The Law of Reflection

Transcription

1 If the surface off which the light is reflected is smooth, then the light undergoes specular reflection (parallel rays will all be reflected in the same directions). If, on the other hand, the surface is rough, then the light will undergo diffuse reflection (parallel rays will be reflected in a variety of directions) Types of Reflection

2 The Law of Reflection For specular reflection the incident angle θ i equals the reflected angle θ r : θ i =θ r The angles are measured relative to the normal, shown here as a dotted line.

3 Forming Images with a Plane Mirror A mirror is an object that reflects light. A plane mirror is simply a flat mirror. Consider an object placed at point P in front of a plane mirror. An image will be formed at point P behind the mirror. d o = distance from object to mirror d i = distance from image to mirror h o = height of object h i = height of image h h For a plane mirror: o i d o d i d o = -d i and h o = h i Image is behind mirror: d i < 0

4 Images An image is formed at the point where the rays of light leaving a single point on an object either actually intersect or where they appear to originate from. If the light rays actually do intersect, then the image is a real image. If the light only appears to be coming from a point, but is not physically there, then the image is a virtual image. We define the magnification, m, of an image to be: m = image height object height = h h i o = d d i o If m is negative, the image is inverted (upside down).

5 Plane Mirrors A plane mirror image has the following properties: The image distance equals the object distance ( in magnitude ) The image is unmagnified image height h m = = object height h The image is virtual: negative image distance d i < 0 m>0, The image is not inverted i o = d d i o d o d i

6 Spherical Mirrors A spherical mirror is a mirror whose surface shape is spherical with radius of curvature R. There are two types of spherical mirrors: concave and convex. We will always orient the mirrors so that the reflecting surface is on the left. The object will be on the left. concave convex

7 Focal Point When parallel rays (e.g. rays from a distance source) are incident upon a spherical mirror, the reflected rays intersect at the focal point F, a distance R/2 from the mirror. (Locally, the mirror is a flat surface, perpendicular to the radius drawn from C, at an angle θ from the axis of symmetry of the mirror). For a concave mirror, the focal point is in front of the mirror (real). For a convex mirror, the focal point is behind the mirror (virtual). The incident rays diverge from the convex mirror, but they trace back to a virtual focal point F.

8 A Focal Length The focal length f is the distance from the surface of the mirror to the focal point. M CF = FA = radius = FM The focal length FM is half the radius of curvature of a spherical mirror. Sign Convention: the focal length is negative if the focal point is behind the mirror. For a concave mirror, f = ½R For a convex mirror, f = ½R (R is always positive)

9 Ray Tracing It is sufficient to use two of four principal rays to determine where an image will be located. M ray The parallel ray (P ray) reflects through the focal point. The focal ray (F ray) reflects parallel to the axis, and the center-of-curvature ray (C ray) reflects back along its incoming path. The Mid ray (M ray) reflects with equal angles at the axis of symmetry of the mirror.

10 Ray Tracing Examples concave Put film here for Sharp image. convex Real image Virtual image

11 Example 1 An object is placed 30 cm in front of a concave mirror of radius 10 cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magnification of the image? f = + R / 2 = + 5cm = d0 di f do = + 30cm = = di f d0 5cm = = di 30cm 30cm d = 6cm i 1 30cm 5 30 cm = 1 6cm d i >0 Real Image m = d i / d o = 1/5

12 Example 2 An object is placed 3 cm in front of a concave mirror of radius 20 cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magnification of the image? f = + R / 2 = + 10cm = d0 di f do = + 3cm = = di f d0 10cm 3cm = = d 30cm 30cm 30cm d i i = 4.29cm m = d i / d o = Virtual image, d i <0 Magnified, m > 1, not inverted. m > 0

13 Example 3 An object is placed 5 cm in front of a convex mirror of focal length 10 cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magnification of the image? f = R / 2 = 10cm = d0 di f do = + 5cm = = di f d0 10cm 5cm = = d 10cm 10cm 10cm d i i = 3.33cm m = d i / d o = 0.66 Virtual image, d i <0 De-Magnified, m < 1, not inverted. m > 0

14 Images - Terminology p: Object Distance q: Image Distance Magnification M Image Height Object Height Real Images: When light rays pass through and diverge from the image point. Virtual Images: When light rays do not pass through but appear to diverge = h h from the image point.

15 Images Formed by Flat Mirrors p = q The image is virtual For flat mirrors, M = 1 The image distance is equal to the object distance. The image is unmagnified, virtual and upright. The image has front-back reversal.

16 Concept Question An observer O, facing a mirror, observes a light source S. Where does O perceive the mirror image of S to be located? Some other location. 6. The image of S cannot be seen by O when O and S are located as shown.

17 Concave Spherical Mirrors Spherical Concave Mirror A real image is formed by a concave mirror Paraxial Approximation: Only consider rays making a small angle with the principal axis Spherical Aberration

18 q h p h = = tanθ p q h h M = = q R h R p h = = tanα R p q R h h = p q R p q R = R q p = + Focal Point 2 R f = f q p = + Image Formation

19 Convex Spherical Mirrors The image formed is upright and virtual M = h h = q p 1 p + 1 q = 1 f

20 Ray Diagrams For Mirrors Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected through the focal point F. Ray 2 is drawn from the top of the object through the focal point and is reflected parallel to the principal axis. Ray 3 is drawn from the top of the object through the center of curvature C and is reflected back on itself.

