# SPEED OF LIGHT

## SPEED OF LIGHT

Physics 320 Fall (12) 2018 SPEED OF LIGHT PURPOSE Measure the speed of light using a rotating mirror setup with a He-Ne laser as the light source. OVERVIEW This experiment is really nothing more than a "time of flight" measurement in which a speed is determined as a distance divided by a time. The equation you will use to calculate the speed of light c can be obtained through a careful analysis of the experimental setup. A diagram of that setup (which corresponds fairly closely to the physical layout) is shown below. Light from the laser goes through the slit (S) (and 45º glass plate D) and reflects from the rotatable mirror labeled MR.

If the mirror is not rotating and is set at just the right position, the light will proceed to mirror M1, then to M2, and then retrace its path until it strikes the 45º glass deflector plate (D). The deflector plate will reflect some of the returning light to the observer who is looking perpendicular to the laser beam.

If MR is rotating, pulses of light get injected into the abd-optical path when MR is in the proper orientation. These make a round trip, in time 2(a+b+d)/c, along the abd-path returning to MR, which will have rotated slightly in this time. Consequently, the light pulse reflected back toward the laser moves along a path at a slight angle θ relative to the original path r (see ray diagrams at the end). The returning light is thus deflected sideways by a small distance s ≈ rθ relative to its original position when the mirror is stationary. It can be shown that this beam deflection is given by € s = 8π f (a + b + d)r c (1) where f is the rotational rate of the mirror in rev/s.

In this experiment Eq. (1) will be used to determine the speed of light.

The Optical System In order to give a good definition to the beam of light, the slit (lower right of diagram) is made very narrow. But when the slit is narrowed, the light coming through forms a single slit diffraction M1 MR S Laser D observer r M2 L d a b glass disk

– 2 – pattern with a broad central bright region. This central region can be thought of as light diverging from the slit. The lens L changes this diverging beam into a converging beam. The slit, the lens L, and the mirror M2 form an optical system for which the slit is the object and the lens forms an image of that object on M2.

It is very important that the image formed at M2 be as sharp as possible because that image becomes the object for the returning light and the observer looks at the image formed from that object. Note that the observer is looking through a viewing scope at a glass disk between the lens and the 45º beam deflector. The deflector-to-glass-disk distance needs to be the same as the deflector-to-slit distance so that the object at M2 (first image of the slit) will be projected as a sharp image (second image of the slit) on the glass disk. The viewing scope must be focused on the glass disk.

The thin lens equation from geometrical optics can be used to establish approximate distances between components. For the formation of an image, the condition of 1/f = 1/(r+a) + 1/(b+d), where f is the focal length of the lens, must hold. For best results in this experiment we want to maximize both the distance a+b+d and the distance r. The lens you will use is labeled as having a focal length of 5 m, however, the actual focal length is closer to 4.75 m. In practice, the lens position required to produce a sharp image at M2 will need to be determined experimentally. As noted above, a careful analysis of the optical system used in this experiment leads to Eq.

(1). You are required to derive this equation as a part of your lab report. Warning: A "not so careful" analysis will lead to a result that is off by a factor of 2. The ray diagrams at the end of these instructions should help you get this right.

PROCEDURE CAUTION: Never look into the direct or reflected beam of the laser. I. Initial Optics Alignment (slit wide open, no 5m lens) 1. Position the laser, adjustable slit, 45º glass deflector plate and rotatable mirror housing so that the laser beam goes through both slit and deflector plate and hits the middle of the rotatable mirror. 2. Use the key on the chain to rotate the mirror so that the beam hits the center of M1, the mirror mounted at the west wall. Never touch the rotatable mirror with your finger. Do not put the 5m focal length lens in the beam path yet.

3. Adjust the wall mirror (M1) so that the light beam heads toward the mirror (M2) held by the finger clamp on the heavy stand.

With M2 back near the east wall, adjust the position of M2 such that the beam hits the mirror center. The base of the heavy stand should be oriented so that one of the legs with a cross-handle screw adjustment is extending away from the wall mirror and is parallel with the rod on the finger clamp. CAUTION: Do NOT touch the surfaces of mirrors M1 or M2. 4. Adjust M2 such that the reflected beam returns to the center of M1. Do this by using the screw adjustment on the back leg of the heavy stand to move the beam up an down and sliding the back leg by very small amounts to move the beam right or left.

