Съдържание

Условие 1.0 Introduction 1.1 Measurement with the laser distance meter (0,4 т.)1.2 Experiment with the fiber optic cable (3 т.)1.3 Laser distance meter at an angle from the vertical (0,6 т.)1.4 Experiment with the optical vessel (4 т.)Отговори Решение

IPhO 2013, Експеримент — Задача 1. Speed of light

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Всички материали от IPhO 2013

44th IPhO Copenhagen Denmark 7-15 July 2013 — Speed of light — E1 · 7 юли 2013 г.

Условие

Notice: All measurements and calculated values must be presented with SI units with an appropriate number of significant digits. Uncertainties required only when explicitly asked for.

1.0 Introduction

Experiments with a laser distance meter (LDM)

A: Laser distance meter

B: Fiber optic cable (approximately 1 m)

C: Self-adhesive black felt pads with hole

D: Tape measure

E: Tape

F: Scissors

G: Lid from the black box

A laser distance meter (LDM, see Fig. 1.2 and Fig. 1.3) consists of an emitter and a receiver. The emitter is a diode laser that emits a modulated laser beam, i.e. a laser beam for which the amplitude varies at a very high frequency. When the laser beam hits an object, light is reflected in all directions from the laser dot. Some of this light returns to the instrument’s receiver which is situated immediately next to the emitter. The instrument’s telescope optics is focused on the laser dot and receives the light returned from the laser dot. The electronics of the instrument measures the time difference in the modulation of the received light signal relative to the emitted light signal. The delay $t$ in the modulation is exactly the time it takes for the light to travel from emitter to receiver. The measured time is then converted to a value

y = \frac{1}{2}ct + k

This value $y$ is shown in the instrument’s display. Here, $c = 2.998\cdot 10^{8}\mathrm{ms}^{-1}$ is the speed of light. The constant $k$ depends on the instrument setting; on the instrument you can switch between measuring the distance either from the rear end or from the front end of the instrument. When the laser distance meter is turned on, the default setting is to measure from the rear. This setting shall be maintained during all measurements.

Due to parallax, the LDM cannot measure any distance shorter than 5 cm. The maximum distance that can be measured is around 25 m. The shape of the instrument is such that the rear side is perpendicular to the laser beam as well as the front side. When the instrument is lying on the table the polarization is vertical (perpendicular to the display)

The diode laser is of class 2 with power $\lt 1$ mW and wavelength 635 nm. Manufacturer uncertainty for measurements is +/- 2 mm.

Warning: The instrument’s diode laser can damage your eyes. Do not look into the laser beam and do not shine it into other people’s eyes!

Settings for LDM

The above calculation of the distance $y$ of course assumes that the light has been travelling at speed $c$ . At the level of accuracy in this experiment, there is no need to distinguish between the speed of light in vacuum and in air, since the refractive index for dry, atmospheric air at normal pressure and temperature is $1.000\ 29 \approx 1.000$ .

A: On/off

B: Switch between measurement from the rear and the front of the instrument.

C: Indicator for measurement from the rear/front

D: Turn on laser/start measurement

E: Continuous measurement

F: Indicator for continuous measurement

A: Receiver: Lens for the telescope focused on the laser dot

B: Emitter: Do not look into the laser beam!

Photograph of the equipment on a grey table, labelled with red letters: A laser distance meter (Bosch), B coiled black fiber optic cable, C two black felt pads, D IKEA tape measure, E roll of tape, F blue scissors, G lid of the black box. — Figure 1.1 Equipment for the first experiments 1.1 and 1.2.

Photograph of the Bosch laser distance meter from the display side showing 0.769 m, with red arrows and letters: A and B at the bottom buttons, C at the rear/front indicator on the display, D at the red start button, E at the continuous-measurement button, F at the continuous-measurement indicator. — Figure 1.2 The unlabeled six buttons are irrelevant (they are used to calculate area and volume). The relevant buttons are:

Photograph of the laser distance meter held in a hand above grass, seen from the front end; arrow A points at the receiver lens, arrow B at the blue emitter window. — Figure 1.3 The laser distance meter seen from the front end:

1.1 Measurement with the laser distance meter (0,4 т.)

The instrument will perform a measurement when you press the button D, see Fig. 1.2.

