L17. Index of Refraction

About your report:  Since you'll be drawing constructions, submit your report in a scanned document.

 Goal

to measure the index of refraction of i) water and ii) of an unknown transparent solid to 3 significant figures

 Preparation

You'll need to have completed P217 and read section 26-5.

 Equipment

 Part A. Index of refraction of water

 

Sight carefully. Make sure that you use your dominant eye. If you're not sure which eye is dominant, hold a finger upright with your arm extended in front of your face. With both eyes open, line your finger up with some distant vertical object. Now blink first one eye and then the other. If, when you blink an eye, the finger remains lined up with the distant object, the open eye is the dominant eye. For the other eye, you'll see your finger shift to the right or left when you blink.

This part serves a a prelab to practice measurement techniques with a substance whose index of refraction is known.

  1. Sharpen your pencil first if you haven't already. You'll use a pencil rather than a pen to draw your construction.

  2. Place the Styrofaom on your desk, and place a sheet of unruled paper on top. Pin the corners of the paper in place to prevent it from shifting. 

  3. Add water to the plastic snap-lid container to a depth of about 2 cm. Close the lid.

  4. Place the box in the center of the paper as shown in Figure 1. While holding the box firmly in place, trace around it with your pencil. Try not to shift the position of the box while you're tracing rays. If the position does shift, line it back up.

  5. Remove two pins from the corners of the paper and stick them in the approximate locations shown in Figure 1. The angle that a line passing through the pins makes with the side of the bottle should be ~45°.

Figure 1 Figure 2

  1. You'll be sighting through the box from the side opposite the one where Pins 1 and 2 are. Get down at table height so that light from Pins 1 and 2 can reach your eyes after passing through the water. Shift your head so that the images of Pins 1 and 2 seen through the water are aligned as nearly as possible. When they're aligned place the other 2 pins along your line of sight (see Figure 2).

  2. Remove the box now so that you can draw on the paper. Line your ruler up carefully with Pins 1 and 2 and draw a straight line to the tracing of the box boundary. Repeat for Pins 3 and 4. Then draw a line inside the box joining the places where the rays intersect the sides of the box.

Figure 3 Figure 4

  1. Use your protractor to construct normals to the boundaries where the rays intersect.

  2. Now you're ready to measure angles of incidence and reflection. There are 4 angles, as indicated on Figure 3. Line up your protractor carefully and measure the angles to the nearest tenth of a degree or as closely as your eyesight will allow. You may need to extend some of the lines. Record your angle measurements directly on the construction.

  3. The construction is now complete, and you may remove it from the cardboard.

  1. First you'll do calculatons for boundary 1. Show your work in space shown in Figure 4 above. Begin by writing Snell's Law in symbolic form. Then, taking the index of refraction of air to be 1.00, use the two angles for boundary 1 (θi and θr) to calculate the index of refraction of water. Then find the experimental error between your calculated index of refraction and the accepted value of nw = 1.33.

  1. Repeat your calculations of index of refraction and experimental error for boundary 2 angles (θi' and θr'), and show your work in the corresponding space on the construction.

  2. Scan your paper and submit it as per the weekly schedule.

 Part B. Index of refraction of an unknown substance

Method

If your experimental error in Part A was more than 5%, you'll need to improve your experimental techniques in order to obtain good results in Part B. Here are some possible improvements:

  1. Draw all lines carefully with a straightedge and a fine-tipped pencil.

  2. Sight carefully, being sure to you use your dominant eye.

  3. When you measure angles, line up the protractor carefully. Then read angles to the nearest 0.1 degree.

The goal of this part is to determine the index of refraction of a block of plastic. This is the transparent block located in the same box as your helical spring. Here's an overview of the method: You'll measure the angle of incidence as a function of the angle of refraction for 5 incident angles. Then you'll re-express variables in order that a plot of the re-expressed variables yields a straight line. From that, you'll determine the index of refraction of the plastic. With careful measurement, the index of refraction can be determined with an accuracy of about 5%.

  1. Begin by placing a clean sheet of typing or copy paper in portrait orientation on your desk. Then use your straightedge to draw a line through the middle of the paper across the width. We'll call this a boundary line. See Figure 5. These are photos of an experiment in progress. Continue as follows.

    1. Use your protractor to construct a normal to the boundary line shown in Figure 5. The normal is labeled N. Construct the normal to intersect the boundary line a few centimeters to the right of center.