Now make adjustments such that the beam exactly returns to the slit. (Since the beam is diverging, the return beam will produce a large circle of light centered on the slit).

II. Lens Positioning and Final Optics Alignment (narrow slit, 5 m lens in place) 1. Put the 5 m focal length lens in the beam and adjust it so that the light goes through approximately the center of the lens. Narrow the slit until the central broad portion of the single slit diffraction pattern spans a distance of about 30 cm at the location of the rotatable mirror. Move the heavy stand with mirror M2 out of the beam path and use a 3x5 card or piece of white paper to determine where the best image of the slit is formed. Do this by moving the card back and forth along the beam (path

– 3 – d in the diagram) until the image line of light is narrowest.

It is also where the image line is vertical (for the vertical slit). An off focus condition results in a blurry spot or tilted line. 2. Move the heavy stand with mirror M2 back into the beam path at the location of best focus that you have determined. Mark the location of the 5 m focal length lens and then remove the lens from the beam path. Now open the slit and repeat step 4 from the above "Initial Optics Alignment". 3. Replace the 5-m focal length lens in the beam path (at the same location it was when you determined the position for the heavy stand). You will see two spots on the lens.

Adjust either the lens or mirror M1 until the two spots fall exactly on top of each other. [You want the plane of the lens to be a bit "off perpendicular" to the beam to avoid any reflected light acting as another source ... you should be able to locate this reflected light on the wall.] III. Deflector Plate and Viewing Scope Setup 1. Adjust the 45º glass deflector plate so that the returning beam is split off perpendicular to the main beam as shown in the diagram. Position the thin glass disk (held in a clamp on a rod stand) so that the surface toward the plate (which should have the cross-hatch scale lines on it) is the same distance from the surface of the plate being hit by the return beam as the slit is from that surface.

Adjust the vertical and sideways position of the glass disk so that the returning laser light hits it. Position the disk so that the spot (there will be two spots) coming from the front surface of the deflector plate (the surface away from the slit) falls at about the middle of the disk. This spot is the second image of the slit (the first image is at the mirror on the heavy stand) and can be viewed with the scope. 2. CAUTION: Never look through the viewing scope at the return beam unless the mirror is rotating. With the Laser OFF adjust the stand holding the positioning stage that has the viewing scope mounted on it so that the scope focuses on the etched scale on the glass disk.

Use the positioning stage to measure the spacing between the etch marks on the glass disk (noting that there is a lot of "play" in the positioning stage dial).

3. Turn on the laser and adjust the viewing scope so that the split-off beam (the one from the front of the glass plate) goes through the middle of the scope as determined by a light spot on the front lens and a light spot on the eyepiece. It may be easiest to use a white card to determine the beam location ... DO NOT LOOK THROUGH THE VIEWING SCOPE FOR THIS ADJUSTMENT. IV. Rotational Speed Monitoring Setup 1. Connect the blue 9-pin connector from the phototransistor unit (mounted on a rod stand) to the detector input on the front of the instrument rack. Connect a BNC tee to the POSITIVE Input on the back of the EG&G ORTEC 871 TIMER AND COUNTER in the rack.

Connect the BNC cable from the phototransistor unit to the tee, and a BNC cable from the tee to an oscilloscope. Turn on instrument rack (toggle switch on front lower right).

2. Check to be sure the switch on the remote speed control box is off and the speed setting is at minimum (dial turned all the way counterclockwise). Plug the rotatable mirror cord into the speed control box. Start the mirror rotating at a moderate speed. 3. Adjust the position of the phototransistor unit so that laser beam swings past the center of the lens when the housing is horizontal and aimed toward the rotating mirror. 4. Adjust the settings on the oscilloscope so that you see several peaks on the trace. (If you do not see any peaks on the oscilloscope trace, adjust the position of the phototransistor housing until you do see them.) The input gain dial of the oscilloscope should be set at 2 V/Div.

