Optics Image formation from Mirrors Refracting Surfaces

Transcription

1 Optics Image formation from Mirrors Refracting Surfaces Lana Sheridan De Anza College June 8, 2018

2 Last time ray diagrams and terminology image formation from mirrors

3 Overview image formation from mirrors refracting surfaces

4 Ray Diagrams for Spherical Mirrors For a ray diagram: draw at least two rays that you know the path of accurately. For Spherical mirrors: 1 Draw a ray from the top of the object parallel to the principle axis reflected through the focal point F. 2 Draw a ray from the top of the object through the focal point and reflected parallel to the principal axis. 3 Draw a ray from the top of the object through the center of curvature C and reflected back on itself. Where the lines meet, an image is formed.

5 When the object is located between the focal point and a concave mirror surface, the image is virtual, upright, and enlarged. Examples of Ray Diagrams 36.2 Images Formed by Sphe When the object is located so that the center of curvature lies between the object and a concave mirror surface, the image is real, inverted, and reduced in size. F 1 O Principal axis 3 2 C I F Front Back Cengage Learning/Charles D. Winters a

6 Principal axis Front Examples of Ray Diagrams a Back Cengage Lear When the object is located between the focal point and a concave mirror surface, the image is virtual, upright, and enlarged. 2 3 C F O I 1 Front Back b When the object is in front of a convex mirror, the image is virtual, upright,

7 Front Back Examples of Ray Diagrams (Convex Mirror) b When the object is in front of a convex mirror, the image is virtual, upright, and reduced in size O I F C Front Back c

8 Cases for Spherical Mirrors f > 0, converging, concave p > 2f real inverted diminished f < p < 2f real inverted enlarged p < f virtual upright enlarged f < 0, diverging, convex any p virtual upright diminished

9 e Formation Concave Mirrors and the Mirror Equation The real image lies at the location at which the reflected rays cross. h O a C a I h' u u V Principal axis q R p Figure 36.9 The image formed by a spherical concave mirror when the object O lies outside the center of curvature C. This geometric construction is used to derive Equation The angles of incidence and reflection are the same magnitude, θ. negative So, sign h h = is introduced q p, and because the image is inverted, so h9 is taken to be negative. Therefore, from Equation 36.1 and M = these q results, we find that the magnification of the image is p

10 Concave Mirrors and the Mirror Equation Looking at the green triangle and the (small) red triangle with angle α: h R q = h p R which rearranges to h h = q R p R Using our magnification expression: q p = q R p R

11 Concave Mirrors and the Mirror Equation q p = q R p R Cross-multiplying and rearranging gives 2 R = 1 p + 1 q However, we already concluded that f = R/2, so 1 f = 1 p + 1 q We have confirmed the mirror equation for spherical concave mirrors, and it follows from simple geometry.

12 not derive any equations for convex spherical mirrors because Eq.4, Convex and 36.6 Mirrors can be used for either concave or convex mirrors if we t sign convention. We will refer to the region in which light rays or e toward the mirror as the front side of the mirror and the other sid We can draw the ray diagrams for convex mirrors in the same way. The image formed by the object is virtual, upright, and behind the mirror. Front Back O I F C q p

13 Mirror Question Quick Quiz Consider the image in the mirror in shown. Based on the appearance of this image, which of the following should you conclude? 1098 Chapter 36 Image (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 (D) the mirror is convex and the image is virtual in vi ro ve Q Q NASA 1 Serway & Jewett, page Figure (Quick Quiz 36.3)

14 Mirror Question Quick Quiz Consider the image in the mirror in shown. Based on the appearance of this image, which of the following should you conclude? 1098 Chapter 36 Image (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 (D) the mirror is convex and the image is virtual in vi ro ve Q Q NASA 1 Serway & Jewett, page Figure (Quick Quiz 36.3)

15 Sign Conventions for Mirrors! 1 f = 1 p + 1 q Variable is Positive is Negative p object in front of mirror q image in front of mirror image behind mirror (real) (virtual) h and M image upright image inverted f and R concave mirror convex mirror

16 Example 36.4: Convex Mirror Image An automobile rearview mirror shows an image of a truck located 10.0 m from the mirror. The focal length of the mirror is 0.60 m. Find the position of the image of the truck and the magnification of the image. le 36.4) An n in a convex of an autoimage of the frame of the onstrates the same locace.. Bo Zaunders/Corbis

17 Example 36.4: Convex Mirror Image An automobile rearview mirror shows an image of a truck located 10.0 m from the mirror. The focal length of the mirror is 0.60 m. Find the position of the image of the truck. (where does it appear to be?)

18 Example 36.4: Convex Mirror Image An automobile rearview mirror shows an image of a truck located 10.0 m from the mirror. The focal length of the mirror is 0.60 m. Find the position of the image of the truck. (where does it appear to be?) q = 0.57 m Find the magnification of the image.

