Chapter 7: Geometrical Optics. The branch of physics which studies the properties of light using the ray model of light.

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1 Chapter 7: Geometrical Optics The branch of physics which studies the properties of light using the ray model of light.

2 Overview Geometrical Optics Spherical Mirror Refraction Thin Lens f u v r and f 2 Ray Diagram Ray Diagram Thin lens equation & Lens maker s equation Snell s Law n n2 n2 n u v r

3 7. Reflection at a Spherical Surface State the law of reflection Sketch and use ray diagrams to determine the characteristics of image formed by spherical mirrors Use f u v for real object only Learning Objectives

4 Law of Reflection The incident ray, the reflected ray and the normal all lie in the same plane. The angle of incidence, i equal the angle of reflection, r as shown in figure below. Normal

5 Reflection at a plane surface u : object distance v :image distance r i h o h i : object height :image height A Object i u i v A' Point object i Vertical (extended) object h o Object u i r r v h i Image

6 Reflection at a plane surface

7 Reflection at a Spherical Surface A spherical mirror is a reflecting surface with spherical geometry. Two types: Convex, if the reflection takes place on the outer surface of the spherical shape. Concave, if the reflecting surface is on the inner surface of the sphere.

8 Ray Diagrams for Spherical Mirrors Ray diagram is defined as the simple graphical method to indicate the positions of the object and image in a system of mirrors or lenses. For concave mirror Focus point, F is defined as a point where the incident parallel rays converge after reflection on the mirror. For convex mirror Focus point, F is defined as a point where the incident parallel rays seem to diverge from a point behind the mirror after reflection.

9 Mirror Approximation Width of the mirror is smaller to the curvature of the mirror Reflected rays make a small angle to the incident rays All rays cross each other at nearly a single point C θ F P C Front back

10 Mirror How to draw When draw the mirror, set the border line first, use straight line for the mirror within the border line When draw the Ray Diagram, adjust the size of the image so that it does not go over the border line. The position of the image has shift significantly

11 Ray Diagram for Concave Mirror Ray A ray parallel to the principal axis passes through or diverges from the focal point F after reflection. Ray 2 A ray which passes through or is directed towards the focal point F is reflected as a ray parallel to the principal axis.

12 Ray Diagram for Concave Mirror Ray 3 A ray which passes through or is directed towards the centre of curvature C will be reflected back along the same path. At least any two rays for drawing the ray diagram.

13 Image Formed by Concave Mirror The characteristics of the images formed by concave mirror depends on object s location. The relationship between the object distance and object size and the image distance and image size are depicted in the diagram below: Object Image

14 Ray Diagram for Convex Mirror Ray A ray parallel to the principal axis passes through or diverges from the focal point F after reflection. Ray 2 A ray which passes through or is directed towards the focal point F is reflected as a ray parallel to the principal axis.

15 Ray Diagram for Convex Mirror Ray 3 A ray which passes through or is directed towards the centre of curvature C will be reflected back along the same path. At least any two rays for drawing the ray diagram.

16 Image Formed by Convex Mirror Unlike concave mirrors, convex mirrors always produce images that share same characteristics: virtual upright diminished (smaller than the object) formed at the back of the mirror (behind the mirror) As the object distance is decreased, the image distance is decreased and the image size is increased. Convex mirror always being used as a driving mirror because it has a wide field of view and providing an upright image.

17 How to Describe an Image? L represents the relative location O represents the orientation (either upright or inverted) S represents the relative size (either magnified, diminished or the same size as the object) T represents the type of image (either real or virtual).

18 The Mirror Equation Spherical mirror s equation: f u v Real object only for spherical mirror (only): f Therefore the equation can also be written as: r 2 2 r u v

19 Linear Magnification, m Linear magnification of the spherical mirror, m is defined as the ratio between image height, h i and object height, h o. m h h i o v u m is a positive value if the image formed is upright and it is negative if the image formed is inverted. Height, h is a positive value if the image formed is upright and it is negative if the image formed is inverted.

20 Sign Convention for Spherical Mirror s Equation: Physical Quantity Positive sign (+) Negative sign (-) Object Distance, u Image Distance, v Real object (in front of the mirror) Real image (same side of the object) Virtual object (at the back of the mirror) Virtual image (Opposite side of the object) Focal length, f (and r) Concave mirror Convex mirror Linear magnification, m Upright image Inverted image Object/ Image Height, h Upright image Inverted image VERY Important!!

