1. Chapter 25 Reflection and Mirrors

2. Geometrical Optics
In the study of how light behaves, it is useful to use “light rays” and the fact that light travels in straight lines.
When light strikes the boundary between two media, three things may happen: reflection, refraction, or absorption.
reflection
refraction
absorption
Water
Air

3. Reflection, Refraction, and Absorption
Water
Air
reflection
refraction
absorption
Water
Air
reflection
Reflection: A ray from air strikes the water and returns to the air.
refraction
absorption
Refraction: A ray bends into the water toward the normal line.
Absorption: A ray is absorbed atomically by the water and does not reappear.

4. The Laws of Reflection
1. The angle of inci- dence qi is equal to the angle of reflection qr :
Water
Air N
reflection qr qi
qi = qr
All ray angles are measured with respect to normal N.
3. The rays are completely reversible.
2. The incident ray, the reflected ray, and the normal N all lie in the same plane.

5. The Plane Mirror
A mirror is a highly polished surface that forms images by uniformly reflected light.
Note: images appear to be equi-distant behind mirror and are right-left reversed.

6. Definitions
Object distance: The straight-line distance p from the surface of a mirror to the object.
Image distance: The straight-line distance q from the surface of a mirror to the image.
Object
Image
Object distance
Image distance =
p = q p q
qi = qr

7. Real and Virtual
Real images and objects are formed by actual rays of light. (Real images can be projected on a screen.)
Real object
Virtual image
Light rays
No light
Virtual images and objects do not really exist, but only seem to be at a location.
Virtual images are on the opposite side of the mirror from the incoming rays.

8. Image of a Point Object Plane mirror p q = p q
Real object p
q = p q
Virtual image
Image appears to be at same distance behind mirror regardless of viewing angle.

9. Image of an Extended Object
Plane mirror p q
q = p
Virtual image
Image of bottom and top of guitar shows forward-back, right-left reversals.

10. Terms for Spherical Mirrors
A spherical mirror is formed by the inside (concave) or outside (convex) surfaces of a sphere.
Concave Mirror R
Axis V C
A concave spherical mirror is shown here with parts identified.
Linear aperture
Center of Curvature C
Radius of curvature R
The axis and linear aperture are shown.
Vertex V

11. The Focal Length f of a Mirror
Since qi = qr, we find that F is mid- way between V and C; we find:
Incident parallel ray qi R qr C V F
axis f
The focal length f is:
Focal point
The focal length, f
The focal length f is equal to half the radius R

12. The Focus of a Concave Mirror
The focal point F for a concave mirror is the point at which all parallel light rays converge.
axis
Incident parallel Rays C
For objects lo- cated at infinity, the real image appears at the focal point since rays of light are almost parallel. F
Focal point

13. The Focus of a Convex Mirror
The focal point for a convex mirror is the point F from which all parallel light rays diverge.
axis C F R
Virtual focus; reflected rays diverge.
Incident Rays
Reflected Rays

14. Image Construction:
Ray 1: A ray parallel to mirror axis passes through the focal point of a concave mirror or appears to come from the focal point of a convex mirror. C F
Concave mirror
Object C F
Convex mirror
Object
Ray 1
Ray 1

15. Image Construction (Cont.):
Ray 2: A ray passing through the focus of a concave mirror or proceeding toward the focus of a convex mirror is reflected parallel to the mirror axis.
Concave mirror C F C F
Convex mirror
Ray 1
Ray 1
Ray 2
Ray 2
Image
Image

16. Image Construction (Cont.):
Ray 3: A ray that proceeds along a radius is always reflected back along its original path.
Concave mirror C F C F
Convex mirror
Ray 2
Ray 1
Ray 3 C F
Ray 2
Ray 1
Image
Ray 3

17. The Nature of Images
An object is placed in front of a concave mirror. It is useful to trace the images as the object moves ever closer to the vertex of the mirror.
We will want to locate the image and answer three questions for the possible positions:
1. Is the image erect or inverted?
2. Is the image real or virtual?
3. Is it enlarged, diminished, or the same size?

18. Object Outside Center C
1. The image is inverted; i.e., opposite of the object orientation.
Ray 1
Ray 3
Ray 2 C F
2. The image is real; i.e., formed by actual light rays in front of mirror.
Concave mirror
3. The image is diminished in size; i.e., smaller than the object.
Image is located between C and F

19. Object at the Center C
1. The image is inverted; i.e., opposite of the object orientation.
Ray 1
Ray 2 C F
2. The image is real; i.e., formed by actual light rays in front of mirror.
Ray 3
3. The image is the same size as the object.
Image is located at C, inverted.

