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Chapter 26 Geometrical Optics 11/16/2011 Chapter 26 Units of Chapter 26 • The Reflection of Light Geometrical Optics • Forming Images with a Plane Mirror • Spherical Mirrors • Rayyg Tracing and the Mirror Eq uation • The Refraction of Light • Ray Tracing for Lenses • The Thin-Lens Equation • Dispersion and the Rainbow Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-1 The Reflection of Light 26-1 The Reflection of Light If a stone is dropped into a pond, circular Spherical waves and plain waves waves emanate from the point where it landed. Rays, perpendicular to the wave fronts, give the direction in which the waves propagate. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-1 The Reflection of Light 26-1 The Reflection of Light The law of reflection states that the angle of Reflection from a smooth surface is called incidence equals the angle of reflection: specular reflection; if the surface is rough, it is diffuse reflection. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 1 11/16/2011 26-2 Forming Images with a Plane Mirror 26-2 Forming Images with a Plane Mirror Light reflected from the flower and vase hits the Properties of Mirror Images Produced by Plane mirror. Obeying the law of reflection, it enters Mirrors: the eye. The eye interprets the ray as having had a straight-line path, and sees the image • A mirror image is upright, but appears reversed behind the mirror. right to left. • A mirror image appears to be the same distance behind the mirror that the object is in front of the mirror. • A mirror image is the same size as the object. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-2 Forming Images with a Plane Mirror 26-3 Spherical Mirrors A corner reflector reflects light A spherical mirror has the shape of a parallel to the incident ray, no section of a sphere. If the outside is matter the incident angle. mirrored, it is convex; if the inside is mirrored, it is concave. What’s the minimum height of mirror in order to see the full image of yours? Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-3 Spherical Mirrors 26-3 Spherical Mirrors Parallel rays hitting a spherical mirror come together at the focal point (or appear to have come from the focal point, if the mirror is convex). For a convex mirror, the focal length is negative, as the rays do not go through the focal point. The opposite is true for a concave mirror. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 2 11/16/2011 26-3 Spherical Mirrors 26-4 Ray Tracing and the Mirror Equation We have made the assumption here that the rays do not hit the mirror very far from the principal axis. If they do, the We use three principal rays in finding the image image is blurred; this is called spherical aberration, and can produced by a concave mirror. be remedied by using a parabolic mirror instead. • The parallel ray (P ray) reflects through the focal point. • The focal ray (F ray) reflects parallel to the axis. • The center-of-curvature rayy( (C ray) reflects back along its incoming path. When the Hubble Space Telescope was first launched, its optics were marred by spherical aberration. This was fixed with corrective optics. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-4 Ray Tracing and the Mirror Equation 26-4 Ray Tracing and the Mirror Equation This image shows how these three rays are used to find The process is similar for a concave mirror, the image formed by a convex mirror. The image is although there are different results depending located where the projections of the three rays cross. The size of the image can also be determined. on where the object is placed. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-4 Ray Tracing and the Mirror Equation Exam 2 We derive the mirror equation using the ray 100 88 66 Score Distribution 100 84 66 diagrams: 12 100 84 65 11 96 83 64 10 95 81 59 8 8 93 78 57 7 6 93 76 55 6 92 76 52 udents in Range 4 t 4 90 74 45 3 90 73 42 2 11 90 70 40 No. of S 00 0 89 70 36 0 102030405060708090 89 69 28 Score Range 88 67 Average: 74.5 Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 3 11/16/2011 26-4 Ray Tracing and the Mirror Equation 26-4 Ray Tracing and the Mirror Equation Using the similar triangles and the fact that f = ½ R, we get the mirror equation: Here, do is the distance from the mirror to the object, di is the distance from the mirror to the image, and f is the focal length. We can also find the magnification: Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-4 Ray Tracing and the Mirror Equation 26-5 The Refraction of Light Light moves at different speeds through different media. Here are the sign conventions for concave and convex When it travels from one medium into another, the mirrors: change in speed causes the ray to bend. The angle of refraction is related to the different speeds: Example: A convex spherical mirror with a radius of curvature of 10.0 cm produces a virtual image one-third the size of the real object. Where is the object? Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-5 The Refraction of Light 26-5 The Refraction of Light The speed of light in a medium We can now write the angle of refraction in is given by the index of terms of the index of refraction: n=1.80 refraction of that medium: Remembering that the speed of a wave is v = ƒ λ, one wonders whether the frequency or the wavelength of light changes upon entering a medium? Here are some typical indices of refraction: Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 4 11/16/2011 26-5 The Refraction of Light Total Internal Reflection and Critical Angle Refraction can make objects immersed in water appear • Total internal reflection can broken, and can create mirages. occur when light attempts to move from a medium with a high index of refraction to one with a lower index of refraction • A particular angle of incidence will result in an angle of refraction of 90°. This angle of incidence is called the critical angle • For angles of incidence greater than the critical angle, the beam is entirely reflected at the boundary n2 sinθc = for n1 > n2 n1 Copyright © 2010 Pearson Education, Inc. Fiber Optics Refraction in a Prism • An application of internal • The amount the ray is bent away from reflection its original direction is called the • Plastic or glass rods are used angle of deviation, δ to “pipe” light from one place • Since all the colors have different to another angles of deviation, they will spread out into a spectrum • Applications include • A prism spectrometer is commonly – MdilMedical use o ffibf fiber op tic ca bles used to study wavelengths emitted by for diagnosis and correction of a light source medical problems – Telecommunications Example: An optical fiber consists of a transparent core (n=1.55) surrounded by cladding (n=1.35), as shown in the figure below. A ray of light in air is incident at the entrance to the fiber with incident angle φ1. (a) What is the speed of light in the fiber core? (b) What is the maximum value for φ1 so that the light will not refract into the cladding? Rainbow Thin Lenses • A thin lens consists of a piece of glass or other primary rainbow transparent substance that can be characterized by the effect it has on light, mostly either converging or diverging the incoming light. • Lenses are commonly used to form images by refraction in optical instruments • Sun light shines on distant raindrops or mist (small water spheres). • Red is observed higher (outside) than violet, with other colors in between. • These angles specify conditions when the refracted/reflected light is most intense (total internal reflection). (a) Converging lenses (b) Diverging lenses Copyright © 2010 Pearson Education, Inc. 5 11/16/2011 Focal Length of Thin Lenses 26-6 Ray Tracing for Lenses The three principal rays for lenses are similar to those for mirrors: • The focal length, ƒ, is the image distance that corresponds to an infinite object distance • The P ray—or parallel ray—approaches the lens parallel to its axis. • A thin lens has two focal points, corresponding to parallel • The F ray is drawn toward (concave) or through (convex) the focal rays from the left and from the right point. • The midpoint ray (M ray) goes through the middle of the lens. Assuming the lens is thin enough, it will not be deflected. This is the thin-litilens approximation. Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-6 Ray Tracing for Lenses 26-7 The Thin-Lens Equation A concave lens always forms a virtual image. A convex We derive the thin-lens lens forms different image types depending on where the equation in the same way we object is located with respect to the focal point. did the mirror equation, using these diagrams: Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. 26-7 The Thin-Lens Equation Examples Sign conventions for thin lenses: A lens produces a real image that is twice as large as the object and is located 15 cm from the lens.
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