Chapter 5: Waves Experiments

Prism Spectrometer -- Dispersion Curve for Glass (Version 2)

References

Crummett and Western, Physics: Models and Applications, Sec. 36-1,4
Halliday, Resnick, and Walker, Fundamentals of Physics (5th ed.), Sec. 34-7,8
Tipler, Physics for Scientists and Engineers (3rd ed.), Sec. 30-4

Introduction

In this experiment, you will have an opportunity to (1) determine the index of refraction of a glass prism as a function of wavelength; (2) become familiar with the Newton-Raphson method, an important numerical method for solving transcendental equations by successive approximations; and (3) practice your computer programming skills on a simple, straightforward problem.

The geometry of the prism spectrometer to be used is shown in Figure 1. Light from a source S is passed through a collimator, in which the light rays are refracted parallel to the axis, and strikes a glass prism. It is refracted at each of the prism surfaces, and enters a telescope through which the refracted light can be viewed. The total angle through which light has been deflected is D in Figure 2. This angle D will depend on the angle of incidence at which light strikes the prism , the prism's apex angle (A), and the wavelength of the light.

Figure 2 shows the geometry used in determining the index of refraction of a glass prism. The incident light consists of several discrete wavelengths which are refracted different amounts at the first and second prism surfaces (the figure shows the path for only one of these wavelengths). In general, the index of refraction of the prism will be different for each different wavelength of light. When you look through the telescope at light that is not monochromatic, therefore, you will see a separate image of the collimator entrance slit in light of each wavelength. (Each image is a vertical line, so one speaks of light containing different discrete wavelengths of light as producing a "line spectrum.")

For a given spectral line the second surface angle of refraction may be found from

(1)

(you will need to prove this). Applying Snell's law at each surface

so (2)

For a given spectral line, the only unknown in the above equation is n. However, the equation is transcendental, and there is no way of solving explicitly for n. Instead we need some kind of systematic method of approximating the value of n that makes F(n) = 0.

The Newton-Raphson method is one such. Suppose that a graph of F(n) vs. n looked something like Figure 3. If you had some initial guess (call it n₁) for the value of n for which F(n) = 0, you could get a better guess (n₂, say) by extrapolating the slope of the graph from n₁ until it hits the line F(n) = 0. (The extrapolation is shown in Fig.3 by a dashed line.) We know the formula for the slope of a straight line:

and so

We can repeat the procedure using n₂, get a still better approximation n₃, and so on and on:

(3)

-- continuing to repeat the process until the change in n as small as we wish. This is the Newton-Raphson method.

Equipment

• Gaertner student spectrometer
• glass prism
• multiple light source

The spectrometer is a frame (see Figure 1) upon which the collimator, prism support, and telescope are mounted in such a way as to allow them to rotate separately around a common axis. An angle scale is provided which allows measurement of the angular position of the telescope to within one minute of arc. The spectrometers should already have been aligned before you use them, so that the objective lens of the telescope brings the parallel light from the collimator into focus exactly in the plane of the cross hairs, and so that telescope, collimator, and prism table are all rotating about the same axis.

A large thumbscrew near the axis of the instrument, about an inch above scale level, clamps the telescope. Large adjustments of the telescope position are made by releasing this screw. A second large thumbscrew at the same height, at the base of the telescope, provides fine adjustment of the telescope position. The prism table assembly rides on a collar which is clamped in place by a smaller thumbscrew, fixing its height. The table is clamped against rotation by another screw about 4 cm below the prism table; loosening this screw allows the prism table to rotate about the spectrometer axis.

At the source end of the collimator, there is a small thumbscrew which adjusts the width of the entrance slit. This lets you use a wide slit, admitting lots of light, to find your way around; and then narrow the slit for precise angle measurements. This, and the four thumbscrews mentioned in the last paragraph, are the only adjustments you should have to make on the spectrometer.

The angular position of the telescope is read on a circular scale fastened to the base of the instrument. Main scale divisions are 0.5 degree apart, and they are subdivided by a 30-part vernier scale (see Sec. 2-1), so you can read the scale to one minute of arc. For example, if the index falls between 153.5 degrees and 154 degrees on the main scale, and the 19th vernier-scale mark lines up, the angle reading is . You may occasionally want to read the scale to half a minute of arc, if two adjacent vernier-scale marks seem equally well aligned. Note that the angle that the scale reads is not D (in Figure 2), but approximately 180 degrees ± D.

Procedure

(1) With no prism on the spectrometer table, swing the telescope around until the light coming directly from the collimator is observed. Set the telescope so that this image of the slit falls on the cross hairs, and record the reading on the angle scale. Repeat this measurement several times.

(2) Place the prism on the spectrometer table so that the apex of the prism points directly at the collimator, as in Figure 4. Re-tighten the screw. Illuminate the slit using the white (incandescent) bulb in your multiple light source, and rotate the telescope until the image of the slit, reflected from the side of the prism, can be seen. (This is position B in Figure 3). If you can't find the image, get your instructor to help you. Clamp the telescope position, and use the fine adjustment screw to set the image of the slit exactly on the cross hairs. Read and record the angular position of the telescope. Unclamp the telescope and rotate it to position B'; re-clamp it and use the fine-adjustment screw to set the cross-hairs on this image of the slit, and read and record the telescope angle. The angle through which the telescope has been rotated between B and B' is twice the apex angle (A) of your prism; you should prove this in your write-up.

(3) Using the mercury source, position the prism on the table in an orientation like that in Fig. 2, one that gives a good line spectrum due to the refraction at the two prism surfaces. Measure the angular position of the rays reflected at the first surface, and use this measurement to calculate the angle of incidence .

(4) Measure the angular positions of each of the spectral lines formed by refracted light. Calculate deviation angles D for each of the spectral lines.

(5) Repeat steps (3) and (4) for light striking the other side of the prism. That is, if in (4) you were looking at light deflected to the left, repeat using light deflected to the right.

Analysis

(1) Justify each of the angular relationships given beside the figure on the first page. Then prove that , and that . Finally, starting from Snell's law, prove that the equation for F(n) is correct.

(2) Write a computer program, in any language, that will calculate an index of refraction n for each of your determinations of each of the spectral lines, using the Newton-Raphson method of successive approximations. Your writeup should include a listing of your program and a table of n values obtained and corresponding wavelengths.

(3) Check the results of your computer program by using one of your indices of refraction to calculate the corresponding deviation angle D, using Snell's law at each surface. Compare this calculated value of D to that measured.

(4) For your prism, plot graphs of index n vs wavelength and n vs . (EXCEL works well.) Do a least squares straight line fit on of the latter graph in order to find the constants in the equation . Also calculate uncertainties in these values.

Chapter 5 -- Waves Experiments -- Ultrasonic Double Source Interference

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