Physics
Spherical Mirror
Spherical mirrors are curved mirrors that have a spherical shape. They are categorized into concave and convex mirrors based on the way they curve. Concave mirrors converge light rays to a focal point, while convex mirrors diverge light rays. These mirrors are used in various optical devices, such as telescopes, microscopes, and vehicle rear-view mirrors.
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8 Key excerpts on "Spherical Mirror"
eBook - PDF- James Shipman, Jerry Wilson, Charles Higgins, Bo Lou, James Shipman(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 180 Chapter 7 ● Optics and Wave Effects Figure 7.16 Spherical Mirror Geometry A Spherical Mirror is a section of a sphere with a center of curvature C. The focal point F is halfway between C and the vertex V. The distance from V to F is called the focal length f. The distance from C to V or to a point on the sphere is the radius of curvature R (the radius of the sphere). And R 5 2f, or f 5 R/2. C F Radius of curvature R Mirror Principal axis V f The inside surface of a spherical section is said to be concave (as though looking into a recess or cave), and when it has a mirrored surface, it is a concave (converging) mirror. The reason for “converging” is illustrated in ●● Fig. 7.17a. Reflecting light rays parallel to the principal axis converge and pass through the focal point. The rays are “focused” at the focal point. (Off-axis parallel rays converge in the focal plane.) Similarly, the outside surface of a spherical section is said to be convex, and when it has a mirrored surface, it is a convex (diverging) mirror. Parallel rays along the principal axis are reflected in such a way that they appear to diverge from the focal point (Fig. 7.17b). In regard to reverse ray tracing, light rays coming to the mirror from the surround- ings are made parallel, and an expanded field of view is seen in the diverging mirror. Diverging mirrors are used on side mirrors of cars and trucks to give drivers a wider rear view of traffic and in stores to monitor aisles (Fig. 7.17c). Ray Diagrams The images formed by Spherical Mirrors can be found graphically using ray diagrams. An arrow is commonly used as the object, and the location and size of the image are determined by drawing two rays: Figure 7.17 Spherical Mirrors (a) Rays parallel to the principal axis of a concave or converging Spherical Mirror converge at the focal point.
eBook - PDF- Vern Ostdiek, Donald Bord(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Telescope mirrors have as their basic functions the gathering of light and the concentration of that light to a point. The ability of a mirror to collect light increases with its surface area. To acquire enough radiation to study faint objects adequately, astronomers have sought to build instru- ments with larger and larger apertures (openings) and, hence, larger light- collecting areas. The quality of the images produced by telescopes is greatly affected by the shapes of the mirrors. The easiest curved mirror to make is one that has a sur- r r face in the shape of portion of a sphere. But such a Spherical Mirror is not perfect for the task of focusing light rays. Figure 9.26a shows that parallel light rays reflecting off a Spherical Mirror are not all focused at the same point. An im- age formed using such a mirror will be somewhat blurred. This phenomenon is called spherical aberration. We will see in Section 9.4 that the same thing happens with lenses. As the name implies, spherical aberration is a defect associated with spherical surfaces. A concave mirror in the shape of a parabola does not have this aberration. (You may recall that we saw the parabola in Section 2.7.) A parabolic mirror will concentrate all the rays coming from a distant source at the same point (Figure 9.26b). Thus, the ideal surface for a telescope mirror (or for that matter, reflectors in auto headlamps and household flashlights) Convex mirror Plane mirror Field of view Field of view Figure 9.23 The field of view of a convex mirror is much larger than that of a plane mirror. Figure 9.24 This convex mirror on a bike path at the University of Minnesota allows quick surveillance of a large area that would normally be hidden from view. Secondary mirror Primary mirror F Figure 9.25 This is a basic design of a large astronomical telescope. The large, concave primary mirror and the small, convex secondary mirror combine to focus incoming light at the focal point F.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
