## Lens "Mini" Experiment
## IntroductionThis experiment began from a small discussion about a book, which was later turned into a film, called "Lord of the Flies." In one scene in the film a character uses his glasses to start a small fire by focusing the Sun's rays, which the stranded boys on the island needed for heat, cooking and other things. We questioned if it was actually possible for regular glasses to burn something. There were two kinds of experiments, those done indoors in the lab and those done outside in bright sunlight. The main purpose of the indoor experiments was to verify the Thin-Lens Equation and use it to determine the focal lengths of several lenses as precisely as possible. ## TheoryA key formula when working with lenses is the Thin-Lens Equation:
where d is the image distance.
From the Thin-Lens Equation we are able to mathematically see and
understand many interesting and valuable situations that arise when
working with lenses. When the object is the same distance from the
lens as the image, for instance, we can easily verify that the focal
length must equal half the image (object) distance or if the object is
very far away from the lens (at infinity) the focal length will equal
the image distance, just to mention a few._{i}## Indoor Measurements
I used a work table to precisely measure the focal lengths of the
magnifying lenses and the reading glasses by placing a flashlight at
one end of the table, a screen at the other and the lenses in between
them. The light was covered with black tape and had a small hole
poked in it, thus making the light a point source. I moved the lenses
at different points between the screen and light and by recording data
at the spots when the light came to a focus on the screen, from the
Thin-Lens Equation the focal length of the lenses could be calculated.
Varying the distance between the screen and source also allowed us to
varify the properties of the Thin-Lens Equation. We see that when
d, the focal length _{i}f
is equal to one-half the image or the object distance (or the distance
between the lens and the object or image is twice the focal length).
If the object moves closer than this point (twice the focal length) we
can predict that we will not be able to view the image on the screen.
Shown below is the data for a 7 cm diameter magnifying lens and the
reading glasses:
From the data it is clear that when the object is far away from the
lens the focal length does not equal the image distance. Why is that?
A major limiting factor in this experiment is the size of the table
that the flash light and the lenses were on. It is true that at
infinity the focal length will equal the image distance, however, in
our case, the object is not at infinity but at a distance much closer
(a few hundred centimeters). When 1/d is not zero and the value of the focal length
is not solely determined by the image distance._{o}## Outdoor MeasurementsWe brought four types of lenses outside on a sunny day, three magnifying lenses and the reading glasses. We placed our screen (a piece of paper) on the ground and held a lens above it, varying the distance until the Sun's light came to a focus; when this point was reached the paper began to burn. The Sun's energy flux density or irradiance is what was responsible for this burning. ## Solar Irradiance
Irradiance is defined as the rate of energy emission per unit area.
The standard and accepted value for the Sun's irradiance at the
surface of the Earth is 1 kW/m
Knowing the solar irradiance we can find the power through a specific
area, such as the power through the lenses. At a point before the
Sun's light hits the lens, the irradiance is simply 1
kW/m When the lens brings the sunlight to a focus on the screen, it begins to burn. One might reason that in order for a hole to burn through the screen the Sun's power hitting the screen must be amplified. Therefore, the bright spot at the screen must contain more power than the light entering the lens. This line of reasoning is on the right track, however it is incorrect. In actuality the power at the lens and the power at the spot is exactly the same. The only thing that differs between them is their irradiances. That is, while the power at the spot is the same as at the lens, the irradiance at the spot is much greater here than at the lens. The reason why the irradiance changes in going from the lens to the spot is the two different areas that the light goes through. It's concept is similar to water moving through a funnel. The rate at which water enters the funnel is relatively slow, depending on how fast or slow someone pours it. In contrast, at the end or tip of the funnel, the rate that the water leaves is fast, however the amount of water that is poured in and that leaves is the same (amount of water in = amount of water out, but rate differs). To find the irradiance at the spot we simply multiply the irradiance at the lens by the ratio of the lenses area to spot's area: Irradiance at Spot = (Irradiance at Lens) x (Area at Lens/Area at Spot).
Below is the data that was collected for each lens:
While taking measurements outside I noticed that while all three
magnifying lenses were able to burn the paper the reading glasses were
not. Comparing the data for lens 1 with the reading glasses it is
clear that both lenses deliver the same power to the spot, however
their irradiances are different. The irradiance for lens 1 is about
20 times larger than the reading glasses! Yet why does the magnifying
lens burn the paper and the reading glasses do not? After all, the
glasses magnify the Sun's irradiance by about 27 times! At first
glance it would seem that 27,000 W/m ## Minimum Irradiance
Bringing a magnifying lens outside to find the minimum distance and
spot size for the paper to burn, we can find the minimum irradiance
needed for burning to occur by moving the lens from a distance above
the screen to a point where burning just begins to occur. Using
magnifying lens 3 we measured the minimum spot size to be 1.3 cm.
Using (2) we found that the minimum irradiance for burning to occur
was 43.6 kW/m ## Spot SizeSo far we have learned how to find the irradiance and power entering a lens and at a spot on a screen and also the minimum irradiance needed to burn a hole in the screen, yet we have not really discussed much about the spot's size that is viewed on the screen. Looking at the data for the three magnifying lenses and the reading glasses we see that the size of the spots change for each lens. What do these different spot sizes represent, do they effect the irradiance or anything we have observed so far and given a specific lens can we actually predict the spot size we will see? By utilizing (1) we can find the magnification of a lens by:
In our experiment ^{11} m), d the lens
to screen distance, _{i}h is the size or diameter of
the Sun (1.39 x 10_{o}^{9} m) and h is the spot
size. From (3) we see that if an object is upright then the image we
see from the lens will be inverted. Since our object is the Sun, a
big "circle" so to speak, we can negate the minus sign in the formula
because the orientation of its image does not matter. Since we know
the Earth to Sun distance, the Sun's size and the distance from the
lens to the screen we can find the spot size by:_{i}
_{i}_{i}_{o}_{o}_{i}_{o}_{o}_{i} |

Thomas Cummings Summer 2003 | My Page Laser Teaching Center |