PHYS112 : Labs
Subsections


11 Geometric Optics

You may need to refer to Appendix A on uncertainties and Appendix B on linear regressions.

Introduction

Light crossing a planar boundary between a medium with index of refraction $n_1$ and a medium with index of refraction $n_2$ changes direction at the boundary. The directions of travel of the light in the two mediums are related by Snell's law,

\begin{displaymath} n_2 \sin(\theta_2) = n_1 \sin(\theta_1) \end{displaymath} (37)

where the angles $\theta_1$ and $\theta_2$ are measured as shown in Figure 24 relative to a line normal (perpendicular) to the boundary.

Figure 24: Refraction of a light ray crossing a boundary between mediums.
\scalebox{0.5}{ \includegraphics{geOptics-Snell.eps} }

Mirrors and lenses form images of objects. The positions of the object and image of a mirror or thin lens are related to its focal length $f$ by

\begin{displaymath} \frac{1}{o} + \frac{1}{i} = \frac{1}{f} \end{displaymath} (38)

where the image distance $i$ and object distance $o$ are measured relative to the position of the mirror or lens. The signs of $i$, $o$, and $f$ follow the conventions in Table 1.


Table 1: Sign conventions for mirrors and thin lenses.
    mirrors lenses
$f$ + concave converging
  - convex diverging
$o$ + real object (in front)
  - virtual object (behind)
$i$ + real image (in front) real image (behind)
  - virtual image (behind) virtual image (in front)
$m$ + upright image
  - inverted image
$h$, $h'$ + object/image oriented up
  - object/image oriented up


The magnification of a lens or mirror is given by

\begin{displaymath} m = -\frac{i}{o} = \frac{h'}{h} \end{displaymath} (39)

where $h$ is the height of the object and $h'$ is the height of the image. The overall magnification of a compound system of lenses and/or mirrors is simply the product of the magnifications of the constituents,
\begin{displaymath} m = m_1 m_2 \ldots \end{displaymath} (40)

The sign conventions for $h$, $h'$, and $m$ are given in Table 1.

Experiments and Analysis

Snell's Law

Figure 25: A light ray refracted at the flat surface of a semicircular container filled with water. The numbered circles represent the pins used to track the light in the experiment.
\scalebox{0.5}{ \includegraphics{geOptics-Snell2.eps} }

You will investigate the refraction of light traveling from air into water using a semicircular plastic container. Light incident at the center of the semicircle, as shown in Figure 25 refracts at the planar boundary. However, no refraction occurs at the circular boundary, because the angle of incidence at that boundary is $0^\circ$. The plastic walls of the container do not change the direction of travel of the light, because the air-plastic and plastic water boundaries are parallel to each other. Hence, the refraction shown in Figure 25 is due solely to the difference between the indices of refraction of air and water.

  1. Place a piece of paper on the cork board, and using a straight edge, draw a line across the center, and then draw another line perpendicular to that line.

  2. Fill the semicircular container with water and place the flat edge along one of the lines you drew, centered on the point at which they cross. Trace the shape of the container in case the container moves from the lines.

  3. Place a pin at some point opposite the flat face of the container (like pin 1 in Figure 25).

  4. Place a second pin at the center of the flat side of the container (like pin 2 in Figure 25). These two pins define the incident ray, which makes an angle $\theta_1$ with the normal to the flat surface of the container.

  5. Position yourself facing the flat surface of the container so that you can look through the container, sighting along pins 1 and 2. Place a third pin on the far side of the container so that it appears to line up with pins 1 and 2 through the container (like pin 3 in Figure 25).

  6. Remove the container, but keep the pins in place. Use a straight edge to draw lines connecting pins 1 and 2 (incident ray) and pins 2 and 3 (refracted ray).

  7. Use the protractor to measure $\theta_1$ and $\theta_2$.

  8. Repeat this procedure for four more angles of incidence, and use excel to make a graph of $\sin \theta_1$ vs. $\sin \theta_2$.

    Note: Trig functions in excel take angles in radians. You can use the radians() function to convert from degrees into radians.

  9. If your data is compatible with a linear model, use the LINEST function to determine the index of refraction of water from the slope of the graph, along with its uncertainty. (The index of refraction of air is 1.00.)

A Converging Lens

  1. Place a convex lens in the lens holder. Place the screen and the object lamp at opposite ends of the optical bench, just within the measurable scale on the bench.

  2. Move the lens until you can clearly see a focused image of the object lamp on the screen. Measure the distance between the object and the lens $(o)$ and the distance between the lens and the screen $(i)$.

  3. Repeat the process for four other screen-bulb distances.

  4. Enter your image and object distances into a spreadsheet and use your data to extract a best value for the focal length of your lens along with its uncertainty.

  5. Working with the group across the room, try to create an image of a light bulb mounted on their optical bench on your screen. It should appear as a small, bright dot. Measure the image distance.

  6. Find by trial and error the minimum screen-bulb distance at which you can focus an image on the screen with the lens.

Virtual Images

  1. Choose either a diverging lens or a convex mirror from the boxes from the bins at the front of the lab.

  2. Turn off the light bulb and place it so that you can see its image when looking into the lens/mirror.

  3. Measure the distance between the bulb and the lens/mirror $(o)$.

  4. Use parallax to determine the distance between the lens/mirror and the virtual image $(i)$. If you are not sure how this works, ask your instructor for help.

  5. Repeat the process for four other object distances.

  6. Enter your image and object distances into a spreadsheet and use your data to extract a best value for the focal length of your lens/mirror along with its uncertainty.

Obstructions

  1. Create a focused image of the filament on the screen with your converging lens.

  2. Use a sheet of paper, placed directly in front of the bulb, to block half of the light from the bulb. Record your observations of the effect this has on the image.

  3. Slowly move the sheet toward the lens along the optical axis, keeping half of the light blocked by the paper. Continue moving the paper until it reaches the lens. Record your observations of the image throughout.

  4. While against the lens, slide the paper slowly back and forth so that you block more than and less than half of the lens. Record your observations of the effect this has on the image.

Before You Leave Lab

Show your instructor your spreadsheet and discuss preliminary answers to the questions below.

Hand In ...

... a printout of your spreadsheet, including the graph and LINEST fit, and answers to the following.

  1. (a) Is your graph of $\sin \theta_1$ vs. $\sin \theta_2$ compatible with a linear model? Is it compatible with Snell's Law? (b) What is your best value for the index of refraction of water? (c) How does it compare with the accepted value $(n_\mathrm{H_2O} = 1.33)$?

  2. In your initial investigation of the converging lens, what did you find to be the minimum screen-bulb distance at which you could focus an image on the screen? Use Eq. 38 to explain this observation.

  3. Report the best values and uncertainties of the focal lengths of the converging lens and diverging lens/mirror, and describe how you determined them.


Copyright © 2006-2009, L.A. Riley, T. J. Carroll, J.S. Scott Updated Sun Apr 26 23:00:14 2009

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