19 Example 36.4: Convex Mirror Image An automobile rearview mirror shows an image of a truck located 10.0 m from the mirror. The focal length of the mirror is 0.60 m. Find the position of the image of the truck. (where does it appear to be?) q = 0.57 m Find the magnification of the image. M = q p =

20 Images Formed by Refraction When light rays change media they are bent. This also can form images.

21 sider two transparent m boundary between the Rays making small angles with the We can find the location principal andaxis We assume the object a size diverge of the from image a point formed by object at O and are refracted Let s consider the parax considering paraxial rays. through the image point I. at the spherical surface Figure shows a n n 1 2 law of refraction applie Images Formed by Refraction n 1 n 2 R O I Because u 1 and u 2 are a tion sin u < u (with ang p q We know that an exteri interior angles, so apply Figure An image formed by refraction at a spherical surface. For paraxial rays, Snell s law becomes: n 1 θ 1 = n 2 θ 2 (1)

22 Images Formed by Refraction 36.3 Images Formed by Refraction n 1 u 1 P n 2 Figure G derive Equation 3 n 1, n 2. d u 2 O a b C g I R p q Looking at the diagram: likewise: Combining all three expressions and eliminating u 1 and u 2 gives 180 θ 1 = 180 α β θ 1 = α + β (2) n 1 a 1 n 2 g 5 (n 2 2 n 1 )b (36.7) Figure shows three right triangles that have a common vertical leg of length d. For paraxial rays (unlike the relatively large-angle ray shown in Fig ), the horizontal legs of these triangles are approximately p for the triangle containing angle a, R for the triangle β containing = γ + angle θ 2 b, and (3) q for the triangle containing angle g. In the small-angle approximation, tan u < u, so we can write the approximate relationships from these triangles as follows: Multiply (2) by n 1 and (3) by n 2, add them and subtract (1): tan a < a < d tan b < b < d tan g < g < n d 2 β p= n 1 α + n 1 β R + n 2 γ q Substituting these expressions into Equation 36.7 and dividing through by d gives Rearranging: n 1 n 1 α + n 2 γ = (n 2 n 1 )β p 1 n 2 q 5 n 2 2 n 1 R (36.8) Relation betw image distanc

23 Images Formed by Refraction For small angles (paraxial approx): α tan α = d p Similarly, β d R and γ d q So, n 1 p + n 2 q = n 2 n 1 R

24 Flat Refracting Surfaces (Like a rectangular fish tank.) 1102 Chapter 36 Image Formation The image is virtual and on the same side of the surface as the object. O p n 1 n 2 n 1 n 2 Figure The image formed q Flat Refracting Surfaces If a refracting surface is flat, then R is infinite an n 1 In this case R. p 52n 2 q n 1 p + n 2 q = 0 q 52 n 2 n 1 p From this expression, And so we see that the sign of q according to Table 36.2, the image formed by a same side of the surface q as = the n 2 object p n as illustrate 1 in which the object is in the medium of index n 1 case, a virtual image is formed between the objec n 2, the rays on the back side diverge from one an in Figure As a result, the virtual image is fo Q uick Quiz 36.4 In Figure 36.16, what happens point O is moved to the right from very far awa

25 Flat Refracting Surfaces Example (Problem 30) A cubical block of ice 50.0 cm on a side is placed over a speck of dust on a level floor. Find the location of the image of the speck as viewed from above. The index of refraction of ice is

26 Flat Refracting Surfaces Example (Problem 30) A cubical block of ice 50.0 cm on a side is placed over a speck of dust on a level floor. Find the location of the image of the speck as viewed from above. The index of refraction of ice is n 1 p + n 2 q = n 2 n 1 R, R q = n 2 n 1 p

27 Flat Refracting Surfaces Example (Problem 30) A cubical block of ice 50.0 cm on a side is placed over a speck of dust on a level floor. Find the location of the image of the speck as viewed from above. The index of refraction of ice is n 1 p + n 2 q = n 2 n 1 R, R q = n 2 n 1 p = 1 (50.0 cm) = 38.2 cm

28 Sign Conventions for Refracting Surfaces! n 1 p + n 2 q = n 2 n 1 R Variable is Positive is Negative p object in front of surface [virtual object] 2 q image behind surface image in front of surface (real) (virtual) h (and M) image upright image inverted R object faces convex surf. object faces concave surf. (C behind surface) (C in front of surface) C is the center of curvature. M = h h = n 1q n 2 p 2 Will be useful in derivations.

29 Summary image formation from mirrors refracting surfaces 4th Collected Homework due Monday June 18. Quiz Monday. Homework Serway & Jewett: Carefully read all of Chapter 36. (over the next few days) Ch 36, onward from page OQs: 13; Probs: 29, 34, 37 (to be set in future, will not be on Monday s quiz) Ch 36. OQs: 1, 3, 5, 11; CQs: 5, 9, 11; Probs: 39, 43, 53, 55, 71, 73, 87, 89