1. Chapter 31: Images and Optical Instruments
Reflection at a plane surface
Image formation
The reflected rays entering eyes look
as though they had come from image P’.
virtual
image P P’
Light rays radiate from a point object
at P in all directions.

2. Reflection and refraction at a plane surface
Image formation

3. Reflection and refraction at a plane surface
i (or s’) is the image distance
s is the object distance:
|s| =|i|
Image formation s’
Sign Rules:
Sign rule for the object distance:
When object is on the same side of the reflecting
or refracting surface as the incoming light, the object
distance s is positive. Otherwise it is negative.
(2) Sign rule for the image distance:
When image is on the same side of the reflecting or
refracting surface as the outgoing light, the image
distance i ( or s’) is positive. Otherwise it is negative.
(3) Sign rule for the radius of curvature of a spherical
surface:
When the center of curvature C is on the same side
as the outgoing light, the radius of the curvature is
positive. Otherwise it is negative.

4. Reflection at a plane surface
Image formation
image is erect
image is virtual
Multiple image due to multiple
Reflection by two mirrors h h’
m = h’/h=1
lateral magnification

5. Reflection at a plane surface
Image formation
When a flat mirror is rotated, how
much is the image rotated?

6. Reflection at a spherical mirror
Concave and convex mirror

7. Reflection at a spherical mirror
Focal points at concave and convex mirror
Focal point or focus: Point F at which rays from a source point are
brought together (focused) to form an image.
Focal length: Distance f from mirror where focus occurs.
f=R/2 where R is the radius of a spherical mirror.

8. Reflection at a spherical mirror
Focal points at a concave mirror
object h d
image s’ If

9. Reflection at a spherical mirror
Image of an extended object at a concave mirror
real image
Principle rays: Light rays that can be traced (more easily) from the source
to the image:
1. Parallel to optical axis
2. Passing through the focal point
3. Passing through the center of curvature
4. Passing through the center of the mirror surface or lens

10. Reflection at a spherical mirror
Magnification of image at a concave mirror h h’
When s,s’ >0 , m<0 inverted
s/s’<0, m>0 upright or
erect

11. Reflection at a spherical mirror
Example with a concave mirror
real image
real image
real image
virtual image

12. Reflection at a spherical mirror
Example with a concave mirror

13. Reflection at a spherical mirror
Image at a convex mirror s s’ f f R
s positive
s’ negative (virtual image)
R negative
f negative

14. Reflection at a spherical mirror
Magnification of image at a convex mirror
For a convex mirror f < 0 s’
m > 1 magnified m < 1 minimized
m > 0 image upright
m < 0 image inverted

15. Refraction at a spherical surface
Refraction at a convex spherical surface q1
q1-q2
For small angles

16. Refraction at a spherical surface
Refraction at a concave spherical surface
For a concave surface, we can use the same formula
But in this case R < 0 and f < 0. Therefore the image is virtual.

17. Refraction at a spherical surface
Relation between source and image distance
at a convex spherical surface s’
Snell’s law
For a convex (concave) surface, R >(<) 0.

18. Refraction at a spherical surface
Example of a convex surface
|s’|

19. Refraction at a spherical surface
Example of a concave surface
|s’|

20. Refraction at a spherical surface
Example of a concave surface

21. Refraction at a spherical surface
Example of a concave surface

22. Convex Lens Sign rules for convex and concave lens: Sign Rules:
Sign rule for the object distance:
When object is on the same side of the reflecting
or refracting surface as the incoming light, the object
distance s is positive. Otherwise it is negative.
(2) Sign rule for the image distance:
When image is on the same side of the reflecting or
refracting surface as the outgoing light, the image
distance i (or s’) is positive (real image). Otherwise it is negative
(virtual image).
(3) Sign rule for the radius of curvature of a spherical
surface:
When the center of curvature C is on the same side
as the outgoing light, the radius of the curvature is
positive. Otherwise it is negative.

23. Convex Lens surface 2 surface 1 Image due to surface 1:
Lens-makers (thin lens) formula
surface 2
surface 1 s’
Image due to surface 1:
s’1 becomes source s2 for surface 2:
R1>0
R2<0
s1 = s and s’2 = s’:
Parallel rays (s=inf.)
w.r.t. the axis converge
at the focal point

24. Convex Lens
Magnification s’
same as for mirrors

25. Convex Lens
Object between the focal point and lens
A virtual image

26. Convex Lens real inverted image m < 1 real inverted image m >1
Object position, image position, and magnification
real inverted image
m < 1
real inverted image
m >1
virtual erect image
m >1

27. Lens
Types of lens

28. Lens
Two lens systems

29. Lens
Two lens systems (cont’d)

30. Lens
Two lens systems (cont’d)

31. Lens
Two lens systems (cont’d)

32. Eyes
Anatomy of eye

33. Eyes farsightedness nearsightedness
Near- and far-sightedness and corrective lenses
farsightedness
nearsightedness

34. Angular size h d In general the minimum distance d=dmin~25 cm at
which an eye can see image of an object comfortably
and clearly.

35. Magnifying glass s’ s hi qi virtual image the eye is most relaxed when
but
the minimum distance at
which an eye can see image
of an object comfortably and clearly.
for human eye.

36. Microscope small magnifier Object is placed near F1 (s1~f1).
q2i
small
Object is placed
near F1 (s1~f1).
Image by lens1
is close to the
focal point of
lens2 at F2.
magnifier
image ang. size

37. Refracting telescope Image by lens1 is at its focal point which is
the focal point of lens 2
image distance
after lens1
image height by lens1 at its focal point
magnifier
angular size of image by lens2; eye
is close to eyepiece

38. Reflecting telescope

39. Aberration
sphere
paraboloid

40. Chromatic aberration

41. Gravitational lens

42. Exercises
Problem 1
What is the size of the smallest vertical plane mirror in which a woman
of height h can see her full-length?
Solution x
x/2
The minimum length of mirror for
a woman to see her full height h
Is h/2 as shown in the figure right.
(h-x)/2
h-x

43. Exercises Problem 2 (focal length of a zoom lens) f1 f2=-|f2| f1 r0 I’
ray bundle f1 r0 I’ Q
r’0 r0 d x s2
d (variable)< f1
s’2
|f2|>f1-d f
(a) Show that the radius of the ray bundle decreases to

44. Exercises Problem 2 (focal length of a zoom lens) f1 f2=-|f2| f1 r0 I’
ray bundle f1 r0 I’ Q
r’0 r0 d x s2
d (variable)< f1
s’2 f
(b) Show that the final image I’ is formed a distance
to the right of the diverging lens.

45. Exercises Problem 2 (focal length of a zoom lens) f1 f2=-|f2| f1 r0 I’
ray bundle f1 r0 I’ Q
r’0 r0 d x s2
d (variable)< f1
s’2 f
(c) If the rays that emerge from the diverging lens and reach the final
image point are extended backward to the left of the diverging lens,
they will eventually expand to the original radius r0 at some point Q.
The distance from the final image I’ to the point Q is the effective focal
length of the lens combination. Find the effective focal length.