1.1 Use the LDM to measure the distance $H$ from the top of the table to the floor. Write down the uncertainty $\Delta H$ . Show with a sketch how you perform this measurement. [0,4 т.]

1.2 Experiment with the fiber optic cable (3 т.)

You have been given a fiber optic cable of length approximately 1 m and diameter approximately 2 mm. The cable consists of two optical materials. The core (diameter approximately 1 mm) is made from a plastic with a high refractive index. The core is surrounded by a cladding made from a plastic with a slightly lower refractive index, and this is covered by a protective jacket of black plastic. Core and cladding serve as a wave guide for light shone into the cable, since the boundary between core and cladding will cause total reflection – and thereby prevent the light from leaving the core – as long as the angle of incidence is larger than the critical angle for total reflection. The light will therefore follow the core fiber, even if the cable bends, as long as it is not bent too much.

The LDM should now be set for continuous measurement (E, see Fig. 1.2), so that the display indication $y$ updates approximately once per second. The LDM will automatically go into sleep mode after a few minutes. It can be reactivated by pushing the red start button.

Carefully and gently cover the lens of the receiver with one small, black felt pad (the other is a backup) with a hole of diameter 2 mm (see figure 1.3A). The adhesive side of the pad should be pressed softly against the lens. Insert a fiber optic cable of length $x$ in the hole in the pad so that it touches the lens,

see Fig. 1.5.

The other end of the cable should be held against the emitter, so that it touches the glass in the middle of the laser beam. Now read off the $y$ -value from the display. The supplied scissors should be used to cut the fiber optic cable into different lengths $x$ .

Think very carefully before cutting the fiber optic cable, as you cannot get any more cable!!

Notice also that the LDM display might show a thermometer icon after a while in the continuous mode due to excessive heating of the electronics. If this happens, turn off the LDM for a while to cool off the instrument.

Drawing of a cut-away fiber optic cable: a light inner core cylinder protruding from a pale cladding layer, which protrudes from a thick dark grey protective jacket; dashed lines continue the jacket outline to the right. — Figure 1.4 Diagram of a fiber optic cable.

Two photographs side by side: (a) fingers holding a round black felt pad with a small central hole and the end of the black fiber optic cable next to it; (b) hands holding a loop of fiber optic cable whose ends touch the front of the laser distance meter, which displays 0.416. — Figure 1.5 (a) Felt pad and fiber optic cable. (b) Attaching the fiber optic cable.

1.2a Measure corresponding values of $x$ and $y$ . Set up a table with your measurements. Draw a graph showing $y$ as a function of $x$ . [1,8 т.]

1.2b Use the graph to find the refractive index $n_{\mathrm{co}}$ for the material from which the core of the fiber optic cable is made. Calculate the speed of light $v_{\mathrm{co}}$ in the core of the fiber optic cable. [1,2 т.]

1.3 Laser distance meter at an angle from the vertical (0,6 т.)

In this part of the experiment you will need the equipment shown in Fig. 1.6.

A: Optical vessel with water and measuring tape

B: Magnet to secure the angle iron on top of the black box. (You find magnet placed on the angle iron).

C: Angle iron with self-adhesive foam pads

D: Self-adhesive foam pads

Remove the black felt pad from the lens. The LDM should now be placed in the following set-up: Place two self-adhesive foam pads on the angle iron, see A on Fig. 1.7.

The LDM should be carefully placed on the angle iron as shown in Fig. 1.8.

The angle iron with the LDM should be mounted on the black box as shown in Fig. 1.9. Secure the angle iron to the box with a magnet placed below inside the box. (The tiny magnet is found on the angle iron). It is important to mount the LDM exactly as in the photo, since the side of the box facing upwards slants by approximately 4 degrees. The laser beam should now be pointing unobstructedly downwards at an angle.