21 Example A dentist uses a small mirror attached to a thin rod to examine one of your teeth. When the tooth is.20 cm in front of the mirror, the image it forms is 9.25 cm behind the mirror. Determine a. the focal length of the mirror and state the type of the mirror used b. the magnification of the image

22 Example Solution

23 Example 2 An upright image is formed 20.5 cm from the real object by using the spherical mirror. The image s height is one fourth of object s height. a. Where should the mirror be placed relative to the object? b. Calculate the radius of curvature of the mirror and describe the type of mirror required. c. Sketch and label a ray diagram to show the formation of the image

24 Example 2 Solution

25 Example 2 Solution

26 Example 2 Solution

27 Example 3 A mirror on the passenger side of your car is convex and has a radius of curvature 20.0 cm. Another car is seen in this side mirror and is.0 m in front of the mirror (behind your car). If this car is.5 m tall, calculate the height of the car image.

28 Example 3 Solution

29 Example 4 A person of.60 m height stands 0.60 m from a surface of a hanging shiny globe in a garden. a. If the diameter of the globe is 8 cm, where is the image of the person relative to the surface of the globe? b. How tall is the person s image? c. State the characteristics of the person s image.

30 Example 4 Solution

31 Example 4 Solution

32 7.2 Refraction at a Plane and Spherical Surface State and use the laws of refraction (Snell s Law) for layers of materials with different densities. Use n n2 n2 n u v r for spherical surface. Learning Objectives

33 Refraction Refraction is defined as the changing of direction of a light ray and its speed of propagation as it passes from one medium into another. Laws of refraction state: The incident ray, the refracted ray and the normal all lie in the same plane. For two given media, sin i sin r n n 2 constant or n sin i n2 sin r where n n 2 : refractive index of medium containing incident ray : refractive index of medium containing refracted ray

34 Refraction at a Plane n < n 2 (Medium is less dense medium 2) n > n 2 (Medium is denser than medium 2) The light ray is bent toward the normal (i > r) The light ray is bent away from the normal (i < r)

35 Refraction at a Plane (Special case) When i = 0, no refraction take place

36 Refraction at a Plane (Special case) i = critical angle r = 90 i > critical angle i = r (Must be from denser to less dense medium)

37 Refractive Index (Index of Refraction) Refractive index is defined as the constant ratio for the two given media. The value of refractive index depends on the type of medium and the colour of the light. It is dimensionless and its value greater than. Consider the light ray travels from medium into medium 2, the refractive index can be denoted by n 2 velocity of velocity of light in medium light in medium 2 v v 2 (Medium containing the incident ray) (Medium containing the refracted ray)

38 Refractive Index (Index of Refraction) If medium is vacuum, then the refractive index is called absolute refractive index, written as n velocity of velocity of light in vacuum light in medium c v Note!! If the density of medium is greater hence the refractive index is also greater

39 Relationship between n and λ: As light travels from one medium to another, its wavelength, changes but its frequency, f remains constant. The wavelength changes because of different material. The frequency remains constant because the number of wave cycles arriving per unit time must equal the number leaving per unit time so that the boundary surface cannot create or destroy waves. By considering a light travels from medium (n ) into medium 2 (n 2 ), the velocity of light in each medium is given by v f or v f 2 2

40 Relationship between n and λ: c n v v c n 2 2 f f 2 2 where v If medium is vacuum or air: n c n n22 and v 2 c n 2 Refractive index is inversely proportional to the wavelength n wavelength wavelength of of light in vacuum light in medium 0

41 Relationship between n and d: Other equation for absolute refractive index in term of depth is given by n real depth apparent depth

42 Example 5 A fifty cent coin is at the bottom of a swimming pool of depth 3.00 m. The refractive index of air and water are.00 and.33 respectively. Determine the apparent depth of the coin.

43 Example 5 Solution

44 Refraction at a Spherical Surface Convex Surface towards the object Normal line i r O P C I n n 2 O P C

45 Refraction at a Spherical Surface Concave Surface towards the object Normal line i r O I C P n n 2 O C P

46 Refraction at a Spherical Surface Equation of spherical refracting surface: n n2 n2 n u v r where n n 2 u v : refractive index of medium containing incident ray : refractive index of medium containing refracted ray : object distance from pole P : image distance from pole P i r O u P C v I n n 2

47 Refraction at a Spherical Surface If the refraction surface is flat (plane): n 2 n v r then 0 u The equation (formula) of linear magnification for refraction by the spherical surface is given by m h h i o nv n u 2

48 Sign Convention for Refraction or Thin Lenses: Physical Quantity Positive sign (+) Negative sign (-) Object Distance, u Image Distance, v Real object (in front of the refracting surface) Real image (opposite side of the object) Virtual object (at the back of the refracting surface) Virtual image (same side of the object) Radius of Curvature, r Convex surface Concave surface Focal length, f Converging lens Diverging lens Linear magnification, m Upright image Inverted image Object/ Image Height, h Upright image Inverted image VERY Important!!

