L18.
Focal Length of a Convex Lens 
About your report:
You'll prepare your report as a Logger Pro file and submit
it to WebAssign.
Goals:
To measure the focal length of a convex lens using two methods and compare the
results
Introduction:
You'll use a converging lens such as a magnifying glass. One method of
measuring the focal length of the lens is to focus an image of the sun on the
sidewalk and simply measure the distance from the lens to the sidewalk.
The other method is to set up an optical bench for projecting an image of an
object onto a screen. The image distance is measured as a function of the
object distance, and the results are fit to the thin lens equation. The
equation of fit is then used to determine the focal length of the lens.
Equipment
Converging lens
(doubleconvex)
2 index cards (sheets of
card stock paper will also do)
Small flashlight
Meter stick
30cm ruler
Books (or similar
objects) to use as supports
Sticky tape
Scissors
Optical bench (optional)
Method
Part A.
Measuring focal length with the Sun as light source
Caution: Don't focus sunlight onto a flammable
material such as grass, paper, or wood. This can start a fire very
quickly.
You'll need a sunny day
for this. If it's cloudy, you can go to Part B first and then do Part A
when the Sun comes out. All you need for this is the lens and a ruler.
You're going to project an image of the Sun onto a nonflammable surface such as
a concrete sidewalk. With the Sun behind you and the sunlight coming over
your shoulder, hold the lens with its principal axis roughly parallel to the
direction of the light from the Sun. Move the lens until the smallest,
sharpest image of the Sun is produced. The image spot can be very bright,
so don't stare at the spot. Measure the distance from the spot to the
lens. Measure to the nearest millimeter.
Part B.
Measuring focal length using the thin lens formula
Note: It will help to darken the room
where your work in order to see the
image more clearly.
Unless you can get an
optical bench from your science department, you'll need to make your own.
It's fairly simple though not as easy to position components and not as accurate
as a manufactured bench. Nevertheless, you can get good results with a
makeshift system. Here's how to set it up.

The first thing to do
is create an object arrow. Use an index card or piece of cardstock for
this. Refer to the figures below. Fold the card down the center as
shown in Figure 1. Then cut out half of an arrow shape along the fold as
shown in Figure 2. The arrow need only be 2 millimeters wide and 2
centimeters high. When you unfold the card as in Figure 3, you'll have a
complete arrow. This will be the object. You'll shine light
through it.

Refer to Figure 5 for
the setup of your optical bench. In order to hold the card with the
object arrow upright, you can fold it under a book or paperweight as shown in
Figure 4 above. You may need a piece of sticky tape to keep the card
vertical. At the opposite end of the bench, use the other card as a
screen. Support it vertically in a manner similar to the object card.
You'll be varying the position of the screen. Place it about 75 cm from
the object card for now. Position a meter stick between the object and
screen. If you can raise the meter stick higher by resting it on top of
the books, that will make for easier readings. The last thing to do is
place the flashlight behind the object arrow. In use, you'll hold the
lens between the object and the screen and adjust the lens position to produce a
sharplyfocused image of the arrow on the screen. Practice this now.
Remember that you're looking for a real image and that it will be inverted.
If you see a fuzzy upright arrow on the screen, that's not an image. You
can get that even without the lens.
Figure 5 

 Prepare a data table like the one below. See Figure 5 for
definitions of the symbols. You'll keep x_{o} fixed during the
experiment, so you need record that only once. The lens and screen
positions will change. The method is to set the screen in a particular
position and then shift the lens one way or the other to produce the sharpest,
real image of the object arrow on the screen. Read the positions as best
you can. It should be easy to read x_{o} and x_{2} to
the nearest millimeter but reading x_{1} will be more difficult.
After taking your readings, move the screen about 10 cm toward the lens.
Readjust the lens position for a sharp image and record all positions.
Get at least five sets of readings. At some point, you'll find that the
image is too small to judge focus easily. And, of course, if the object
distance is less than the focal length of the lens, you won't get a real image
at all.
x_{o}
(cm) 
x_{1}
(cm) 
x_{2}
(cm) 















Analysis
You'll calculate object and image distances and fit them to
the thin lens formula. The focal length will be related to one of the
fit coefficients.

Begin by entering your data for x_{1}
and x_{2} into two columns of an LP file.

Since you need object and image
distances, create two new calculated columns for d_{o} and d_{i}.
Remember to select the appropriate number of decimal digits.

The thin lens formula relates the
reciprocals of d_{o} and d_{i} in a linear relationship.
Therefore, it makes sense to plot 1/d_{i} vs. 1/d_{o}.
That means you'll need two more calculated columns for the inverses of the
object and image distances.

Plot a graph of 1/d_{i}
vs. 1/d_{o} and do a linear fit. (You should know by now not
to do a fit to the
proportion y = Ax, but some people still insist on doing fits without an
intercept. We guarantee that you'll get the wrong answer if you do
that. Fit to y = Ax + B instead.)

As always, put your
matching table in the text box and write the equation of fit with
values, units, significant figures, and physics variables.

Your fit probably isn't anywhere close to the best linear
fit you've ever obtained. You probably noticed that there was some
guessing in determining where the focus was the best. Also, you
probably found that you had some difficulty in reading the lens position.
In order to see that the reading errors can explain the errors in the fit,
put error bars on both the variables. Here's how to go about
estimating the errors. For d_{o}, scan your column of values
and pick one in the middle. Estimate mentally the uncertainty in your
measurements of d_{o}. Then calculate what percentage that
uncertainty is of your middle value of d_{o}. That same
percentage will apply to your values of 1/d_{o}. Add error
bars that size to 1/d_{o}. Repeat for 1/d_{i}.
Interpretation
Write the interpretation
and conclusion on a new page of your file.

Write out the thin
lens formula in standard symbolic form. Based on this formula, what do
you expect the slope of your fit of 1/d_{i} vs. 1/d_{o} to be?
Calculate the % difference between the expected value and the value from your
fit.

Explain why the intercept of
your fit should be the focal length of the lens. Then calculate the focal length
using the intercept.

Explain why your
measurement in Method A gives the focal length of the lens.

Find the
experimental error between the values of the focal length that you measured in Parts A
and B of the method. Take the value from Part A as the accepted value.
Conclusion: In addition to the usual items in the conclusion,
provide a qualitative discussion of the errors in the experiment. Also
summarize the results of your previous error calculations. 