While watching the trace, adjust the position of the phototransistor housing until the spikes are at their maximum height (about 10 volts above zero level). Keep adjusting until the peaks are at their maximum and the line between the peaks is fairly smooth (except for the undershoot). Run the motor speed to maximum to be certain the maximum peak height does not fall below a few volts. Do not leave the motor running at top speed for very long.

5. We will use the scaler (i.e., the Timer/Counter) to count mirror rotations in ten second time intervals. For this, set up the scaler as follows: TIMER AND COUNTER set to COUNTER mode. TIME BASE set to 0.1 SEC with M=1 and N=2 (so count time is 0.1×102 seconds). Turn the knob labeled

– 4 – DWELL to about "11 o'clock". Now press sequentially the buttons labeled STOP, RESET, COUNT. You should see counts being recorded on the display when the mirror is rotating and peaks are displayed on the oscilloscope. The rate at which they are being recorded will change when you change the speed of the motor.

The counter will turn off after 10 seconds and then automatically reset and start for another 10 second interval. The DWELL knob adjusts the amount of time the recorded count value is displayed before counting is resumed. Note that for this type of periodic pulse input, the oscilloscope will measure a frequency (bottom right of the scope display) that should agree with the counter result.

Note that the rotating mirror reflects from both sides. Consequently, two counts will be recorded for a single rotation of the mirror. Thus the oscilloscope frequency must be divided by 2 and the counter values must be divided by 20 to get the actual rotational rate of the mirror in rev/s. V. Final Checkout 1. With the rotatable mirror stopped, narrow the slit until the central bright region of the diffraction pattern spans about 30 cm. Start the mirror rotating and be sure good stable spikes are still seen on the oscilloscope. Briefly run the motor speed to maximum and be sure the spikes do not reduce very much.

(If they do, they probably will not be recorded on the digital counter.) 2. With the motor running at a moderate speed look through the viewing scope. You should see an elongated spot of light. (This is the second slit image. It will never really look like a sharp image of the slit.) As you watch the light spot though the scope, increase the speed of the motor. You should see the spot move to the right. Since the scope inverts the viewed image, the light spot on the disk is actually moving to the left. Once you confirm movement of the return beam with changing mirror speed, the setup is ready for data collection.

You might want to make some final adjustments to the slit width while viewing the light spot to get as small a spot as possible (although if the slit is too narrow the beam intensity at the photodetector may become too low to register). VI. Data Collection 1. Measure the path distances r, a, b, and d and use these to compute the actual lens focal length f. 2. Start the mirror rotating (slowly) and adjust the speed such that the spot of light is centered on one of the etch marks. Record this speed, then increase the rotation rate (spot of light should move right) until the spot is centered on the next etch mark.

Continue this procedure until the motor reaches maximum speed. Now start slowing down the motor, continuing to take data until the spot is recentered at its original position. Taking data in this fashion allows you to estimate just how reproducible your results are since you will get two speed measurements for each position (etch line). As you take data try to estimate the measurement uncertainty in the position of the spot. A smaller/better-defined spot will result in more precise measurements. 3. Repeat the procedure above several times alternating which person is viewing the spot. DATA ANALYSIS 1.

Since the starting reference line was set for a condition of slow speed and not the mirror at rest, Eq. (1), which was derived assuming s is the shift from the rest condition, cannot be used directly. Obtain an expression that can be used with the shift of s2–s1 where s1 is spot position at the slow rotational rate and s2 is the position at the fast rate. Show how you obtained the expression.

2. Obtain an average value for c (including an uncertainty estimate) from your several values calculated using the f and s values of each trial together with the path length measurements.

– 5 – RAY DIAGRAMS FOR SPEED OF LIGHT ROTATING MIRROR Initial mirror orientation sending a light pulse into the optical system ... angle of reflection is β: Mirror orientation when pulse completes the round-trip through the optical system ... mirror has rotated by angle α and the angle of reflection is now β–α : Use these two diagrams to determine how the beam deflection angle θ is related to the mirror rotation angle α.

β β π–2β to mirror M1 β–α β–α θ α return from M1

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