Refraction of Light by Glass & Converging and Diverging Lenses©98
Experiment 10
Objective: To determine the law of refraction of light by glass using geometric raytracing; to become familiar with the characteristics of converging and diverging lenses.
DISCUSSION:
When a ray of light in the air strikes the surface of a transparent medium, such as
glass, at some angle θ1 to the normal of a surface, that part of the ray which passes into
the glass is bent or refracted so that it leaves the surface at some other angle θ2. The
situation is shown in Fig. 1.
N
The ancients were aware of these phenomena and found that
it was roughly true that
1
2
1
(1)
k
Glass
2
where k is a constant unique to the medium the light entered
from air. However, a better law (discovered in the early 17th
century by Willebord Snell) is that
(2)
n sin  n sin
1
1
Air
2
2
where n1 is a constant unique to medium 1 and is known as
the index of refraction of medium 1 relative to
medium 2. Similarly, n2, is the index of refraction of
medium 2 relative to medium 1. When a ray of
refracted light enters the eye, the observer interprets the
ray as having traveled along a straight line from its
source. The direction of this line is the direction at
which light enters the eye. Consider Fig. 2. Medium 2
is a rectangular glass plate, and medium 1 is air. Pins
are placed at P1 and P2. The observer views these pins
through the glass, and then places a third pin at P3 so
that, when seen through the glass, the 3 pins are
aligned. Actually the line of light connecting P1, P2 and
P3 is bent at the first and second surfaces of the glass.
These positions P1, P2 and P3, along with the normal to
the first surface, enable one to measure the angles θ1
and θ2.
Figure 1: Refraction of light in glass.
Image of P1
P1
1
P2
Air
Glass
2
P3
3
Air
Figure 2. The measurement of angles of
incidence and refraction.
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If n1 < n2, as in going from air to glass, the beam of light is bent toward the
normal at the first surface as in Fig. 2. However if n1 > n2, as in going from glass to air,
the beam of light is bent away from the normal, as shown at the second surface of Fig. 3.
Sometimes, when a light ray hits the boundary, it does so at such an angle that it cannot
be refracted. The boundary also cannot absorb the light, and so it must either be carried
along the boundary or reflected back into the medium as seen in Fig. 4. The critical
angle θc is the angle at which the light will follow the boundary of the medium, making
θ2 of Snell's Law equal to 90º. Since sin 90º = 1.0, Snell's Law then becomes n2sin θ2 =
n2 for this particular case. But, we know that the sine function has a maximum value of
1.0, and so the term n1sin θ1 must be equal or greater than n2 in order to make the
statement true. Angles larger than the critical angle will cause total internal reflection
while angles less than the critical angle will cause the light ray to be refracted as usual.
P1
Air
Air
P2
P2
P3
P3
P1
Glass
Figure 3. Refraction Through A Prism
Glass
1<
Figure 4: Total
internal reflection.
Note: The index of refraction of a medium with respect to air is commonly called n. Since nair is about
1.0003, the common measure of n will indicate the index of refraction of a given material with respect to
air, instead of a vacuum.
A converging lens is a piece of transparent optics, usually glass or plastic, that
causes parallel rays of light to converge toward the central axis of the lens. For example,
a simple converging lens can concentrate enough sunlight (striking the Earth as parallel
rays) to start a fire on a dry sheet of paper. In contrast, a diverging lens causes parallel
rays of light to diverge or spread away from the central axis of the lens. For example, a
simple diverging lens would spread out sunlight that went through it as though the
sunlight were coming from a point along the central axis in front of the lens.
The optical axis of a lens is a line through the center of the lens and perpendicular
to the plane of the lens. The focal point of the converging lens is that point on the optical
axis at which rays of light from an infinitely remote source on the optical axis (like the
sun) converge. The focal point of the diverging lens is that point on the optical axis from
which the rays of light passing through the lens, from an infinitely remote source on the
optical axis, seem to diverge. It is also the point from which the rays of light that enter the
lens parallel to the optical axis seem to diverge after passing through the lens. Figs.1 and
2 show how rays of light travel through a converging and a diverging lens, respectively.
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Diverging Lens
Converging Lens
so
Object
Parallel Ray
Cen
tral
Ray
Parallel Ray
Focal Point
Virtual Image
Object
Optical Axis
Real
Image
si
Cen
tral Optical Axis
Ray
Focal Point
f
so
f
si
The rules for constructing the image of an object, or a "ray diagram" are as
follows:
1. From a point on the object lying off the optical axis, draw a straight line to the lens,
parallel to the optical axis, as shown in Figs. 1 and 2. Change the direction of this
line on the other side of the lens so that it:
a. passes through the focal point, if the lens is converging
b. seems to originate from the focal point, if the lens is diverging.
2. From the same point on the object, draw a second straight line through the center of
the lens, continuing it on the other side of the lens with its direction unchanged.
3. If the extensions of the two lines constructed above converge after passing through
the lens, draw the real image of the object- point at the intersection of the two lines.
Or if the extensions of the two lines diverge, draw the virtual image of the object –
point at the point from which the extended lines seem to diverge.
The images formed by diverging lenses are always virtual because the transmitted rays of light always
diverge. But the images formed by converging lenses may be real or virtual. They are real if the object is
farther from the lens than is the focal point, and they are virtual if the object is nearer the lens than is the
focal point.
It can be shown from the geometry of the rays in Figs. 1 and 2 that in general
1
f