If the inside surface of the mirror is polished, it is a concave mirror. If the outside surface is polished, it is a convex mirror. The principal axis of a mirror is a straight line drawn through the center of curvature and the middle of the mirror’s surface. Rays that are close to the principal axis are known as paraxial rays. Paraxial rays are not necessarily parallel to the principal axis. The radius of curvature R of a mirror is the distance from the center of curvature to the mirror. The focal point of a concave Spherical Mirror is a point on the principal axis, in front of the mirror. Incident paraxial rays that are parallel to the principal axis converge to the focal point after being reflected from the concave mirror. The focal point of a convex Spherical Mirror is a point on the principal axis, behind the mirror. For a convex mirror, incident paraxial rays that are parallel to the principal axis diverge after reflecting from the mirror. These rays seem to originate from the focal point. The fact that a Spherical Mirror does not bring all rays parallel to the principal axis to a single image point after reflection is known as spherical aberration. The focal length f indicates the distance along the principal axis between the focal point and the mirror. The focal length and the radius of curvature R are related by Equations 25.1 and 25.2. 25.5 The Formation of Images by Spherical Mirrors The image produced by a mirror can be located by a graphical method known as ray tracing. For a concave mirror, the following paraxial rays are useful for ray tracing (see Figure 25.17): Ray 1. This ray leaves the object traveling parallel to the principal axis. The ray reflects from the mirror and passes through the focal point. Ray 2. This ray leaves the object and passes through the focal point. The ray reflects from the mirror and travels parallel to the principal axis.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The radius of curvature R of a mirror is the distance from the center of curvature to the mirror. The focal point of a concave Spherical Mirror is a point on the principal axis, in front of the mirror. Incident paraxial rays that are parallel to the principal axis converge to the focal point after being reflected from the concave mirror. The focal point of a convex Spherical Mirror is a point on the principal axis, behind the mirror. For a convex mirror, incident paraxial rays that are parallel to the principal axis diverge after reflecting from the mirror. These rays seem to originate from the focal point. The fact that a Spherical Mirror does not bring all rays parallel to the principal axis to a single image point after reflection is known as spherical aberration. The focal length f indicates the distance along the principal axis between the focal point and the mirror. The focal length and the radius of curvature R are related by Equations 25.1 and 25.2. 25.5 The Formation of Images by Spherical Mirrors The image produced by a mirror can be located by a graphical method known as ray tracing. For a concave mirror, the following paraxial rays are useful for ray tracing (see Figure 25.17): Ray 1. This ray leaves the object traveling parallel to the principal axis. The ray reflects from the mirror and passes through the focal point. Ray 2. This ray leaves the object and passes through the focal point. The ray reflects from the mirror and travels parallel to the principal axis. f 5 1 2 R (Concave mirror) (25.1) f 5 2 1 2 R (Convex mirror) (25.2) Image distance d i is 1 if the image is in front of the mirror (real image). d i is 2 if the image is behind the mirror (virtual image). Magnification m is 1 for an image that is upright with respect to the object. m is 2 for an image that is inverted with respect to the object. Check Your Understanding (The answers are given at the end of the book.) 13.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Rays that are close to the principal axis are known as paraxial rays. Paraxial rays are not necessarily parallel to the principal axis. The radius of curvature R of a mirror is the distance from the centre of curvature to the mirror. The focal point of a concave Spherical Mirror is a point on the principal axis, in front of the mirror. Incident paraxial rays that are parallel to the principal axis converge to the focal point after being reflected from the concave mirror. The focal point of a convex Spherical Mirror is a point on the prin- cipal axis, behind the mirror. For a convex mirror, incident paraxial rays that are parallel to the principal axis diverge after reflecting from the mirror. These rays seem to originate from the focal point. The fact that a Spherical Mirror does not bring all rays parallel to the principal axis to a single image point after reflection is known as spherical aberration. The focal length f indicates the distance along the principal axis between the focal point and the mirror. The focal length and the radius of curvature R are related by equations 25.1 and 25.2. f = 1 2 R (Concave mirror) (25.1) f = - 1 2 R (Convex mirror) (25.2) 25.5 Perform ray tracing for Spherical Mirrors. The image produced by a mirror can be located by a graphical method known as ray tracing. For a concave mirror, the following paraxial rays are useful for ray tracing (see figure 25.17). Ray 1. This ray leaves the object travelling parallel to the principal axis. The ray reflects from the mirror and passes through the focal point. Ray 2. This ray leaves the object and passes through the focal point. The ray reflects from the mirror and travels parallel to the principal axis. Ray 3. This ray leaves the object and travels along a line that passes through the centre of curvature. The ray strikes the mirror perpendicularly and reflects back on itself. For a convex mirror, the following paraxial rays are useful for ray tracing (see figure 25.21a): Ray 1.
