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 (double-convex)

2 index cards (sheets of card stock paper will also do)
Small flashlight
Meter stick

30-cm 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 non-flammable 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.

1. 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.

 Figure 1 Figure 2 Figure 3 Figure 4

Note:  The instructions below for positioning the components of the optical bench are for a lens of about 10 cm focal length.

1. 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 sharply-focused 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
1. Prepare a data table like the one below.  See Figure 5 for definitions of the symbols.  You'll keep xo 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 xo and x2 to the nearest millimeter but reading x1 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.
 xo (cm) x1 (cm) x2 (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.

1. Begin by entering your data for x1 and x2 into two columns of an LP file.

2. Since you need object and image distances, create two new calculated columns for do and di.  Remember to select the appropriate number of decimal digits.

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

4. Plot a graph of 1/di vs. 1/do 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.)

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

6. 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 do, scan your column of values and pick one in the middle.  Estimate mentally the uncertainty in your measurements of do.  Then calculate what percentage that uncertainty is of your middle value of do.  That same percentage will apply to your values of 1/do.  Add error bars that size to 1/do.  Repeat for 1/di.

Interpretation

Write the interpretation and conclusion on a new page of your file.

1. 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/di vs. 1/do to be?  Calculate the % difference between the expected value and the value from your fit.

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

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

4. 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.