Geometric Optics Rules and Image Characteristics

Summary of Geometric Optics Rules 1. Object distances, p are always positive (except in the case of more than one lens or mirror when the first image is on the far side of the second lens or other cases where you have a virtual object like object behind mirror). 2. Image distances, q, are positive for real images and negative for virtual images. 3. Real images form on the same side of the object for mirrors and on the opposite side for refracting surfaces (lenses). Virtual images form on the opposite side of the object for mirrors and on the same side for refracting surfaces. 4. When an object faces a convex mirror or concave refracting surface the radius of curvature, R, is negative. When an object faces a concave mirror or convex refracting surface the radius of curvature is positive. Object object location location type orientation image image image sign of f sign of R (R1 lens) sign of q sign of m for Plane mirror anywhere opposite object opposite virtual same object same as negative =+1 f= Concave mirror concave mirror convex mirror converging lens (convex) converging lens diverging lens inside f virtual positive positive negative positive outside f same real inverted positive positive positive negative anywhere opposite virtual same negative negative negative positive inside f same virtual same positive positive negative positive outside f opposite real inverted positive positive positive negative anywhere same virtual same negative negative negative positive

Your eyes tell you where/how big an object is Mirrors and lenses can fool your eyes Images Ch 36 Flat mirror images Place a point light source P in front of a mirror If you look in the mirror, you will see the object as if it were at the point P , behind the mirror As far as you can tell, there is a mirror image behind the mirror For an extended object, you get an extended image The distances of the object from the mirror and the image from the mirror are equal Flat mirrors are the only perfect image system (no distortion) P P Image Object Mirror p q = p q

Image Characteristics and Definitions h h h h M p q Image Object Mirror The front of a mirror or lens is the side the light goes in The object distance p is how far the object is in front of the mirror The image distance q is how far the image is in front* of the mirror Real image if q > 0, virtual image if q < 0 The magnification M is how large the image is compared to the object Upright if positive, inverted if negative *back for lenses If you place an object in front of a flat mirror, its image will be A) Real and upright B) Virtual and upright C) Real and inverted D) Virtual and inverted

CT - 1 In the morning you look at yourself in the mirror and you cannot see your feet. In order to see them in the mirror you should A.Move closer to the mirror and look down. B.Move backward. C.Give it up - you will never be able to see your feet in the mirror.

Spherical Mirrors Typical mirrors for imaging are spherical mirrors sections of a sphere It will have a radius R and a center point C We will assume that all angles involved are small Optic axis:an imaginary line passing through the center of the mirror Vertex: The point where the Optic axis meets the mirror The paths of some rays of light are easy to figure out A light ray through the center will come back exactly on itself A ray at the vertex comes back at the same angle it left Let s do a light ray coming in parallel to the optic axis: The focal point F is the place this goes through The focal length f = FV is the distance to the mirror A ray through the focal point comes back parallel FC FX = 2 2 CX R = f FV CV FC = = = sin tan X C F 1 1 V f f R 1 2 R

Spherical Mirrors: Ray Tracing 1. Any ray coming in parallel goes through the focus 2. Any ray through the focus comes out parallel 3. Any ray through the center comes straight back Let s use these rules to find the image: = f R 1 2 F Do it again, but harder A ray through the center won t hit the mirror So pretend it comes from the center Similarly for ray through focus Trace back to see where they came from C F P C

Spherical Mirrors: Finding the Image The ray through the center comes straight back The ray at the vertex reflects at same angle it hits Define some distances: X = = = = PX QY h h VP VQ p q = CV R h Q V P h C Y 1 p 1 q 1 f = f R 1 2 + = Magnification Since image upside down, treat h as negative h h q p q p = = M

Convex Mirrors: Do they work too? Up until now, we ve assumed the mirror is concave hollow on the side the light goes in Like a cave A convex mirror sticks out on the side the light goes in The formulas still work, but just treat R as negative The focus this time will be on the other side of the mirror Ray tracing still works = f R 1 2 Summary: F C A concave mirror has R > 0; convex has R < 0, flat has R = Focal length is f = R Focal point is distance f in front of mirror p, q are distance in front of mirror of image, object Negative if behind 1 p 1 q 1 f + = q p = M

