Experiment 8: Refraction of Light

Experiment 8: Refraction of Light

$Learning Objectives$

Describe how a light ray changes direction when it crosses interfaces between materials of different refractive index.
Trace the incident, refracted, and emergent rays through a triangular prism using the pin-alignment method.
Determine the refractive index of the prism material from measured angles and compare independent entry- and exit-face estimates.
Determine the total angular deviation produced by a prism and relate it to the prism apex angle.
Determine the focal length of a converging lens using image formation and the thin-lens equation.
Determine focal length using the two-position displacement method for a fixed object-screen separation.
Determine the equivalent focal length of two thin converging lenses in contact and compare it with the individual focal lengths.
Evaluate uncertainty caused by parallax, pin alignment, angle measurement, lens placement, screen focusing, and non-ideal optics.

This experiment combines two geometric-optics investigations. In the first, sighting pins are used to reconstruct the path of a ray through a triangular prism and test Snell's law. In the second, a real image formed by one or two convex lenses is projected onto a screen so that focal length can be obtained by several independent methods. The procedures are based on the supplied laboratory-manual experiment, with the measurement sequence clarified and the optical assumptions stated explicitly.

$Target Learning Outcome$

Describe refraction through a prism, apply the laws governing refraction, determine an experimental refractive index, and determine the focal length of a convex lens by independent experimental methods.

I. Discussion of Theory

$Refraction$

Refraction is the change in propagation direction that occurs when light crosses an interface at which its speed changes. The frequency of the light remains unchanged at the interface; its wavelength and speed change with the optical medium.

$Refractive Index$

The absolute refractive index $n$ of a material is the ratio of the speed of light in vacuum to the speed of light in that material. For ordinary transparent materials in visible light, $n$ is greater than 1 and depends weakly on wavelength and temperature.

Absolute refractive index

Refractive index relates wave speed in vacuum to wave speed in a material.

n=\frac{c}{v}

Variables

Symbol	Description	Unit
$n$	absolute refractive index	dimensionless
$c$	speed of light in vacuum	m/s
$v$	speed of light in the material	m/s

$Normal Line$

The normal is an imaginary line perpendicular to the optical surface at the point where the ray crosses the interface. Angles of incidence and refraction are measured from the normal, not from the surface.

Snell's law

The incident and refracted rays obey this relationship at a plane interface.

n_1\sin i=n_2\sin r

Variables

Symbol	Description	Unit
$n_1$	refractive index of incident medium	dimensionless
$n_2$	refractive index of transmitted medium	dimensionless
$i$	angle of incidence measured from the normal	degrees or radians
$r$	angle of refraction measured from the normal	degrees or radians

$Direction of bending$

When light enters a medium with a larger refractive index, the refracted ray bends toward the normal. When it enters a medium with a smaller refractive index, it bends away from the normal. This statement concerns the ray direction relative to the local surface normal.

Experimental prism index from the entry face

For air incident on a prism, take the refractive index of air as approximately 1.000.

n_{\text{prism}}\approx\frac{\sin i}{\sin r_1}

Experimental prism index from the exit face

At the prism-to-air interface, the internal ray emerges into air.

n_{\text{prism}}\approx\frac{\sin e}{\sin r_2}

$Ray geometry inside a prism$

For a prism with apex angle $A$ , the two internal refraction angles satisfy

r_1+r_2=A.

If $i$ is the first incidence angle and $e$ is the emergence angle, the total deviation $\delta$ is

\delta=i+e-A.

The two independently calculated refractive-index values should agree within experimental uncertainty if the ray construction and angle measurements are consistent.

$Critical Angle$

The critical angle is the incident angle inside the higher-index material for which the refracted ray in the lower-index material is exactly parallel to the interface. At larger internal incidence angles, total internal reflection occurs.

Critical angle

This form applies when light travels from medium 1 to a lower-index medium 2.

\sin\theta_c=\frac{n_2}{n_1},\qquad n_1>n_2

Convex-Lens Theory

$Converging Lens$

A converging or convex lens brings paraxial rays that enter parallel to the principal axis to a focus on the opposite side of the lens. A real object placed beyond the focal point can form a real, inverted image that can be projected onto a screen.