49 Example 6 A cylindrical glass rod in air has a refractive index of.52. One end is ground to a hemispherical surface with radius, r = 3.00 cm as shown in figure below. Calculate, a. the position of the image for a small object on the axis of the rod, 0.0 cm to the left of the pole as shown in figure. b. the linear magnification. (Given the refractive index of air, n a =.00)

50 Example 6 Solution

51 Example 6 Solution

52 Example 7 Figure below shows an object O placed at a distance 20.0 cm from the surface P of a glass sphere of radius 5.0 cm and refractive index of.63. Determine a. the position of the image formed by the surface P of the glass sphere, b. the position of the final image formed by the glass sphere. (Given the refractive index of air, n a =.00)

53 Example 7 Solution a) u = 20.0 cm, r = +5.0 cm, n glass.63, n air =.00 n n2 u v v v n2 n r cm Light n n 2 O P C I u 20.0 cm r v 2.5 cm

54 Example 7 Solution Light O P C Q First surface n b) u 2 =.5 cm, r = 5.0 cm, n glass.63, n air =.00 n 2 I 2 u 2 Second surface 2.5 cm.5 cm = O 2 I n n2 u v v v n2 n r cm The image is 3.74 cm at the back of the second surface Q.

55 Try it now! A point source of light is placed at a distance of 25.0 cm from the centre of a glass sphere of radius 0 cm. Find the image position of the source. (Given refractive index of glass =.50 and refractive index of air =.00) Answer : 27.5 cm at the back of the concave surface (second refracting surface)

56 7.3 Thin Lenses Sketch and use ray diagrams to determine the characteristic of image formed by concave and convex lenses Use thin lens equation for real object only Use lens maker s equation Use the thin lens formula for a combination of two convex lenses Learning Objectives

57 Thin Lens Thin lens is defined as a transparent material with two spherical refracting surfaces whose thickness is thin compared to the radii of curvature of the two refracting surfaces. Convex (Converging) lens which are thicker at the centre than the edges Concave (Diverging) lens which are thinner at the centre then at the edges

58 Thin Lenses For converging (convex) lens Focal point is defined as the point on the principal axis where rays which are parallel and close to the principal axis converges after passing through the lens. Its focus is real (principal). For diverging (concave) lens Focal point is defined as the point on the principal axis where rays which are parallel to the principal axis seem to diverge from after passing through the lens. Its focus is virtual.

59 Ray Diagram for Convex Lens Ray Ray which is parallel to the principal axis will be deflected by the lens towards/ away from the focal point F. Ray 2 Ray passing through the optical centre is un-deflected.

60 Ray Diagram for Convex Lens Ray 3 Ray which passes through the focal point becomes parallel to the principal axis after emerging from lens. At least any two rays for drawing the ray diagram.

61 Image Formed by Convex Lens The characteristics of the images formed by convex lens depends on object s location. The relationship between the object distance and object size and the image distance and image size are depicted in the diagram below: Object Image

62 Ray Diagram for Concave Lens Ray Ray which is parallel to the principal axis will be deflected by the lens towards/ away from the focal point F. Ray 2 Ray passing through the optical centre is un-deflected.

63 Ray Diagram for Concave Lens Ray 3 Ray which appear to converge to the focal point becomes parallel to the principal axis after emerging from lens. At least any two rays for drawing the ray diagram.

64 Image Formed by Concave Lens Unlike convex lens, concave lens always produce images that share same characteristics: virtual upright diminished (smaller than the object) formed in front of the lens (between focal point and lens) As the object distance is decreased, the image distance is decreased and the image size is increased.

65 Thin Lens Equation & Lens Maker s Equation Thin lens equation: u v f For real object only Lens Maker s Equation: f n n medium r material r 2 where f : focal length r : radius of curvature of first refracting surface r 2 : radius of curvature of second refracting surface n material : refractive index of lens material n medium : refractive index of medium

66 Thin Lens Equation & Lens Maker s Equation If the medium is air, then the lens maker s equation can be written as f n r r2 The radius of curvature of flat refracting surface is infinity, r =. For thin lens formula and lens maker s equation, Use the sign convention for refraction.