Mirrors: Formulas and Conventions: A concave mirror has R > 0; convex has R < 0, flat has R = Focal length is f = R Focal point is distance f in front of mirror p, q are distance in front of mirror of object/image Negative if behind For all mirrors (and lenses as well): The radius R, focal length f, object distance p, and image distance q can be infinity, where 1/ = 0, 1/0 = Light from the Andromeda Galaxy bounces off of a concave mirror with radius R = 1.00 m. Where does the image form? A) At infinity B) At the mirror C) 50 cm left of mirror D) 50 cm right of mirror 1 1 1 q f p = f R 1 2 1 p 1 q 1 f + = 1 = = 0 Concave, R > 0 50 cm = = 50 cm f R 1 2 q =+ p = 50 cm 2 Mly

Ex- (Serway 36-25) A spherical mirror is to be used to form, on a screen located 5 m from the object, an image 5 times the size of the object. (a) Describe the type of mirror required (concave or convex). (b) What s the required radius of curvature of the mirror? (c) Where should the mirror be placed relative to the object? Solve on board

Images of Images: Multiple Mirrors You can use more than one mirror to make images of images Just use the formulas logically Light from a distant astronomical source reflects from an R1 = 100 cm concave mirror, then a R2 = 11 cm convex mirror that is 45 cm away. Where is the final image? f = 50 cm f = 1 1 p 1 q 1 f 5.5 cm + = 2 1 1 1 1 p 1 q 1 f + = 2 1 2 2 5 cm 45 cm 10 cm 1 q 1 + = p = 5 cm 2 50 cm 1 1 1 q 1 + = q = q = 50 cm 55 cm 5 cm 5.5 cm 1 2 2

Refraction and Images Now let s try a spherical surface between two regions with different indices of refraction Region of radius R, center C, convex in front: Two easy rays to compute: Ray towards the center continues straight Ray towards at the vertex follows Snell s Law = sin sin n n 1 1 2 2 R X n1 q h P n p n q n n 1 C Q + = 1 2 2 1 R h p 2 n2 Y Magnification: n q n p = M 1 2

Comments on Refraction R is positive if convex (unlike reflection) R > 0 (convex), R < 0 (concave), R = (flat) n1 is index you start from, n2 is index you go to Object distance p is positive if the object in front (like reflection) Image distance q is positive if image is in back (unlike reflection) We get effects even for a flat boundary, R = Distances are distorted: X 1 2 0 p q n q p n 1 n q M n p n p n q n n + = 1 2 2 1 R R n1 n n q + = h P Q p = 2 2 n2 = 1 Y 2 n n p No magnification: = = 1 M 1 2 n n p 2 1

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Flat Refraction n n = q p 2 A fish is swimming 24 cm underwater (n = 4/3). You are looking at the fish from the air (n = 1). You see the fish A) 24 cm above the water C) 32 cm above the water E) 18 cm above the water 1 B) 24 cm below the water D) 32 cm below the water F) 18 cm below the water R is infinity, so formula above is valid Light comes from the fish, so the water-side is the front Object is in front Light starts in water For refraction, q tells you distance behind the boundary p = 24 cm = = 18 cm 4 3 1 n n 1 24 cm 2 ( ) 1 24 cm 4 3 = = q 18 cm

CT 2 A parallel beam of light is sent through an aquarium. If a convex glass lens is held in the water, it focuses the beam A.closer to the lens than B.at the same position as C.farther from the lens than outside the water.

Double Refraction and Thin Lenses Just like with mirrors, you can do double refraction Find image from first boundary Use image from first as object for second We will do only one case, a thin lens: Final index will match the first, n1 = n3 The two boundaries will be very close n1 n2 n3 p Where is the final image? First image given by: n n n p q n1 n2n1 n + = 1 2 2 1 R 1 1 This image is the object for the second boundary: Final Image location: Add these: p ( ) 2 1 n n p q R = q p n n q 1 R n n 1 2 + = 2 1 1 2 R 1 p 2 2 1 q 1 R 1 R n n 1 n n + = + = 1 2 1 1 1 1 2 1 2

Thin Lenses (2) 1 p 1 q 1 R 1 R 1 f 1 R 1 R n n n n + = 1 1 2 2 1 1 2 1 1 2 Define the focal length: This is called lens maker s equation Formula relating image/object distances Same as for mirrors Magnification: two steps Total magnification is product Same as for mirrors 1 p 1 q 1 f + = 1 1 n q n p n q n p = = M M 2 1 2 2 1 2 qq pp q p = M M M = = M 1 1 2 2 = q p 1 2

