is one that refracts and converges parallel light to a real focus beyond the lens. It is thicker near the middle. F The principal focus is F F F denoted by the red F. A diverging lens is one that refracts and diverges parallel light which appears to come from a virtual focus in front of the lens.  Summary: Lensmaker’s Equation R1 and R2 are R1, R2 = Radii interchangeable + n= index of glass R1 R2 f = focal length - 1 1 1   (n  1)    f  R1 R2  1. R1 and R2 are positive for convex outward surface and negative for concave surface. 2. Focal length f is positive for converging and negative for diverging lenses.  Summary of Math Approach y F 2F 2F F -y’ f p q Lens Equation: Magnification: 1 1 1 y ' q   M  p q f y p  Summary of Sign Convention 1. Object p and image q distances 1 1 1 are positive for real and images   negative for virtual images. p q f 2. Image height y’ and magnifi- y ' q cation M are positive for erect M  negative for inverted images y p 3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.  Fiber Optics Fiber optic lines are strands of glass or transparent fibers that allows the transmission of light and digital information over long distances. They are used for the telephone system, the cable TV system, the internet, medical imaging, and mechanical engineering spool of optical fiber inspection. Optical fibers have many advantages over copper wires. They are less expensive, thinner, lightweight, and more flexible. They aren’t flammable since they use light signals instead of electric signals. Light signals from one fiber do not interfere with signals in nearby fibers, which means clearer TV A fiber optic wire reception or phone conversations. Continued…  Fiber Optics Cont. Fiber optics are often long strands of very pure glass. They are very thin, about the size of a human hair. Hundreds to thousands of them are arranged in bundles (optical cables) that can transmit light great distances. There are three main parts to an optical fiber: Core- the thin glass center where light travels. Cladding- optical material (with a lower index of refraction than the core) that surrounds the core that reflects light back into the core. Buffer Coating- plastic coating on the outside of an optical fiber to protect it from damage. Continued… Light travels through the core of a fiber optic by continually Fiber Optics (cont.) reflecting off of the cladding. Due to total internal reflection, the cladding does not absorb any of the light, allowing the light to There are two types of optical travel over great distances. Some fibers: of the light signal will degrade Single-mode fibers- transmit over time due to impurities in the one signal per fiber (used in glass. cable TV and telephones). Multi-mode fibers- transmit multiple signals per fiber (used in computer networks).  Inferior Mirages A person sees a puddle ahead on the hot highway because the road heats the air above it, while the air farther above the road stays cool. Instead of just two layers, hot and cool, there are really many layers, each slightly hotter than the layer above it. The cooler air has a slightly higher index of refraction than the warm air beneath it. Rays of light coming toward the road gradually refract further from the normal, more parallel to the road. (Imagine the wheels and axle: on a light ray coming from the sky, the left wheel is always in slightly warmer air than the right wheel, so the left wheel continually moves faster, bending the axle more and more toward the observer.) When a ray is bent enough, it surpasses the critical angle and reflects. The ray continues to refract as it heads toward the observer. The “puddle” is really just an inverted image of the sky above. This is an example of an inferior mirage, since the cool are is above the hot air.  Rainbows A rainbow is a spectrum formed when sunlight is dispersed by water droplets in the atmosphere. Sunlight incident on a water droplet is refracted. Because of dispersion, each color is refracted at a slightly different angle. At the back surface of the droplet, the light undergoes total internal reflection. On the way out of the droplet, the light is once more refracted and dispersed. Although each droplet produces a complete spectrum, an observer will only see a certain wavelength of light from each droplet. (The wavelength depends on the relative positions of the sun, droplet, and observer.) Because there are millions of droplets in the sky, a complete spectrum is seen. The droplets reflecting red light make an angle of 42o with respect to the direction of the sun’s rays; the droplets reflecting violet light make an angle of 40o. Rainbow images  Human eye The human eye is a fluid-filled object that focuses images of objects on the retina. The cornea, with an index of refraction of about 1.38, is