21 Image is real, inverted and smaller than the object

22 Image is virtual, upright and larger than the object

23 Image is virtual, upright and smaller than the object

24 The law of reflection states that the angle of incidence (θ i ) equals the angle of reflection (θ r ). This is true for specular reflection. (Specular reflection is mirror-like reflection.)

25 Normal Incident Ray Reflected Ray θ i θ r Specular Reflecting Surface Included in the law of reflection is the fact that the incident ray, the normal, and the reflected ray all lie in the same plane.

26 Plane Mirror Plane Mirror Ray Diagramming

27 Plane Mirror Plane Mirror Ray Diagramming

28 Plane Mirrors Using ray diagramming one finds that the image is 1. Upright 2. Same size as the object 3. Virtual.

29 Magnification of a surface Diffuse Reflection Colored lines are for the purpose of distinguishing reflected rays from incident rays.

30 Reflection from Curved Surfaces (Concave shown here) Ray Diagramming Principal axis f Reflecting Surface The law of specular reflection is still obeyed.

31 Reflection from Curved Surfaces (Concave shown here) Ray Diagramming f Principal axis Reflecting Surface The law of specular reflection is still obeyed.

32 Reflection Most objects we see reflect light rather than emit their own light.

33 Law of Reflection The angle of incidence equals the angle of reflection. This is true for both flat mirrors and curved mirrors.

34 Normal Line A Angle of Incidence = Angle of Reflection B MIRROR

35 Tangent Incidence Normal Reflection C F

36 Types of Reflection Specular Reflection - images seen on smooth surfaces (e.g. plane mirrors) Diffuse Reflection - diffuse light coming from a rough surface (cannot see a reflection of yourself)

37 Locating the Image for Plane Mirrors 1. Draw the image the same distance behind the mirror as the object is in front. 2. Draw a connector line from each object to each image. 3. If the connector line passes through the mirror, the image will be seen.

38

39 Concave Mirrors

40 Light from Infinite Distance C F Focuses at the focal point

41 Two Rules for Locating the Image for Concave Mirrors Any incident ray traveling parallel to the principal axis on the way to the mirror will pass through the focal point upon reflection

42 C F

43 Two Rules for Concave Mirrors Any incident ray traveling parallel to the principal axis on the way to the mirror will pass through the focal point upon reflection Any incident ray passing through the focal point on the way to the mirror will travel parallel to the principal axis upon reflection

44 C F

45 C F

46 C F

47 Virtual Image C F

48 Real vs. Virtual Image When a real image is formed, it still appears to an observer as though light is diverging from the real image location only in the case of a real image, light is actually passing through the image location Light does not actually pass through the virtual image location it only appears to an observer as though the light was emanating from the virtual image location

49 Real Image C F Virtual Image C F

50 Types of Images for Mirrors and Lenses A real image is one in which light actually passes through the image point Real images can be displayed on screens A virtual image is one in which the light does not pass through the image point The light appears to come (diverge) from that point Virtual images cannot be displayed on screens

51 More About Images To find where an image is formed, it is always necessary to follow at least two rays of light as they reflect from the mirror. The image formed by the flat mirror is a virtual image Object distance Image distance

52 Flat Mirror Simplest possible mirror Properties of the image can be determined by geometry One ray starts at P, follows path PQ and reflects back on itself A second ray follows path PR and reflects according to the Law of Reflection p=q!

53 23.2 Spherical Mirrors A spherical mirror has the shape of a segment of a sphere A concave spherical mirror has the silvered surface of the mirror on the inner, or concave, side of the curve A convex spherical mirror has the silvered surface of the mirror on the outer, or convex, side of the curve

54 Focal Length If an object is very far away, then p and 1/p 0; q=r/2 Incoming rays are essentially parallel In this special case, the image point is called the focal point The distance from the mirror to the focal point is called the focal length The focal length is ½ the radius of curvature f = R/2

55 Focal Length Shown by Parallel Rays

56 23.3 Convex Mirrors A convex mirror is sometimes called a diverging mirror The rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror The image is virtual because it lies behind the mirror at the point where the reflected rays appear to originate In general, the image formed by a convex mirror is upright, virtual, and smaller than the object

57 Image Formed by a Convex Mirror

58 Ray Diagrams A ray diagram can be used to determine the position and size of an image They are graphical constructions which tell the overall nature of the image They can also be used to check the parameters calculated from the mirror and magnification equations

59 Drawing A Ray Diagram To make the ray diagram, you need to know The position of the object The position of the center of curvature Three rays are drawn They all start from the same position on the object The intersection of any two of the rays at a point locates the image The third ray serves as a check of the construction

60 The Rays in a Ray Diagram Ray 1 is drawn parallel to the principle axis and is reflected back through the focal point, F Ray 2 is drawn through the focal point and is reflected parallel to the principle axis Ray 3 is drawn through the center of curvature and is reflected back on itself 1 2 3

61 Ray Diagram for Concave Mirror, p > R The image is real The image is inverted The image is smaller than the object

62 Ray Diagram for a Concave Mirror, p < f The image is virtual The image is upright The image is larger than the object

63 Ray Diagram for a Convex Mirror The image is virtual The image is upright The image is smaller than the object