Using the Lens Makers Equation If f > 0, called a converging lens Thicker in middle If f < 0, called a diverging lens Thicker at edge 1 f 1 R 1 R n n 1 2 1 1 2 If you are working in air, n1 = 1, and we normally call n2 = n. By the book s conventions, R1, R2 are positive if they are convex on the front You can do concave on the front as well, if you use negative R Or flat if you set R = If you turn a lens around, its focal length stays the same If the lenses at right are made of glass and are used in air, which one definitely has f < 0? Light entering on the left: We want R1 < 0: first surface concave on left We want R2 > 0: second surface convex on left D A B C

Ray Tracing With Converging Lenses Unlike mirrors, lenses have two foci, one on each side of the lens Three rays are easy to trace: 1. Any ray coming in parallel goes through the far focus 2. Any ray through the near focus comes out parallel 3. Any ray through the vertex goes straight through F F f f Like with mirrors, you sometimes have to imagine a ray coming from a focus instead of going through it Like with mirrors, you sometimes have to trace outgoing rays backwards to find the image

Ray Tracing With Diverging Lenses With a diverging lens, two foci as before, but they are on the wrong side Still can do three rays 1. Any ray coming in parallel comes from the near focus 2. Any ray going towards the far focus comes out parallel 3. Any ray through the vertex goes straight through F F f f Trace purple ray back to see where it came from

Lenses and Mirrors Summarized The front of a lens or mirror is the side the light goes in R > 0 p > 0 Object in front q > 0 Image in front f mirrorsConcave = f R 1 2 front 1 f 1 R 1 R n n Convex front Object in front Image in back 1 2 lenses 1 1 2 1 p 1 q 1 f h h q p + = = M Variable definitions: f is the focal length p is the object distance from lens q is the image distance from lens h is the height of the object h is the height of the image M is the magnification Other definitions: q > 0 real image q < 0 virtual image M > 0 upright M < 0 inverted

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Ex- A transparent sphere of unknown composition is observed to form an image of the Sun on the surface opposite to the Sun. What is the refractive index of the sphere? Ex - A transparent photographic slide is placed in front of a converging lens that has a focal length of 2.44 cm. The lens forms an image of the slide 12.9 cm from the slide. How far is the lens from the slide if the image is (a) real and (b) virtual. Solve on Board

Imperfect Imaging With the exception of flat mirrors, all imaging systems are imperfect Spherical aberration is primarily concerned with the fact that the small angle approximation is not always valid F Chromatic Aberration refers to the fact that different colors refract differently F Both effects can be lessened by using combinations of lenses There are other, smaller effects as well

Angular Size & Angular Magnification To see detail of an object clearly, we must: Be able to focus on it (25 cm to for healthy eyes, usually best) Have it look big enough to see the detail we want How much detail we see depends on the angular size of the object h = h d 0 0 d Two reasons you can t see objects in detail: 1. For some objects, you d have to get closer than your near point Magnifying glass or microscope 2. For others, they are so far away, you can t get closer to them Telescope Goal: Create an image of an object that has Larger angular size At near point or beyond (preferably ) Angular Magnification: how much bigger the angular size of the image is m = 0

The Simple Magnifier The best you can do with the naked eye is: d is near point, say d = 25 cm Let s do the best we can with one converging lens To see it clearly, must have |q| d 1 1 1 p q f = = f = h d 0 ( ) = h q p h + = h -q F 1 f 1 q 1 1 q h h p = + = h h d f d q q = + = m 0 Maximum magnification when |q| = d Most comfortable when |q| = To make small f, need a small R: And size of lens smaller than R To avoid spherical aberration, much smaller Hard to get m much bigger than about 5 d f d f = m = + 1 m max

The Microscope A simple microscope has two lenses: The objective lens has a short focal length and produces a large, inverted, real image The eyepiecethen magnifies that image a bit more Fe Fo Since the objective lens can be small, the magnification can be large Spherical and other aberrations can be huge Real systems have many more lenses to compensate for problems Ultimate limitation has to do with physical, not geometric optics Can t image things smaller than the wavelength of light used Visible light 400-700 nm, can t see smaller than about 1 m

The Telescope A simple telescope has two lenses sharing a common focus The objective lens has a long focal length and produces an inverted, real image at the focus (because p = ) The eyepiecehas a short focal length, and puts the image back at (because p = f) fe fo F Angular Magnification: Incident angle: Final angle: The objective lens is made as large as possible To gather as much light as possible In modern telescopes, a mirror replaces the objective lens Ultimately, diffraction limits the magnification (more later) Another reason to make the objective mirror as big as possible 0 h f h f = = m 0 o 0 = m f f = o e e