    2. Mark a point P on the normal a distance of about a centimeter above the boundary line.

    3. Next you'll mark a row of points (labeled 1 to 5). These points are about 5 centimeters above the boundary line. Note that the horizontal distances of the points from the normal progressively increase. Mark them at these distances from the normal line: 0.5, 1.2, 2.5, 3.9, 5.7 cm.
Figure 5 Figure 6
  1. Place the paper on the styrofoam sheet and pin the corners down. Then lay the plastic box with one of the longer edges aligned with the boundary line. Tape the block in place. Figure 6 above shows the block taped with blue tape. This is to provide contrast for the photo. You may use transparent tape if that's more convenient. You should have a roll in your lab kit. Insert pins at point P and point 1.
Figure 7 Figure 8

  1. Now you're ready for the first sighting. Sight through the block as you did in Part A for the bottle of water. Place two pins several centimeters apart to mark the sight line. See Figure 7. Label each of the pin locations 1. You now have three points marked 1. These three points and point P will determine the path of light for Sight Line 1.

  2. Repeat the process to mark points for Sight Line 2. See Figure 8. Note that the pin at point P is the only pin whose location is fixed. This point will be on all sight lines.

  3. Do the remaining sight lines. When you've done all five, your paper should look similar to that in Figure 9. Note that two of the points labeled 5 were cut off in the photo.

  4. Before you remove the block, draw a line along the lower side of the block to mark the second boundary line.
Figure 9 Figure 10

  1. After removing the block, use your straightedge to connect corresponding points on each side of the block. See Figure 10.

  2. Now draw the paths of the light rays inside the block as shown in Figure 11.
Figure 11 Figure 12

  1. See Figure 12. Prepare a table in the upper left-hand corner of your paper to record data. Note that i and r represent the angles that the incident and refracted light rays make with the normal to the boundary. Therefore, (90o - i) and (90o - r) represent the angles that the incident and refracted light rays make with the boundary itself. It will be easier to measure angles with respect to the boundary rather than the normal. That way, you won't have to clutter your construction with several more normal lines. Later, you can calculate the values of i and r for the analysis. Note that in Figure 12, the angle that Sight Line 5 makes with the boundary has been highlighted in red to indicate what angle must be measured.

  2. Next you'll measure the complements of the angles of refraction, (90o - r), with the protractor. Figure 13 shows one method of doing this. The refracted light rays are extended backward in red to distinguish them from the penciled lines. Once you've extended the lines, you can measure the angles that the refracted rays make with upper boundary and record the values in the table. An alternate method of measuring the needed angles is shown in Figure 14. You may find this method simpler, as it doesn't require extensions of lines. (This latter method was contributed by Charles Zhao, NCSSM, 2013.)

Figure 13 Figure 14
alternate method of measuring complement of angle of refraction

11. Write your name on your construction, scan it, and submit as per the weekly schedule as assignment L17D. The analysis will be due later.

Analysis

  1. Open Logger Pro. Enter the data for the angles (90o - i) and (90o - r) from your construction.

  2. Create a new calculated column for the angle of incidence i. This is simply the complement of (90o - i). However, convert the units to radians in your formula. The reason you need units of radians is because LP assumes those units as the arguments of trig functions.

  3. Repeat step 2 for the angle of refraction r.

  4. Now check your data table to make sure that all columns are labeled appropriately with the name of the variable and the units and that the Displayed Precision is appropriate. Angles in degrees should be displayed to tenths. Angles in radians should have significant figures consistent with the degree measures.

  5. From Snell's Law, you know that the sine of the angle of refraction is proportional to the sine of the angle of incidence. Therefore, in order to obtain a linear fit, it makes sense to re-express both angles before plotting. Create two more calculated columns for sin(i) and sin(r). Again, make sure that the values have the appropriate Displayed Precision.

  6. You can now plot a graph of sin(r) vs. sin(i) and perform a linear fit.

  7. Insert a textbox. Prepare the matching table and the equation of the fit.

  8. Auto-arrange the page.

Interpretation

Create a 2nd page in your LP file. Label the two pages Analysis and Interpretation. Answer the following on the Interpretation page.

  1. Show how you use the slope of the fit to determine the index of refraction of the plastic to 3 significant figures.

  2. Sighting errors, errors in lining up the ruler to draw sight lines, and errors in lining up the protractor are obvious sources of uncertainty. These tend to be random, so it's possible that some of these errors cancel each other. A systematic error, on the other hand, could affect all the data points in a similar way. This might be evidenced by a graph that displayed a significant non-zero intercept. Whether or not your graph had a significant non-zero intercept, describe one possible source of systematic error. Tell why you think the error would be systematic rather than random.

  3. Suppose that the plastic block were completely submerged in a rectangular box of water. Consider a light ray incident from the water on the plastic block at a non-zero angle of incidence. Which way does the ray bend as it enters the plastic from the water? Is the bend more or less than if the light ray entered the plastic from air at the same angle? Explain your answers.

Submitting your file

This concludes the experiment. Submit your Logger Pro file to WebAssign L17.


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