L17.  Refraction

About your report:  Since you'll be drawing constructions, submit your report on paper.

Goals:   1) to measure the index of refraction of water, and 2) to investigate total internal reflection

 

Introduction:   You'll trace the path of a light ray through air, into a box of water, and back out into air.  Then you'll use Snell's law to determine the index of refraction of the water.  Afterwards, you'll trace a light ray through the corner of the box and adjust the angle of incidence so that the ray is totally-internally reflected (TIR) inside the box.  You'll predict the value of the angle of incidence in order to produce TIR, and you'll check your result experimentally.

 

Preparation:  You'll need to have completed P24 and read section 26-5.

 

Equipment

 

Transparent hard plastic box that can hold water (dimensions ~1" deep and 2-3" in length and width)

4 straight pins

~1-ft square piece of corrugated cardboard (or other surface into which pins can be stuck)

Ruler

Protractor

Pencil (sharpened)

Typing or copy paper (unruled)

Laser penlight or pointer

 

Method

 

Part A.  Index of refraction of water

 

With careful construction techniques and measurements, you can achieve 3 significant figure accuracy in measuring the index of refraction of water.

  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 cardboard 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. Fill the plastic box with water but not so high that the water will easily spill. 

  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. 

Figure 1 Figure 2

  1. You'll be sighting through the box from the side opposite that 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 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

 

  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.

Calculations

  1. Show your calculations in the body of your report.  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 (qi and qr) to calculate the index of refraction of water.  Repeat your calculation for boundary 2 angles (qi' and qr'). 

  2. Average your two values of index of refraction.  You'll use this result as the index of refraction of water for future calculations in this lab.

  3. Find the percentage difference between your average index of refraction and the value in the textbook. 

Part B.  Total internal reflection

 

Open this applet to view an animation of the situation that you will reproduce in the lab.  Read the applet description.  Note that for the initial situation, the angle of refraction from the right side of the box is 90°.  This, of course, is the greatest it can be.  Your goals in this part of the lab are a) to determine the minimum angle of incidence for which a refracted ray leaves the box on the right side, and b) compare the experimental value for the angle to that obtained using Snell's Law and the geometry of the situation.

  1. Use a laser pointer for this. Refer to the diagrams below as you follow the instructions.  Begin as in Part A with a clean sheet of unruled paper on the cardboard and the box of water in the center of the paper.  Trace around the box.

 

Figure 4 Figure 5
  1. Use your ruler to draw a dashed line coincident with the right side of the box and extending to the edge of the paper as in Figure 4.  Place a pin a small distance to the right of the line.  Now slowly sweep the laser pointer through a range of incident angles as shown in the diagram until you get a spot of laser light hitting the pin.  This is a matter of trial and error until you get a distinct spot on the pin.  It's important that the light path be like the one shown.  If the angle of incidence is too small, the light will refract across to the upper side of the box, missing the right side altogether.
  2. Once you're successful, mark the path of the incident laser light.  Use two pins as in Figure 5 to do so.
  3. Now you can remove the box.  Draw the normal as in Figure 6 and measure the angle of incidence.
Figure 6

That's all there is to the experimental part, but there's something else we recommend that you do if you have an aquarium at home or at school.  Aquariums are great for studying total internal reflection.  Try shining your laser pointer through a corner similar to what you did above.  The light beam can't get out if the angle of incidence isn't great enough.  Also try looking up at the under surface of the water through the side of the tank.  The under surface will look like a mirror, because the angle at which light must strike the surface from below in order to reach your eyes is such that the light is totally-internally reflected.  The under surface is, in effect, a perfect mirror.  You can also see this effect when underwater in a swimming pool if you view the under surface of the water at particular angles.

  1. The next thing to do is develop the theory that will allow you to calculate the value of the angle that you measured experimentally.  You'll have to apply Snell's Law at both the bottom boundary and the right boundary.  Carefully identify the pairs of angles of incidence and refraction.  You can use geometry to relate the angle of refraction from the bottom boundary to the angle of incidence on the right boundary.  You'll also need to use the fact that the angle of refraction from the right boundary is 90°.  (While the angle wasn't quite 90° in the experiment, it was close enough to obtain good results.  The reason we had you use an angle slightly less than 90° was so that the laser beam wouldn't be distorted by grazing the side of the box.)  Start your proof with a drawing in which you define symbols for the four angles.  Then show your applications of Snell's Law, and carry out the algebra to solve for the minimum angle of incidence, qi, in terms of the index of refraction, n. The best responses will avoid the use of nested trig and inverse trig functions. qi can be expressed in terms of a single inverse sine function.
  2. Using the value of the index of refraction of water that you found in Part A, calculate the minimum value of the angle of incidence.  Then calculate the percentage difference between this theoretical value and the value that you measured.
  3. Explain why there can be no refracted ray from the right side of the box if the index of refraction of the material in the box is greater than the square root of two.

Conclusion:   In addition to the usual components of the conclusion, address this question:  Why were you able to successfully ignore the influence of the plastic box itself?  The plastic refracts light, and its index of refraction isn't necessarily the same as that of the water it contains.

Submitting your report:  Fax your report and your two constructions by the due date.