67 Sign Convention for Refraction or Thin Lenses: Physical Quantity Positive sign (+) Negative sign (-) Object Distance, u Image Distance, v Real object (in front of the refracting surface) Real image (opposite side of the object) Virtual object (at the back of the refracting surface) Virtual image (same side of the object) Radius of Curvature, r Convex surface Concave surface Focal length, f Converging lens Diverging lens Linear magnification, m Upright image Inverted image Object/ Image Height, h Upright image Inverted image VERY Important!!

68 Lens Maker s Equation (Example) A convex meniscus lens is made from a glass with a refractive index n =.50. The radius of curvature of the convex surface is 22.4 cm and the concave surface is 46.2 cm. What is the focal length of the lens? 22.4 cm lens inside the circle 46.2 cm lens outside the circle

69 Lens Maker s Equation (Example) Method r = cm ; r 2 = cm Light 2 r r n f cm f f

70 Lens Maker s Equation (Example) Method 2 r = 46.2 cm ; r 2 = 22.4 cm Light 2 r r n f cm f f

71 Linear Magnification, m Linear magnification of the spherical mirror, m is defined as the ratio between image height, h i and object height, h o. hi v m h u o Since u v f, the linear magnification equation can be written as u v f v m f v

72 Example 8 A biconvex lens is made of glass with refractive index.52 having the radii of curvature of 20 cm respectively. Calculate the focal length of the lens in a. water, b. carbon disulfide. (Given n w =.33 and n c =.63)

73 Example 8 Solution

74 Example 9 A person of height.75 m is standing 2.50 m in from of a camera. The camera uses a thin biconvex lens of radii of curvature 7.69 mm. The lens made from the crown glass of refractive index.52. a. Calculate the focal length of the lens. b. Sketch a labeled ray diagram to show the formation of the image. c. Determine the position of the image and its height. d. State the characteristics of the image.

75 Example 9 Solution

76 Example 9 Solution

77 Example 9 Solution

78 Example 9 Solution

79 Example 0 An object is placed 90.0 cm from a glass lens (n =.56) with one concave surface of radius 22.0 cm and one convex surface of radius 8.5 cm. Determine a. the image position. b. the linear magnification.

80 Example 0 Solution

81 Example 0 Solution

82 Combination of Two Convex Lenses Many optical instruments, such as microscopes and telescopes, use two converging lenses together to produce an image. In both instruments, the st lens (closest to the object)is called the objective and the 2 nd lens (closest to the eye) is referred to as the eyepiece or ocular. The image formed by the st lens is treated as the object for the 2 nd lens and the final image is the image formed by the 2 nd lens. The position of the final image in a two lenses system can be determined by applying the thin lens formula to each lens separately.

83 Combination of Two Convex Lenses The overall magnification of a two lenses system is the product of the magnifications of the separate lenses. m m m 2 where m : magnification due to the first lens m 2 : magnification due to the second lens

84 Example An object is 5.0 cm from a convex lens of focal length 0.0 cm. Another convex lens of focal length 7.5 cm is 40.0 cm behind the first. Find the position and magnification of the image formed by a. the first convex lens b. both lenses

85 Example Solution O u d f f f 2 f2 F F F 2 F 2 a) f = 0.0 cm, u = 5.0 cm f v v u 0 v cm m v u m 2 (at the back of the st lens)

86 Example Solution O u d f f f 2 f2 F F F 2 F 2 u 2 I v u2 d v cm Note!! I = O 2 BUT v u 2

87 Example Solution b) f 2 = 7.5 cm, u 2 = 0.0 cm f v v u v cm m 2 v u m2 3 (at the back of the 2 nd lens) 2 2 Total linear magnification: m m m2 6.0

88 Summary Geometrical Optics Spherical Mirror Refraction Thin Lens f u v r and f 2 Ray Diagram Snell s Law Ray Diagram n n2 n2 n u v r Thin lens equation & Lens maker s equation f n n material medium r r 2

89 IMPORTANT! Real object Real image Virtual object Virtual image Front Back Real object Virtual image Virtual object Real image Front Back

90 IMPORTANT! Object Convex towards the object r +ve Object Concave towards the object r -ve

91 IMPORTANT! r +ve Light r 2 ve r -ve Light r 2

92

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