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Transcript
Chapter 34 – Geometric Optics and Optical Instruments I. Introduction – find the position of the image due to reflections and refractions II. Reflections from Plane Surfaces A. Ray diagram showing where the image of an object is located object y Optic Axis s reflecting surface Observations: 1. image is called a virtual image – the light appears to diverge from the image. (a real image is one to where the light converges) B. 2. image distance = object distance 3. image size = object size Lateral Magnification, m m = C. image size object size Image appears to undergo a left to right reversal, but no top to bottom reversal. Why? 1 III. Reflection from a Spherical Surface A. Types of spherical reflecting surfaces (spherical mirrors) 1. Concave mirror (converging) C 2. Convex mirror (diverging) C B. Location of image for a spherical mirror object y C V R s 2 1. Lateral magnification m image size object size 2. Mirror equation 3. Focal point and focal length a. Focal point, F – position of the image when the object is at infinity Focal length, f – the image distance when the object is at infinity b. Graphical interpretation of the focal point and focal length 1) concave mirror 3 2) C. convex mirror Graphical method of locating images - three (four) rays whose paths are known 1. ray parallel to the optic axis > reflects back through the focal point 2. ray through the center of curvature, C > reflects back through the center of curvature 3. ray through focal point > reflects back parallel to the optic axis 4. ray to vertex > reflects back ( = ’) concave mirror, I: concave mirror, II: 4 concave mirror, III: convex mirror: D. Using the Mirror Equation: 1 1 2 1 s s' R f and m s' s Sign convention – the signs associated with object distances, images distances, etc. are based on where the object, image, etc. is located relative to the incident and reflected rays. MEMORIZE THIS! +s: real object – object on same side as incoming light rays -s: virtual object – object not on same side as incoming light rays +s’: real image – image on same side as outgoing light rays -s’: virtual image – image not on same side as outgoing light rays +R: concave mirror – center of curvature, C, on same side as outgoing light rays -R: convex mirror – center of curvature, C, not on same side as outgoing light rays +f: focal length – focal point, F, on same side as outgoing light rays -f: focal length – focal point, F, not on same side as outgoing light rays +m: upright (erect) image -m: inverted image 5 E. Examples 1. A 3 cm high object is located at a position 80 cm away from a concave mirror of 20 cm radius. Find the position and size of the image. 2. A shaving mirror is a concave mirror of 80 cm radius. A person’s nose is located 20 cm in front of the mirror. Where is the image located and what is its magnification? 3. A person is standing 50 cm in front of a 20 cm convex shaped doorknob. Where is the image of the person located and what is the magnification? 4. Can these equations be used for a plane mirror? 6 IV. Refractions at spherical surfaces n1 object n2 y C Optic Axis R s A. Lateral magnification m B. image size object size Refraction equation 7 C. Sign Convention – same as for mirrors (page 34-5) +s (real object): object on same side as incoming light rays +s’ (real image): image on same side as outgoing light rays +R: C on same side as outgoing light rays +m: upright or erect image D. Examples 1. How deep does a coin on the bottom of a two feet deep pond appear to be? 2. A fish is at the center of a spherical fishbowl whose radius is R. Where does the fish appear to be and how large is the fish? Assume you are a distance of 2R from the surface of the fishbowl. 3. What does the fish in question 2 see? 8 V. Multiple optical surfaces A. Analysis object surface 1 surface 2 In general, the image for one optical surface becomes the object for the next optical surface. B. Examples: 1. A glass rod 40 cm long and index 1.50 has one end that is convex of radius 20 cm and the other end has a convex surface of radius 30 cm. An object is placed 100 cm in front of the 20 cm convex surface. Where, what size, and what orientation is the final image? n = 1.50 100 cm 40 cm 9 2. A one centimeter high object is placed a distance 4R to the left of a glass sphere of radius R and index 1.50. The right half of the sphere is coated with a reflecting material. Where, what size, and what orientation is the final image? 3. Two identical spherical mirrors with radius R are separated by a distance equal to their focal length, f = R/2. An object is placed on the surface of one of the mirrors. After undergoing two reflections, where will the image be located, and what kind of image is it? If a small hole is made in the middle of the top mirror, will the image exist? hole f = R/2 object 10 V. Thin lenses – the thickness of the lens is small compared to the radii of curvature of the surfaces. A. Types of thin lenses B. 1. Converging: double convex, plano-convex, converging meniscus 2. Diverging: double concave, plano-concave, diverging meniscus Derivation of thin lens equation. The lens has radii R1 and R2, thickness t, and index n. R1 R2 n O1 I1 -> O2 s1 I2 s1’ s2 s2’ t Surface 1 Surface 2 Note: If s’ for surface 1 is positive, then s for surface 2 is negative; or if s’ for surface 1 is negative, then s for surface 2 is positive. That is s2 = - s1’. 11 1. Write the refraction equations for the two surfaces, and derive the thin lens equation. Surface 1: Surface 2: 2. The focal point of a lens is the image position when the object is at infinity. Rewrite the lens equation including the focal length. 3. What is the magnification of a thin lens? 4. Graphical interpretation of the focal point: a. Converging lens: b. Diverging lens: 12 C. Graphical method to find the position of images – three rays: 1. 2. 3. ray parallel to the optic axis > travels through the focal point ray through the vertex of lens > travels straight through the lens ray through the other focal point > travels parallel to the optic axis Converging lens: Converging lens: Diverging lens: D. Examples: 1. Find the focal length of a thin, plano-convex lens with a radius of 30 cm and made of glass on index n = 1.50. Is the focal length the same if the lens is reversed? 13 2. Find the focal length of a diverging meniscus lens with radii of 20 cm and 30 cm. The index of the glass is n = 1.50. 3. Where must a converging lens of focal length f be placed between an object and screen a distance D apart so that the image is focused on the screen? (conjugate foci) lens object screen x D 4. An object is placed 40 cm in front of a converging lens with a focal length of 20 cm. A diverging lens with a focal length of 30 cm is placed 25 cm behind the converging lens. Find the position of the final image, its magnification, and its orientation. 14 VI. Lens Aberrations – problems that produce blurred images of objects when light passes through lens. A. Spherical aberration – light far from the optic axis focuses at a different point from light traveling close to the optic axis. B. Chromatic aberration – light of different wavelengths focuses at different points. white light white light C. Astigmatism – the curvature of the lens is different in different planes. 15 VII. Optical Instruments – the basics A. Pinhole Camera B. Camera C. Projector D. Eye 16 E. Magnifier F. Microscope G. Telescope 17