Flat Mirrors Consider an object placed in front of a flat mirror Object O is placed a distance p in front of the mirror (p = object distance) Light reflecting from mirror appears to originate at point I An image of object O is formed at a distance q behind the mirror (q = image distance) It is a virtual image (light does not really converge there) When light actually converges at an image point, it is a real image (can be projected on a screen) We can use (at least 2) rays to determine the image orientation and position Notice p = q for a flat mirror Image is upright and virtual

Flat Mirrors The height h of the object equals the image height h’ Lateral magnification M: General definition for any type of mirror For flat mirrors, M = 1 “Magnification” can mean enlargement or reduction in size in optics A flat mirror also produces an apparent left – right reversal A waving left hand appears to be a waving right hand in the mirror

Example Problem #23.1 Does your bathroom mirror show you older or younger than your actual age? Compute an order-of-magnitude estimate for the age difference, based on data that you specify. Solution: Class interactive – Solution will be determined in class

Concave Spherical Mirrors Consider light from an object O striking a spherical concave mirror If rays diverge at small angles, they all reflect through same image point Large diverging angles mean rays intersect principle axis at different points, resulting in a blurred image Geometry of concave mirrors (from yellow and blue similar triangles)

Concave Spherical Mirrors Additional triangle geometry yields the mirror equation: If an object is very far from the mirror, then 1 / p ≈ 0 and q ≈ R / 2 In this case, the image point is called the focal point F and the image distance q is called the focal length f = R / 2 Note that focal point  focus point The mirror equation can be written in terms of focal length:

Convex Spherical Mirrors Geometry of a convex mirror We can use the same mirror equation as for concave mirrors, but we just need to use a particular sign convention for each quantity in the equation

Sign Conventions for Mirrors Also, R > 0 (R < 0) for concave (convex) mirrors

Ray Diagrams for Concave Mirrors

Ray Diagrams for Convex Mirrors Note that the sideview mirror on the passenger side of a car is a convex mirror (hence the warning given on the mirror)

Example Problem #23.16 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? Solution (details given in class): 10.0 cm in front of the mirror

Example Problem #23.19 A spherical mirror is to be used to form an image, five times as tall as an object, on a screen positioned 5.0 m from the mirror. Describe the type of mirror required. Where should the mirror be positioned relative to the object? Solution (details given in class): Concave 1.0 m in front of the mirror

Images Formed by Refraction Consider light from object O refracting at a spherical surface between 2 transparent media From Snell’s law and geometry: Note that real images are formed on the side opposite that of the incident light

Images Formed by Refraction If the refracting surface is flat, R   and we get: Now the (virtual) image is on the same side of the surface as the object

Interactive Example Problem: Lens Design 101 Animation and solution details given in class. (PHYSLET Physics Exploration 34.3, copyright Pearson Prentice Hall, 2004)

Example Problem #23.25 A transparent sphere of unknown composition is observed to form an image of the Sun on its surface opposite the Sun. What is the refractive index of the sphere material? Solution (details given in class): 2.00

Thin Lenses A thin lens forms an image by refraction of light Has 2 refracting surfaces Examples of thin lenses shown at right Lenses in group (a) converge parallel rays to a focal point on the opposite side of the lens Lenses in group (b) diverge parallel rays so they appear to originate from a focal point on the same side of the lens (converging lens) (diverging lens)

Geometry of Thin Lenses Magnification by a lens Thin–lens equation Lens maker’s equation (same as for a mirror) (same as for a mirror) ( f = focal length in air) ( n = index of refraction of the lens material) (R1 (R2) = radius of curvature of front (back) surface)

Sign Conventions for Thin Lenses

Ray Diagrams for Thin Lenses Converging lenses

Ray Diagrams for Thin Lenses Diverging lenses

Example Problem #23.35 A certain LCD projector contains a single thin lens. An object 24.0 mm high is to be projected so that its image fills a screen 1.80 m high. The object-to-screen distance is 3.00 m. (a) Determine the focal length of the projection lens. (b) How far from the object should the lens of the projector be placed in order to form the image on the screen? Solution (details given in class): (b) 39.5 mm (a) 39.0 mm

Interactive Example Problem: Building a Converging Lens Animation and solution details given in class. (PHYSLET Physics Problem 35.11, copyright Pearson Prentice Hall, 2004)

Combinations of Thin Lenses If 2 thin lenses are used to form an image, use the following procedure: The image produced by the first lens is calculated as though the second lens were not present The image formed by the first lens is treated as the object for the second lens The image formed by the second lens is the final image of the system The overall magnification is the product of the magnifications of the separate lenses This procedure can be extended to 3 or more lenses

Example Problem #23.40 An object is placed 20.0 cm to the left of a converging lens of focal length 25.0 cm. A diverging lens of focal length 10.0 cm is 25.0 cm to the right of the converging lens. Find the position and magnification of the final image. Solution (details given in class): Position = 9.26 cm in front of the 2nd lens Magnification = +0.370

Example Problem #23.50 Front of mirror Back of mirror Back of lens Front of lens The object shown above is midway between the lens and the mirror. The mirror’s radius of curvature is 20.0 cm, and the lens has a focal length of –16.7 cm. Considering only the light that leaves the object and travels first towards the mirror, locate the final image formed by this system. Is the image real or virtual? Is it upright or inverted? What is the overall magnification of the image? Solution (details given in class): The image is virtual, upright, located 25.3 cm behind the mirror, with an overall magnification of +8.05.

HW Problems #23.51, 23.59 #23.51: #23.59:

Aberrations Spherical aberration Chromatic aberration Light passing through lens at different distances from the principal axis is focused at different points Apertures are used to help narrow the incoming beam of light Chromatic aberration Different wavelengths of light refracted by a lens focus at different points Can be reduced by using combination of converging and diverging lenses