20. Object Between C and F
1. The image is inverted; i.e., opposite of the object orientation.
Ray 1
Ray 3 C
2. The image is real; i.e., formed by actual light rays in front of mirror. F
Ray 2
3. The image is enlarged in size; i.e., larger than the object.
Image is outside of the center C

21. The parallel reflected rays never cross.
Object at Focal Point
When the object is located at the focal point of the mirror, the image is not formed (or it is located at infinity).
Ray 1
Ray 3 C F
Reflected rays are parallel
The parallel reflected rays never cross.
Image is located at infinity (not formed).

22. Object Inside Focal Point
1. The image is erect; i.e., same orientation as the object. C
2. The image is virtual; that is, it seems to be located behind mirror. F
Virtual image
Erect and enlarged
3. The image is enlarged; bigger than the object.
Image is located behind the mirror

23. Observe the Images as Object Moves Closer to MirrorF
Ray 2
Ray 1
Ray 3
Concave mirror C F
Ray 3
Ray 2
Ray 1 C F
Erect and enlarged
Virtual image C F
Ray 1
Ray 3
Ray 2 C F
Ray 3
Reflected rays are parallel
Ray 1

24. Image gets larger as object gets closer
Convex Mirror Imaging C F
Convex mirror C F
Convex mirror
Ray 1
Ray 1
Ray 2 2
Image
Image gets larger as object gets closer
All images are erect, virtual, and diminished. Images get larger as object approaches.

25. Converging and Diverging Mirrors
Concave mirrors and converging parallel rays will be called converging mirrors from this point onward.
Convex mirrors and diverging parallel rays will be called diverging mirrors from this point onward. C F
Converging Mirror
Concave C F
Diverging Mirror
Convex

26. The Plane Mirror = Object distance Image distance p = q p q
Image is virtual
Object distance: The straight-line distance p from the surface of a mirror to the object.
Image distance: The straight-line distance q from the surface of a mirror to the image.

27. Definition of Symbols
By applying algebra and geometry to ray-tracing diagrams, such as the one below, one can derive a relationship for predicting the location of images. y Y’ R q p f
Object dist. p
Image dist. q
Focal length f
Radius R
Object size y
Image size y’

28. Mirror Equation y Y’ R q p f

29. Sign Convention
1. Object distance p is positive for real objects and negative for virtual objects.
2. Image distance q is positive for real images and negative for virtual images.
3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.

30. Example 1. A 6 cm pencil is placed 50 cm from the vertex of a 40-cm diameter mirror. What are the location and nature of the image? C F p q f
Sketch the rough image.
p = 50 cm; R = 40 cm

31. Example 1 (Cont. ). What are the location and nature of the image
Example 1 (Cont.). What are the location and nature of the image? (p = 50 cm; f = 20 cm) C F p q f
q = cm
The image is real (+q), inverted, diminished, and located 33.3 cm from mirror (between F and C).

32. Alternative Solutions
It might be useful to solve the mirror equation algebraically for each of the parameters:
Be careful with substitution of signed numbers!

33. Example 2: An arrow is placed 30 cm from the surface of a polished sphere of radius 80 cm. What is the location and nature of image?
Draw image sketch:
p = 30 cm; R = -80 cm
Solve the mirror equation for q, then watch signs carefully on substitution:

34. Example 2 (Cont.) Find location and nature of image when p = 30 cm and q = -40 cm.
The image is virtual (-q), erect, and diminished. It appears to be located at a distance of 17.1 cm behind the mirror.

35. Magnification of Images
The magnification M of an image is the ratio of the image size y’ to the object size y.
Magnification:
Obj.
Img.
M = +2
M = -1/2
y y’
y and y’ are positive when erect; negative inverted.
q is positive when real; negative when virtual.
M is positive when image erect; negative inverted.

36. Example 3. An 8-cm wrench is placed 10 cm from a diverging mirror of f = -20 cm. What is the location and size of the image? Y’ Y p q
Virtual image
Converging mirror F C
Wrench
q = cm
Virtual !
Magnification:
M =
Since M = y’/y y’ = My or:
y’ = cm