where most of the refraction occurs. Some of this light will then passes through the pupil opening into the lens, with an index of refraction of about 1.44. The lens is flexi- Human eye w/rays ble and the ciliary muscles contract or relax to change its shape and focal length. When the muscles relax, the lens flattens and the focal length becomes longer so that distant objects can be focused on the retina. When the muscles contract, the lens is pushed into a more convex shape and the focal length is shortened so that close objects can be focused on the retina. The retina contains rods and cones to detect the intensity and frequency of the light and send impulses to the brain along the optic nerve. Example 1 A 2.0 cm high object is placed 7.10cm from a concave mirror whose radius of curvature is 10.20 cm. A) find the location of the image B) find the size of the image Example 1 A 2.0 cm high object is placed 7.10cm from a concave mirror whose radius of curvature is 10.20 cm. f=1/2R = ½(10.20)=5.10cm, so the object is located between the center of curvature and the focal point. Example 1 Example 1 Example 2. A 6 cm pencil is placed 50 cm from the vertex of a 40-cm diameter mirror. What are the location and nature of the image? p Sketch the rough image. p = 50 cm; R = 40 cm f C F R 40 cm q f   ; f  20 cm 2 2 1 1 1 1 1 1     p q f 50 cm q 20 cm  Example 2 (Cont.). What are the location and nature of the image? (p = 50 cm; f = 20 cm) 1 1 1 p   50 cm q 20 cm f 1 1 1   C q 20 cm 50 cm F q q = +33.3 cm The image is real (+q), inverted, diminished, and located 33.3 cm from mirror (between F and C).  Working With Reciprocals: The mirror equation can easily be solved by using the reciprocal button (1/x) on most calculators: 1 1 1   p q f Possible sequence for finding f on linear calculators: Finding f: P 1/x q + 1/x = 1/x Same with reverse notation calculators might be: Finding f: P 1/x q Enter 1/x + 1/x Example 3: An arrow is placed 30 cm from the surface of a polished sphere of radius 80 cm. What is the location and nature of image? Draw image sketch: p = 30 cm; R = -80 cm R -80 cm f   ; f  40 cm 2 2 Solve the mirror equation for q, then watch pf q signs carefully on substitution: p f Example 3 (Cont.) Find location and nature of image when p = 30 cm and q = -40 cm. (30 cm)(-40 cm) q 30 cm - (-40 cm) q = -17.1 cm The image is virtual (-q), erect, and diminished. It appears to be located at a distance of 17.1 cm behind the mirror.  Example 4. An 8-cm wrench is placed 10 cm from a diverging mirror of f = -20 cm. What is the location and size of the image? q pf  (10cm)(-20cm) Virtual p  f 10 cm - (-20 cm) image Y q = - 6.67 cm Virtual ! Y’ p q F C q (6.67 cm) Converging M  Wrench mirror p 10 cm Magnification: Since M = y’/y y’ = +5.34 cm M = +0.667 y’ = My or: Example 5. How close must a girl’s face be to a converging mirror of focal length 25 cm, in order that she sees an erect image that is twice as large? (M = +2) q M  2  ; q  2 p p pf Also, q p f pf Thus,  2 p f = -2(p - f) = -2p + 2f p f f 25 cm f = -2p + 2f p  p = 12.5 cm 2 2 Example 6. A glass meniscus lens (n = 1.5) has a concave surface of radius –40 cm and a convex surface whose radius is +20 cm. What is the focal length of the lens. +20 cm R1 = 20 cm, R2 = -40 cm 1 -40 cm 1 1   (n  1)    f  R1 R2  n = 1.5 1  1 1   2 1   (1.5  1)     f  20 cm (40 cm   40 cm  f = 20.0 cm Converging (+) lens. Example 7: What must be the radius of the curved surface in a plano-convex lens in order that the focal length be 25 cm? R1 = , R2 = 25 cm R2=? R1= 0 1 1 1   (n  1)    f=? f   R2  1  1  0.500  (1.5  1)    R2 = 0.5(25 cm) 25 cm  R2  R2 R2 = 12.5 cm Convex (+) surface. Example 8. A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What are the nature, size, and location of image. F p = 15 cm; f = 25 cm 1 1 1 F   p q f pf (15 cm)(25 cm) q  q = -37.5 cm p f 15 cm - 25 cm The fact that q is negative means that the image is virtual (on same side as object). Example 8 Cont.) A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What are size of image. p = 15 cm; q = -37.5 cm y’ y F y ' q F M  y p y' ( 37.5 cm)  Y’ = +20 mm 8 mm 15 cm The fact that y’ is positive means that the image is erect. It is also larger than object. Example 9: What is the magnification of a diverging lens (f = -20 cm) the object is located 35 cm from the center of the lens? First we find q... then M 1 1 1 y ' q F   M  p q f y p pf (35 cm)(-20 cm) q  q = +12.7 cm p  f 35 cm - (-20 cm) q (12.7 cm) M  M = +0.364 p 35 cm  Example 10: Derive an expression for calculating the magnification of a lens when the object distance and focal length are given. 1 1 1 pf y ' q   q M  p q f p f y p From last equation: q = -pM Substituting for q in second equation gives...  pM  pf f Thus,... M p f p f Use this expression to verify answer in Example 4.