eBook - PDF- David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
The largest single reflecting telescopes have diameters around 10 m, and thus have about 100 times the light-gathering capability of the largest refracting telescopes. Even larger reflecting tele- scopes can be constructed by combining the light from many individual mirrors into a single image. Earth-bound optical telescopes are limited in their abil- ity to form sharp images by atmospheric distortion; the nat- ural turbulence in the atmosphere distorts the (nearly) plane wavefronts that reach the Earth from distant objects. One cure for this problem has been obtained through the devel- opment of adaptive optics; by sensing the atmospheric dis- tortion, the shape of a flexible mirror can be modified to compensate for the distortion and thus produce a sharp im- age. An alternative way to eliminate the effects of the at- mosphere is to place the telescope above the atmosphere. Figure 40-31 shows the Hubble Space Telescope, a reflect- ing telescope that was launched into Earth orbit by a space shuttle in 1990. 930 Chapter 40 / Mirrors and Lenses FIGURE 40-31. The Hubble Space Telescope. 5. No matter how far you stand from a certain mirror, your im- age appears upright. What type of mirror is this? (A) Concave (B) Convex (C) Plane (D) Either (B) or (C) (E) There is not enough information to answer this ques- tion. 6. While standing in front of a certain mirror, you notice that your image appears enlarged. What type of mirror is this? (A) Concave (B) Convex (C) Plane (D) Either (A) or (B) (E) There is not enough information to answer this question. 7. A mirror produces a real image at i from a real object at o i. What can you conclude about the focal length of the mirror? (A) f 0 (B) 0 f i (C) i f o (D) o f 40-4 Spherical Refracting Surfaces 8. Light shines through a spherical air bubble underwater. What type of optical device does the bubble act like? (A) Converging (B) Diverging (C) Planar 9. A diver in a glass bubble helmet looks at a fish as in Fig.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
In a convex mirror, the image of an object is always virtual, upright, and reduced in size as shown in Figure 35.13c. In this case, as the object distance decreases, the virtual image increases in size and moves away from the focal point toward the mir- ror as the object approaches the mirror. You should construct other diagrams to verify how image position varies with object position. Q UICK QUIZ 35.2 You wish to start a fire by reflecting sunlight from a mirror onto some paper under a pile of wood. Which would be the best choice for the type of mirror? (a) flat (b) concave (c) convex Q UICK QUIZ 35.3 Consider the image in the mirror in Figure 35.14. Based on the appearance of this image, would you conclude that (a) the mirror is concave and the image is real, (b) the mirror is concave and the image is virtual, (c) the mirror is convex and the image is real, or (d) the mirror is convex and the image is virtual? Figure 35.14 (Quick Quiz 35.3) What type of mirror is shown here? Example 35.3 The Image Formed by a Concave Mirror A Spherical Mirror has a focal length of 1 10.0 cm. (A) Locate and describe the image for an object distance of 25.0 cm. S O L U T I O N Conceptualize Because the focal length of the mirror is positive, it is a concave mirror (see Table 35.1). We expect the possi- bilities of both real and virtual images. continued PITFALL PREVENTION 35.4 Choose a Small Number of Rays A huge number of light rays leave each point on an object (and pass through each point on an image). In a ray diagram, which displays the characteristics of the image, we choose only a few rays that fol- low simply stated rules. Locating the image by calculation comple- ments the diagram. NASA Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
eBook - PDF- James A. Samson, David L. Ederer(Authors)
- 1998(Publication Date)
- Academic Press(Publisher)
The curve required for point-to-point focusing is an ellipse (Fig. 23). If one conjugate is at infinity, the curve required is a parabola, and for a virtual image a hyperbola. These are internally reflecting (concave) surfaces. Since most spectroscopic applications have real objects at finite distances and require converging optics, we derive the parameters of the ellipse here. Referring to Fig. 23, if the object is distance u, the image distance u, and the glancing angle 0 at the mirror center C, we wish to find the semi-axes a and b in the ellipse equation: x2 y2 -+ -= I . a2 b2 Noting that the focal properties of the ellipse give a = (u + v)/2 FF‘ = 2 d m , (29) where FF‘ is the intrafocal distance, the law of cosines can be applied to the triangle FCF’ to find b b = /? [l - cos(n - 20)l. 2 In a spectroscopic system consisting of a spherical (circular) mirror and a grating, it is also possible to use the line spacing variation of the grating to correct the terms in Eq. (23). This technique, analogous to the use of a phase corrector plate in visible light optics, is discussed in Vol. 32, Chapter 3. “Aspherizing” the circular mirror does not reduce coma, which is even more severe for an elliptical mirror. Coma correction is achieved 9.5.5.3 Coma. 172 SPHERICAL AND NONSPHERICAL OPTICS by satisfying the Abbk sine condition, that is, by ensuring that the magnification stays constant for all paths across the aperture. This can be achieved by using systems with multiple mirrors, in particular, pairs of mirrors as depicted in Fig. 11. This method of correction is discussed in more detail in Section 9.6, which deals with toroidal and Wolter optics. We note that the system depicted in Fig. 11 is not corrected for astigmatism, and so the equivalent Kirkpatrick-Baez imaging system consists of four mirrors (two acomatic pairs). 9.5.5.4 Astigmatism. This topic was treated in Section 9.5.3.
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