Chapter 24: Geometrical Optics
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Nature of Light Ancient times & Middle Ages: light consist of stream of particles corpuscles 19th century: light is a wave (EM wave) Geometrical optics 20th century: several effects associated with the emission and absorption of light revel that it also has a particle aspect and that the energy carried by light waves is packaged in Chapter 24 discrete bundles called photons. Quantum electrodynamics – a comprehensive theory that includes both wave and particle properties. The propagation of light is best described by a wave model The emission and absorption by atoms and nuclei requires a particle approach Wave Fronts Rays Wave Front – a convenient concept to describe wave It is very convenient to represent propagation a light wave by rays rather than A wave front is the locus of all adjacent points at which by wave fronts the phase of vibration of the wave is the same A ray is an imaginary line along the direction of travel of the wave The branch of optics for which the ray description is adequate is called geometric optics The branch dealing specifically with wave behavior is called physical optics spherical and planar wave fronts and rays Reflection and Refraction When a light ray travels from one medium to another, Part 1 part of the incident light is reflected and part of the light is transmitted at the boundary between the two media. The transmitted part is said to be refracted in the The reflection of light second medium. incident ray reflected ray refracted ray 1 Types of Reflection If the surface from which the light is reflected is smooth, then the light undergoes specular reflection (parallel rays will all be reflected in the same directions). If, on the other hand, the surface is rough, then the light will undergo diffuse reflection (parallel rays will be reflected in a variety of directions) Eduard Manet – A Bar at the Foleis-Bergère (1882) Types of Images for Mirrors and Lenses The Law of Reflection A real image is one in which light actually passes through the image point For specular reflection the incident angle θi equals the reflected angle θ : Real images can be displayed on screens r A virtual image is one in which the light does not pass through the image point θi = θr The light appears to diverge from that point Virtual images cannot be displayed on screens The angles are measured relative to the normal as shown. Plane Mirrors Forming Images with a Plane Mirror A mirror is a surface that can reflect a beam of light in A plane mirror is simply a flat mirror. one direction instead of either scattering it widely in Consider an object placed at point P in front of a plane many directions or absorbing it. mirror. An image will be formed at point P´ behind the mirror. do = distance from object to mirror di = distance from image to mirror ho = height of object hi = height of image ho hi For a plane mirror: do di do = di and ho = hi 2 Plane Mirrors What is wrong with the painting? A plane mirror image has the following properties: 9 The image distance equals the object distance 9 The image is unmagnified 9 The image is virtual 9 The image is not inverted Conceptual Checkpoint To save expenses, you would like to buy the shortest mirror that will allow you to see your entire body. Should the mirror be equal to your height? Part 2 Does the answer depend on how far away from the mirror you stand? Spherical Mirrors Spherical Mirrors Concave Mirror, Notation A spherical mirror is a mirror concave whose surface shape is • The mirror has a radius spherical with radius of curvature of curvature of R • Its center of curvature is R. There are two types of the point C spherical mirrors: concave and • Point V is the center of convex. the spherical segment We will orient the mirrors so that • A line drawn from C to V is called the principle the reflecting surface is on the axis of the mirror left side of the mirror. The object will be to left of the mirror itself. convex 3 Focal Point Focal Length Shown by Parallel Rays When parallel rays are incident upon a spherical mirror, the reflected rays intersect at the focal point F. For a concave mirror, the focal point is in front of the mirror. For a convex mirror, the focal point is behind the mirror. The incident rays diverge from the convex mirror, but they trace back to the focal point F. Focal Length Ray Tracing The focal length f is the distance from the surface of the We will use three mirror to the focal point. It can be shown that the focal principal rays to length is half the radius of curvature of the mirror. determine where an image will be Sign Convention: the focal length is negative if the focal located. point is behind the mirror. For a concave mirror, f = ½R The parallel ray (P ray) reflects For a convex mirror, f = −½R (R is always positive) through the focal point. The focal ray (F ray) reflects parallel to the axis, and the center-of-curvature ray (C ray) reflects back along its incoming path. Ray Tracing – Examples More examples: Images from spherical mirrors concave convex Real images form on the side of the mirror where the object is, and virtual images form on the Virtual image Real imageopposite side 4 Ray Diagram for Concave Mirror, p > R • The image is real • The image is inverted • The image is smaller than the object Ray Diagram for a Concave Mirror, p < f Ray Diagram for a Convex Mirror • The image is virtual • The image is virtual • The image is upright • The image is upright • The image is larger than the object • The image is smaller than the object Spherical Aberration Image Formed by a Concave Mirror • Mirror is not exactly spherical • Geometry shows the • This results in a blurred relationship between image the image and object • This effect is called distances spherical aberration 112 + = pq R This is called the mirror equation 5 problem The Mirror Equation Sign Conventions: di is positive if the image is in Example 1 The ray tracing technique front of the mirror (real image) shows qualitatively where the di is negative if the image is image will be located. The behind the mirror (virtual An object is placed 30 cm in front of a concave mirror of distance from the mirror to the image) radius 10 cm. Where is the image located? Is it real or image, di, can be found from virtual? Is it upright or inverted? What is the magnification f is positive for concave mirrors the mirror equation: of the image? f is negative for convex mirrors 1 1 1 + = m is positive for upright images 11111 1 += += d 6 do di f m is negative for inverted images m =−i =− =−0.2 dd0 ii f30 d⎛⎞10 ⎜⎟+ d0 30 2 image magnification. If m is ⎝⎠ Image is smaller, real and inverted. do = distance from object to negative, the image is inverted 111 530× mirror =−or di = =6 cm (upside down). di 530 305− di = distance from image to mirror image height h d m = = i = − i f = focal length object height ho do problem problem Example 2 Example 3 An object is placed 5 cm in front of a convex mirror of An object is placed 3 cm in front of a concave mirror of radius focal length 10 cm. Where is the image located? Is it real 20 cm. Where is the image located? Is it real or virtual? Is it or virtual? Is it upright or inverted? What is the upright or inverted? What is the magnification of the image? magnification of the image? 11111 1 d 3.33 + =+=− m =−i = =0.67 11111 1 d 4.3 dd0 ii f510 d += += i d0 5 20 m =− = =1.43 dd0 ii f3 d⎛⎞ d 3 111 ⎜⎟ 0 or Image is smaller, virtual, and upright. ⎝⎠2 =− − Image is larger, virtual, and upright. di 10 5 111 310× = −or d =− =−4.3 cm 510× d 10 3i 7 d =− =−3.33 cm i i 15 problem Problem A concave mirror produces a virtual image that is three times as tall as the object. (a) If the object is 22 cm in front of the mirror, what is the image distance? (b) What Part 3 is the focal length of this mirror? The Refraction of Light d i 111111 am.) =− b.) += −= d0 dd0 i f22 66 f d 366=−i d =− cm66× 22 22 i f ==33 cm 66− 22 6 The Refraction of Light Snell’s Law In general, when light enters a new material its direction will change. The angle of refraction θ2 is related to the angle of The speed of light is different in different materials. We incidence θ by Snell’s Law: 1 sinθ sinθ define the index of refraction, n, of a material to be the ratio 1 = 2 of the speed of light in vacuum to the speed of light in the v1 v2 where v is the velocity of light in the medium. material: c n = Normal line v θ When light travels from one medium to another its velocity Snell’s Law can also be written as 1 and wavelength change, but its frequency remains n1 sinθ1 = n2 sinθ2 Air constant. For a vacuum, n = 1 The angles θ1 and θ2 are Glass measured relative to the line For other media, n > 1 θ normal to the surface between 2 n is a unitless ratio the two materials.