Thin-lens equation

For a real object and real image, use positive object and image distances as measured magnitudes in this laboratory convention.

\frac{1}{f}=\frac{1}{D_o}+\frac{1}{D_i}

Variables

Symbol	Description	Unit
$f$	focal length of the converging lens	cm or m
$D_o$	object distance from the lens	cm or m
$D_i$	real image distance from the lens	cm or m

Linear magnification

The negative sign denotes inversion for a real image in the standard Cartesian sign convention.

m=-\frac{D_i}{D_o}=\frac{h_i}{h_o}

$Distant-object approximation$

If the object is sufficiently far from the lens, $D_o\gg f$ and $1/D_o$ becomes small. The image distance then approaches the focal length. A distant building, lamp across a large room, or another effectively remote target gives a better approximation than a nearby light-box object.

Two-position displacement method

With the object and screen fixed a distance L apart, two lens positions produce sharp real images when L is at least 4f.

f=\frac{L^2-d^2}{4L}

Variables

Symbol	Description	Unit
$L$	fixed object-to-screen separation	cm or m
$d$	distance between the two sharp-image lens positions	cm or m
$f$	lens focal length	cm or m

$Condition for two sharp-image positions$

The displacement method requires $L\ge4f$ . When $L<4f$ , there are no two real lens positions between the fixed object and screen that satisfy the thin-lens equation.

Two thin lenses in contact

For thin lenses in air with negligible separation, optical powers add.

\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}

Variables

Symbol	Description	Unit
$F$	equivalent focal length	cm or m
$f_1$	focal length of lens 1	cm or m
$f_2$	focal length of lens 2	cm or m

II. Equipment and Materials

Equipment or material	Laboratory use
Triangular glass prism	Produces refraction at two plane faces.
Bond paper or clean white paper	Records the prism outline and pin locations.
Drawing board or plywood board	Provides a flat surface into which pins can be placed.
Optical pins and pin hammer	Establish incident and emergent sight lines.
Ruler and meterstick	Constructs rays and measures distances.
Protractor	Measures incidence, refraction, emergence, and deviation angles.
Light box or illuminated object	Provides a bright object for lens focusing.
Two convex lenses with different focal lengths	Used for individual and combined focal-length measurements.
Screen or white board	Receives the real image formed by a convex lens.
Lens holders or stable supports	Keep the optical elements upright and aligned.

$Safety and apparatus care$

Handle optical pins with their sharp ends directed away from eyes and hands. Press pins only into the intended board. Do not look at the Sun through a lens or prism. Incandescent lamps and light boxes can become hot; allow hot components to cool before handling. Hold lenses and prism by their edges and keep optical surfaces clean.

III. Pre-Laboratory Checks

$Before collecting data$

Verify that the drawing board is flat and that the prism does not rock on the paper.
Use sharp, straight pins and keep them approximately vertical to reduce alignment error.
Check the protractor zero and confirm that all optical angles will be measured from a normal line.
Identify the prism apex angle $A$ from the traced prism geometry.
Clean the prism and lens surfaces using an appropriate lens cloth.
Align the illuminated object, lens center, and screen center at approximately the same height.
Record which convex lens is lens 1 and which is lens 2; do not exchange them during data reduction.

IV. Experimental Procedure

$Part A1 — Locate the refracted ray through the triangular prism$

STEP-BY-STEP

Fasten a clean sheet of bond paper to the drawing board and place the prism near the center.
Trace the prism outline accurately with a fine pencil. Label its vertices and the face through which the ray will enter.
Choose an incidence point $O$ on the first face and draw an incident line approaching $O$ . Place two pins, A and B, on this incident line outside the prism. Keep the pins well separated, preferably several centimeters apart, so small alignment errors produce a smaller angular error.
Replace the prism exactly on its traced outline.
View pins A and B through the opposite prism face with your eye close to the level of the board. Move your head slightly left and right to detect parallax.
Place pin C so that C appears collinear with the images of A and B. Place pin D farther from the prism on the same sight line. Recheck alignment by moving your eye slightly; the four pin images should remain aligned with minimal relative motion.
Mark the pin positions before removing the pins. Remove the prism only after all points are clearly labeled.
Draw the line through A and B to the first prism face and the line through C and D backward to the second face. Join the two interface points to reconstruct the internal ray.
Draw the normal to each prism face at the interface point. Label the first incidence angle $i$ , first refraction angle $r_1$ , second internal incidence angle $r_2$ , and emergence angle $e$ .
Repeat the complete construction for at least three different incidence angles rather than reusing the same pin holes.

