Dr. Neal, WKU
MATH 117
Refraction
Pierre de Fermat (1601 – 1665) discovered that light does not always travel in a straight
line; it takes the quickest possible path. Hence, when passing from medium to medium,
such as from water to air, light rays refract. The principle that explains the resulting
angular relationship is called Snell's Law of Refraction.
Snell’s Law: Let v be the velocity of light in Medium 1 and let w be the speed of light
in Medium 2. If light travels from Medium 1 to Medium 2 where ! is the angle of
incidence in Medium 1 and ! is the angle of refraction in Medium 2 (both measured
from the normal), then
sin ! sin "
=
v
w
Medium 1
Vel = v
!
"
Vel = w
Medium 2
Often, the actual velocities v and w are not given, but instead the ratio of v to w is
sin !
sin "
=
given, such as w is only 90% of v . Then we could say
.
v
0.90 v
Index of Refraction
The speed of light in vacuum is about 186,171 miles/sec. It slows down slightly
through air, and slows down even more through liquid substances and through
substances such as ice or glass. A substance’s index of refraction is the ratio
r=
Speed of Light in Vacuum
.
Speed of Light in Substance
Dr. Neal, WKU
Because light is fastest in vacuum, we have always have r ! 1. For instance, the
index for air is r ≈ 1.00029. We then can say that the speed of light in vacuum is 1.00029
times faster than the speed of light in air. Below are some other indices of refraction.
Indices of Refraction (with respect to Vacuum)
Water (68º F)
1.33
Liquid CO2
Ice
1.31
Alcohol
Ethanol
1.361
Butane
30% Sugar Solution
1.38
Turpentine
30% Sugar Solution
1.49
Benzene
Glycerin
1.473
Sodium Chloride
Crown Glass
1.5
Methylene iodide
1.2
1.329
1.42
1.4721
1.5012
1.54
1.74
If we divide any index above by 1.00029, then we obtain the refractive index of the
substance with respect to air. But these values will be only slightly smaller and may
round to the same numbers as above.
Example. The refractive index of water with respect to air (at 68º F) is still about 1.33.
So if v is the speed of light in air and w is the speed of light in water, then v = 1.33 w .
(a) Suppose a light ray travels from air to water with an angle of incidence of 45º. What
is the angle of refraction? Draw and completely label a diagram.
(b) Suppose a light ray travels from water to air with an angle of incidence of 20º. What
is the angle of refraction? Draw and completely label a diagram.
Solution. The diagrams illustrate the light rays moving from one medium to the other.
In Part (a), the 45º angle of incidence is in the originating medium of air and the angle of
refraction ! is in water. In Part (b), the 20º angle of incidence is in the originating
medium of water. All angles are measured from the normal (i.e., vertical line).
Part (a)
Part (b)
Air: Vel = v
Air: Vel = v
v = 1.33w
v = 1.33w
45º
!
!
Water
Vel = w
20º
Water
Vel = w
Dr. Neal, WKU
sin ! sin "
, where ! is
=
v
w
the angle in the medium where the speed of light is v , and ! is the angle in the medium
where the speed of light is w .
In our case, we have that v is the speed of light in air and w is the speed of light in
water, and v = 1.33 w . So for Part (a) we have
In order to find the angles of refraction, we use Snell’s Law
sin 45º sin !
=
v
w
!
sin 45º sin !
=
1.33w
w
!
sin ! =
sin 45!
.
1. 33
!'
$
#1& sin 45 )
"
=
sin
Thus, the angle of refraction in water is
& 1.33 ) * 32.12º (which is smaller than
%
(
the 45º angle in air).
For Part (b) we have
!
sin ! sin 20º
=
v
w
!
sin !
sin 20º
=
1.33w
w
! sin ! = 1. 33sin 20º
So the angle of refraction in air is ! = sin "1 (1.33 sin 20º ) # 27º (which is larger than the
20º angle in water).
Note: When measured from the normal (i.e. vertical line), the larger angle will always
be in the medium where light is faster, which will always be vacuum. When using
mediums such as air and some liquid, light will be faster in air and thus the larger angle
will be in the air medium.
Exercise. Let v be the speed of light in air and let w be the speed of light in Benzene.
Then v ≈ 1.5 w .
(a) Light travels from Benzene to air with an angle of incidence of 30º. Find the angle of
refraction. Draw and completely label an illustration showing the angles and mediums.
(b) Light travels from air to Benzene with an angle of incidence of 60º. Find the angle
of refraction. Draw and completely label an illustration showing the angles and
mediums.
Answers: (a) ≈ 48.59º
(b) ≈ 35.26º