Draw a diagram like Fig. 31.10, but showing a ray from the arrowhead through the center of curvature. Using the fact that this ray reflects back on itself, draw similar triangles with object and image as their vertical sides, and show that the center of curvature is twice as far from the mirror as the focal point—that is, R = 2f, with R the curvature radius.
Want to see the full answer?
Check out a sample textbook solutionChapter 31 Solutions
Modified Mastering Physics With Pearson Etext -- Standalone Access Card -- For Essential University Physics (3rd Edition)
Additional Science Textbook Solutions
Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)
Sears And Zemansky's University Physics With Modern Physics
Conceptual Physics (12th Edition)
University Physics with Modern Physics (14th Edition)
Life in the Universe (4th Edition)
University Physics (14th Edition)
- Figure P23.28 shows a curved surface separating a material with index of refraction n1 from a material with index n2. The surface forms an image I of object O. The ray shown in red passes through the surface along a radial line. Its angles of incidence and refraction are both zero, so its direction does not change at the surface. For the ray shown in blue, the direction changes according to n1 sin 1 = n2 sin 2. For paraxial rays, we assume 1 and 2 are small, so we may write n1 tan 1 n2 tan 2. The magnification is defined as M = h/h. Prove that the magnification is given by M = n1q/n2p. Figure P23.28arrow_forwardConsider a beam of light from the left entering a prism of apex angle as shown in Figure P34.34. Two angles of incidence, 1, and 3, are shown as Hell as two angles of refraction, 2 and 4. Show that = 1 + 3. Figure P34.34arrow_forwardThe end of a solid glass rod of refractive index 1.50 is polished to have the shape of a hemispherical surface of radius 1.0 cm. A small object is placed in air (refractive index 1.00) on the axis 5.0 cm to the left of the vertex. Determine the position of the image.arrow_forward
- Two rays travelling parallel to the principal axis strike a large plano-convex lens having a refractive index of 1.60 (Fig. P23.54). If the convex face is spherical, a ray near the edge does not pass through the local point (spherical aberration occurs). Assume this face has a radius of curvature of R = 20.0 cm and the two rays are at distances h1 = 0.500 cm and h2 = 12.0 cm from the principal axis. Find the difference x in the position where each crosses the principal axis. Figure P23.54arrow_forwardTwo rays travelling parallel to the principal axis strike a large plano-convex lens having a refractive index of 1.60 (Fig. P23.54). If the convex face is spherical, a ray near the edge does not pass through the local point (spherical aberration occurs). Assume this face has a radius of curvature of R = 20.0 cm and the two rays are at distances h1 = 0.500 cm and h2 = 12.0 cm from the principal axis. Find the difference x in the position where each crosses the principal axis. Figure P23.54arrow_forwardAn object, represented by a gray arrow, is placed in from of a plane mirror. Which of the diagrams in Figure OQ36.14 correctly describes the image, represented by the pink arrow?arrow_forward
- Figure P23.28 shows a curved surface separating a material with index of refraction n1 from a material with index n2. The surface forms an image I of object O. The ray shown in red passes through the surface along a radial line. Its angles of incidence and refraction are both zero, so its direction does not change at the surface. For the ray shown in blue, the direction changes according to n1 sin 1 = n2 sin 2. For paraxial rays, we assume 1 and 2 are small, so we may write n1 tan 1 n2 tan 2. The magnification is defined as M = h/h. Prove that the magnification is given by M = n1q/n2p. Figure P23.28arrow_forwardA physicist directs a laser beam through a transparent medium, toward one surface of an equilateral prism. (The beam travels, and remains in, the plane of the page.) Incident on Surface 1 at an angle ?1, the beam then encounters Surface 2 from within the prism. If the angle of incidence at Surface 2 equals ?c , the critical angle for this prism, what is the original incidence angle, ?1 (in degrees)? The critical angle in this case is ?c = 40.5°. 40.5° 40.5° A prism in the shape of an equilateral triangle is shown. The triangle is upside down such that the base of the triangle is at the top of the figure and the apex is at the bottom. A laser beam comes in from the top left, moves down and to the right and is incident on the center of the base of the triangle. This surface is labeled Surface 1. The incident beam makes an angle of ?1 with the vertical. Within the prism, the beam continues to move down and to the right but at a slope which is steeper than the initial beam. It is incident…arrow_forwardA fish that is d=d= 2.9 m below the surface looks up and sees a child fishing from the shore. What angle of incidence (θ1) does the ray from the person’s face make with the perpendicular to the water at the point where the ray enters? The angle of refraction (θ2) between the ray in the water and the perpendicular to the water is 35.8°. θ1 = 51.1 What is the height of the person’s head above the water? Assume the person is standing L= 7.4 m away from the point where the incident ray intersects the water. h = marrow_forward
- Why is the following situation impossible? While at the bottom of a calm freshwater lake, a scuba diver sees the Sun at an apparent angle of 38.0° above the horizontal.arrow_forwardOne side of a lens is convex with radius of curvature 30 cm. The other side is concave with radius of curvature 50 cm. The index of refraction of the lens is 1.6. What is the focal length of the lens? Group of answer choices 130 cm, converging 130 cm, diverging 31 cm, converging 31 cm, divergingarrow_forwardA light ray strikes a plane mirror lying flat on a horizontal floor, making an angle of incidence of θi= 61.8°. After reflecting from the mirror, the ray makes a spot on a vertical wall. The wall is a horizontal distance d= 1.08 m from the point of reflection, as shown. How high from the floor is the spot on the wall? (Neglect the thickness of the mirror.)arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
- College PhysicsPhysicsISBN:9781285737027Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningCollege PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning