Ex. 26.2 A concave mirror has a 30 cm radius of curvature.
If an object is placed 10 cm from the mirror, where will the
image be found?
f = R/2 = 15 cm, p = 10 cm
Case 5: p < f
1/p + 1/q = 1/f  1/10 + 1/q = 1/15
3/30 + 1/q = 2/30
1/q = -1/30
q = -30 cm
Real or Virtual
M = -q/p = 3
q < 0
Magnified or Reduced
Up-right or Upside-down
Q. An upright image that is one-half as large as an object is
needed to be formed on a screen in a laboratory experiment
using only a concave mirror with 1 m radius of curvature.
If you can make this image, I will give you $10. If you can’t
you should pay me $10. Deal or no deal? Why?
1/p + 1/q = 1/f = 2/R > 0
M = -q/p = ½ > 0
should be a real image: q > 0
M = -q/p cannot be positive, if q > 0.
No deal!!!
Refraction and Lenses
Optical illusion: the pencil is not bent at the air-water boundary.
caused by non-trivial passage of light rays.
Refraction Details, 1


Light may refract
into a material
where its speed is
lower
The angle of
refraction is less
than the angle of
incidence
• The ray bends
toward the normal
Refraction Details, 2


Light may refract
into a material
where its speed is
higher
The angle of
refraction is
greater than the
angle of incidence
• The ray bends away
from the normal
The Index of Refraction


When light passes from one medium
to another, it is refracted because
the speed of light is different in the
two media
The index of refraction, n, of a
medium can be defined
speed of light in a vacuum c
n

speed of light in a medium v
Index of Refraction, cont



For a vacuum, n = 1
For other media, n > 1
n is a unitless ratio
Frequency Between Media

As light travels from
one medium to
another, its frequency
does not change
• Both the wave speed
and the wavelength do
change
• The wavefronts do not
pile up, nor are created
or destroyed at the
boundary, so ƒ must
stay the same
Index of Refraction Extended



The frequency stays the same as the wave
travels from one medium to the other
v=ƒλ
The ratio of the indices of refraction of the
two media can be expressed as various
ratios
c
1 v 1
n1 n2



2 v 2 c
n1
n2
Snell’s Law
All three beams (incident, reflected, and refracted) are in one plane.
sin q1 v1

sin q 2 v2
q1
q1
n = c/v :index of refraction
v: speed of light in a medium
n > 1
v1 = c/n1, v2 = c/n2
n1sinq1 = n2sinq2
q2
Index of Refraction
material
n = c/v
for  = 589 nm
vacuum
1.00
air
1.00029
water
1.33
ice
1.31
typical glass
1.52
polycarbonate
1.59
diamond
2.42
n depends on .
Dispersion
q1
q1
q2
water
q1> q2
Total Internal Reflection

Total internal
reflection can occur
when light
attempts to move
from a medium
with a high index
of refraction to one
with a lower index
of refraction
• Ray 5 shows
internal reflection
Critical Angle

A particular angle
of incidence will
result in an angle
of refraction of 90°
• This angle of
incidence is called
the critical angle
n2
sin q 
for n1  n2
n1
Critical Angle, cont

For angles of incidence greater than the
critical angle, the beam is entirely
reflected at the boundary
• This ray obeys the Law of Reflection at the
boundary

Total internal reflection occurs only when
light attempts to move from a medium of
higher index of refraction to a medium of
lower index of refraction
Total internal reflection
q2
n2
n1 (> n2)
q1
n1sin(q1) = n2sin(q2)
Total internal reflection when q2 = 90
sin(qc) = n2/n1 < 1
qc: critical angle
How could fish survive from spear fishing?
Fish vision
qf = 2qc
qc = sin-1(1/1.33)
= 49
ncore >nclad
Q. What is the critical angle for a glass to air surface if the
Index of refraction for glass is 1.5.
sinqc = na/ng
= 1.0/1.5
= 0.667
qc = 42
q1()
10
20
30
40
50
60
70
80
q2()
7.5
14.9
22.0
28.8
35.0
40.5
44.8
47.6
sinq1/sinq2
1.33
1.33
1.33
1.33
1.33
1.33
1.33
1.33
sin q1
v1
 const 
sin q 2
v2
v1 = c/n1, v2 = c/n2
n1sinq1 = n2sinq2
q1
q1
air
water
q2
A fish swims below the surface of the water. Suppose an observer is
looking at the fish straight above the fish. The observer sees
1.
2.
3.
4.
the fish at a greater depth
than it really is.
the fish at the same depth.
the fish at a smaller depth
than it really is.
no fish due to total internal
reflection.
Q. There are three layers of different media as shown in the figure.
A beam of light bends as shown in the figure when it passes through
the media. What can we say about the materials?
nIsinqI = nIIsinqII
qII > qI  nI > nII
I
nIIsinqII = nIIIsinqIII
qIII > qII  nII > nIII
II
III
nI > nII > nIII
Dispersion of Index of Refraction for Glass
wavelength (nm)
361 (near UV)
434 (dark blue)
486 (green)
589 (yellow)
656 (red)
768 (dark red)
1200 (IR)
2000 (far IR)
n
1.539
1.528
1.523
1.517
1.514
1.511
1.505
1.497
In glass
n (red) ≈ 1.51
n (purple) ≈ 1.53
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