$Part A2 — Determine refractive index and angular deviation$

STEP-BY-STEP

Measure $i$ , $r_1$ , $r_2$ , and $e$ with the protractor. Read angles from the normal, not from the prism face.
For each trial, compute $n_{\text{entry}}=\sin i/\sin r_1$ .
Independently compute $n_{\text{exit}}=\sin e/\sin r_2$ .
Check the prism-geometry relation $r_1+r_2\approx A$ . A large discrepancy usually indicates ray-construction or angle-reading error.
Compute the average experimental refractive index from the valid entry- and exit-face estimates.
Determine the total deviation $\delta$ as the angle between the original incident direction extended forward and the emergent ray. Verify it independently from $\delta=i+e-A$ .
If a reference refractive index for the prism material is supplied by the instructor, compute percent error. Do not assume all glass has the same refractive index.

$About the apparent-depth construction in the supplied manual$

The photographed manual also uses a pin construction that locates an apparent position of the prism vertex and forms line ratios such as real-to-apparent distances. That construction is useful as a second geometric estimate when performed exactly as drawn on the original laboratory sheet. The angle-based Snell-law method above is the primary method here because its geometry and assumptions can be stated unambiguously for any chosen incident ray.

$Part B1 — Estimate focal length using a distant object$

STEP-BY-STEP

Mount lens 1 vertically between a distant bright object and the screen.
Move the screen until the image is as sharp as possible. Use a high-contrast detail of the object to judge focus.
Measure the lens-to-screen distance. Record this as the distant-object estimate of $f_1$ .
Repeat at least three times, approaching best focus from both directions to reduce observer bias.
Repeat the procedure for lens 2 to obtain an estimate of $f_2$ .
If the available object is not distant compared with the focal length, record both $D_o$ and $D_i$ and use the thin-lens equation instead of treating $D_i$ as exactly equal to $f$ .

$Part B2 — Determine focal length using the fixed object-screen displacement method$

STEP-BY-STEP

Place the illuminated object and screen on the same optical axis and measure their fixed separation $L$ .
Choose $L$ comfortably greater than four times the approximate focal length of the lens.
Place lens 1 between the object and screen. Move it slowly until the first sharp real image is obtained. Record the lens position $x_1$ , object distance $D_{o1}$ , and image distance $D_{i1}$ .
Continue moving the lens toward the screen until the second sharp image is obtained. Record $x_2$ , $D_{o2}$ , and $D_{i2}$ .
Confirm that $D_o+D_i\approx L$ for both positions.
Compute $f$ from the thin-lens equation for each sharp position.
Compute the lens-position separation $d=|x_2-x_1|$ and calculate $f=(L^2-d^2)/(4L)$ .
Repeat for lens 2 using a suitable value of $L$ .

$Focusing criterion$

A bright image is not automatically a sharp image. Judge focus using a small edge, line, or printed feature and locate the screen position that minimizes blur. Record distances from the optical center of the lens holder as consistently as possible.

$Part B3 — Determine the equivalent focal length of two lenses in contact$

STEP-BY-STEP

Mount lens 1 and lens 2 together so their principal planes are as close as the holders safely permit.
Place the pair between the illuminated object and screen and obtain a sharp real image.
Measure $D_o$ from the lens-pair reference plane to the object and $D_i$ to the screen.
Compute the measured equivalent focal length $F$ using $1/F=1/D_o+1/D_i$ .
Compute the predicted equivalent focal length from the separately measured $f_1$ and $f_2$ using $1/F=1/f_1+1/f_2$ .
Compare the measured and predicted values. Discuss the effect of nonzero lens separation and uncertainty in the chosen reference plane.