1
s
o

1
s
(1)
i
where so is the distance of the object from the lens, si is the distance of the image from the
lens, and f is the distance of the focal point from the lens. The following sign
conventions are followed by optical scientists:
1. The focal length f is positive for converging lenses and negative for diverging lenses.
2. The object distance so is positive when the object is on the incoming side of the lens,
negative otherwise.
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3. The image distance si is positive when the image is on the outgoing side of the lens,
negative otherwise.
If two lenses are used in combination and are placed in contact with one another, the
image that would be produced by the first lens acting alone becomes the object for the
second lens. In this case, the effective focal length f of the combination is given by;
1
1
1


(2)
f
f
f
1
2
where f1 and f2 are the focal lengths of the separate lenses.
EXERCISES FOR PRISMS:
1. Refraction in a square rectangular piece of glass:
a. Using the pins as objects, trace several independent rays of light through the
rectangular prism. Suggested angles of θ1 are 5˚, 10˚, 15˚, 20˚, 30˚, and 70˚.
b. For each ray of light, determine the index of refraction of the glass using Snell's
Law. Is this index of refraction a constant? Determine the percent deviation of
each n2 value from the average n2 value.
c. Calculate the value of k in Eq. (1) for each ray of light. Determine the percent
deviation of each k value from the average k value.
2. Reflection and Refraction in a prism:
a. Trace several independent rays of light through the glass triangle.
b. Using the appropriate angles, compute the index of refraction of the glass.
c. Calculate the "critical angle" and verify that your observations of total internal
reflections are valid.
EXERCISES FOR LENSES:
1. Measure the focal lengths of the lenses marked red and yellow using a distant object
outside the window.
a. Place a screen on the end of the optical bench opposite the window.
b. Place one lens holder with a lens on the optical bench between the screen and the
window.
c. Move the lens back and forth on the optical bench until you have the best image
of a distant object.
d. Lock the lens in place, then measure and record the distance between the lens and
the screen. This is the focal length of the lens.
e. Repeat for the other lens.
2. Determine the focal lengths of the same lenses by imaging nearby objects.
a. Add the light source to the optical bench on the opposite end from the screen.
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b. Calculate the focal lengths of the same lenses by using Eq. (1) and measuring the
image and object distances due to an object placed in the neighborhood of the
lens.
c. Repeat this exercise for different object distances and calculate an average focal
length for each lens.
d. How does this calculated average focal length compare to the measured focal
length from Exercise 1?
e. Repeat for remaining lens.
f. Using so and si found above and the calculated f, draw a to-scale ray diagram for
each lens (in cm!).
3. Determine the focal length of the diverging lens by combining it with a suitable
converging lens and using Eq. (2). (note: you may have to hold the lenses together as they will
not both fit in the lens holder).
4. Observe the interchangabilty of object and image distances.
a. Set the object and the screen at least four focal lengths apart for a particular lens.
b. Find the two lens positions that will give a clear image on the screen.
c. How are the two images similar/different?
d. Explain this phenomenon based on Eq. (1).
5. Build a simple telescope.
a. Remove the screen and light source from the optical bench.
b. Add a second lens holder.
c. Put the longer focal length lens in the lens holder closer to the window. This is
the objective of the telescope.
d. Put the shorter focal length lens in the other holder. This lens is the eyepiece.
e. While keeping the objective lens stationary, move the eyepiece until you can see a
clear image through the two lenses. (note: the eyepiece should not be right
against your eye.)
f. Measure and record the distance between the two lenses.
g. Calculate the magnification of your telescope using the following equation:
M
fo
fe
10-5