V. Data and Results

Table 8.1 — Prism Ray-Tracing Measurements

Trial	Prism apex $A$	$i$	$r_1$	$r_2$	$e$	$n_{entry}$	$n_{exit}$	$\delta$ measured	$\delta=i+e-A$
1
2
3

Table 8.2 — Prism Refractive-Index Summary

Quantity	Value
Mean entry-face index
Mean exit-face index
Combined experimental index
Reference index, if supplied
Percent error

Table 8.3 — Distant-Object Focal-Length Estimates

Lens	Trial 1	Trial 2	Trial 3	Mean estimate
Lens 1
Lens 2

Table 8.4 — Fixed Object-Screen Method

Lens	$L$	$x_1$	$x_2$	$d$	$D_{o1}$	$D_{i1}$	$D_{o2}$	$D_{i2}$	$f$ from displacement
Lens 1
Lens 2

Table 8.5 — Two Lenses in Contact

$D_o$	$D_i$	Measured $F$	$f_1$	$f_2$	Predicted $F$	Percent difference

VI. Computation and Data-Quality Requirements

$Required calculations and checks$

Show one complete Snell-law substitution with angle units and the resulting dimensionless refractive index.
Compare entry-face and exit-face values before averaging; investigate any outlier rather than averaging blindly.
Verify $r_1+r_2\approx A$ for each prism trial.
Verify measured prism deviation against $i+e-A$ .
For each lens, show at least one thin-lens calculation using measured $D_o$ and $D_i$ .
For the displacement method, show the checks $L\ge4f$ and $d=|x_2-x_1|$ .
Compare independent focal-length methods using percent difference or percent error when a defensible reference is available.
Keep all distances in one unit system throughout each calculation.

VII. Uncertainty and Sources of Error

$High-value error analysis$

Pin parallax: Pins that only appear aligned from one eye position can produce a wrong ray direction. Recheck by moving the eye laterally.
Pin spacing: Closely spaced pins amplify angular uncertainty; greater spacing improves the line definition.
Prism replacement: If the prism is not returned exactly to its traced outline, both interface locations and normals shift.
Protractor resolution: A one-degree reading error can noticeably affect $\sin i/\sin r$ .
Lens optical center: Distance readings referenced to a thick holder or to different lens surfaces introduce systematic offset.
Subjective focus: The best-focus screen position has a finite interval. Approach focus from both directions and use a fine image feature.
Aberration: Spherical and chromatic aberrations prevent all rays and wavelengths from meeting at one exact point.
Lens separation: The simple in-contact formula assumes negligible spacing between the two thin lenses.

VIII. Post-Laboratory Questions

$Analysis questions$

State the three experimentally relevant laws or rules of refraction: coplanarity of incident/refracted ray and normal, Snell's law, and reversibility of the optical path for reciprocal media.
Starting with Snell's law for air entering a prism, show why $n\approx\sin i/\sin r$ .
For light traveling from glass with $n=1.54$ into water with $n=1.33$ , calculate the critical angle and explain what happens above it.
A ray travels from oil with $n=1.45$ into water with $n=1.33$ . For a specified incidence angle, determine the water-side refraction angle and state whether the ray bends toward or away from the normal.
Explain why a fixed object-screen separation smaller than $4f$ cannot produce the two Bessel lens positions.
Explain why the two displacement-method images have reciprocal magnifications.
Determine the image location, magnification, orientation, and relative size for a convex lens when $D_o$ and $f$ are given.
Explain why two positive thin lenses in contact have an equivalent focal length shorter than either individual focal length.

IX. Conclusion Guide

$Your conclusion should state$

the mean experimental refractive index and whether entry- and exit-face estimates agreed;
whether the prism deviation relationship was satisfied within experimental resolution;
the focal length of each lens from at least two methods;
the measured and predicted equivalent focal length of the lens pair;
the dominant experimental uncertainty and how the procedure could be improved.

Key Takeaways

Refraction angles must be measured from the normal, and Snell's law connects those angles to refractive index.
Pin alignment converts an optical sight line into a physical ray construction, so parallax control and pin spacing are essential.
A prism produces two refractions; its internal angles sum to the apex angle and its deviation satisfies $\delta=i+e-A$ .
A convex lens focal length can be checked by several independent methods rather than trusting a single measurement.
The two-position displacement method is valid only when the fixed object-screen separation is at least four focal lengths.
Agreement among independent methods is stronger evidence than a single result that happens to match a reference value.

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