A: Important: The bottom of the black box must face forward as shown. The side that faces upwards is slanting approximately 4 degrees with respect to the horizontal plane. Make sure that the angle $\boldsymbol{\theta_1}$ is the same all the time

When the LDM is turned on and mounted as explained above, the laser beam will form an angle $\theta_1$ with respect to the vertical direction. This angle, which must be the same throughout this experiment, must now be determined. The optical vessel is not needed here, so put it aside so far.

Photograph of the equipment for experiment 1.3 on a white table: a water bottle, a transparent optical vessel with an IKEA measuring tape (arrow A), a small magnet (arrow B), an angle iron with foam pads (arrow C), foam pad strips (arrow D), a roll of tape, scissors and the laser distance meter. — Figure 1.6 Equipment for experiment 1.3 shown in the figure:

Photograph of an L-shaped angle iron with two white foam pads, one on the upright leg and one on the base, each marked by a red arrow labelled A; a hand holds the laser distance meter above it. — Figure 1.7 How to place the two self-adhesive foam pads on the angle iron.

Two photographs from different sides of the laser distance meter leaning against the upright leg of the angle iron, resting on the foam pads. — Figure 1.8 How to place the laser distance meter on the angle iron.

Photograph of the set-up: the laser distance meter mounted on the angle iron on top of the black box, with the transparent optical vessel with measuring tape in front, a water bottle and unused equipment behind; a red arrow labelled A points to the bottom of the black box facing forward. — Figure 1.9 The experimental set-up. (The black box only serves as a support. The equipment behind the bottle is not used, though).

1.3a Measure with the LDM the distance $y_1$ to the laser dot where the laser beam hits the table top. Then move the box with the LDM horizontally until the laser beam hits the floor. Measure the distance $y_2$ to the laser dot where the laser beam hits the floor. State the uncertainties. [0,2 т.]

1.3b Calculate the angle $\theta_1$ using only these measurements $y_1,\ y_2$ and $H$ (from problem 1.1). Determine the uncertainty $\Delta\theta_1$ . [0,4 т.]

1.4 Experiment with the optical vessel (4 т.)

Place the optical vessel so that the laser beam hits the bottom of the vessel approximately in the middle, see Fig. 1.10. Pour some water into the vessel. The depth of the water is $x$ . Read off $y$ on the display of the LDM.

Diagram of a vessel partly filled with water of depth x. A red beam from the LDM at the upper right hits the water surface at angle θ1 to the dashed vertical normal; a dashed red reflected ray leaves at θ1 on the other side; the refracted ray continues at angle θ2 to the bottom, where it scatters in several directions. A vertical dashed arrow marks the height h from the bottom to the LDM, and another marks the water depth x. — Figure 1.10 Diagram of laser beams in the optical vessel with water of depth $x$ .

1.4a Measure corresponding values of $x$ and $y$ . Set up a table with your measurements. Draw a graph of $y$ as a function of $x$ . [1,6 т.]

1.4b Use equations to explain theoretically what the graph is expected to look like. [1,2 т.]

1.4c Use the graph to determine the refractive index $n_{\mathrm{w}}$ for water. [1,2 т.]

Отговори

Решение

In this document decimal comma is used instead of decimal point in graphs and tables

1.1 Measurement with the laser distance meter

1.1 Use the LDM to measure the distance $H$ from the top of the table to the floor. Write down the uncertainty $\Delta H$ . Show with a sketch how you perform this measurement. 0.4

$H = 907\ \mathrm{mm}\ \pm 2\ \mathrm{mm}$ . See the sketch in the figure corresponding to 1.3b. It must appear how the height is measured with the LDM in the rear mode.

1.2 Experiment with the fiber optic cable

1.2a Measure corresponding values of $x$ and $y$ . Set up a table with your measurements. Draw a graph showing $y$ as a function of $x$ . 1.8

Here, a 2 m cable is used, but 1 m is sufficient. There should be about 8 lengths evenly distributed in the interval from 0 m to 1 m.

x	y
m	m
0,103	0,177
0,176	0,232
0,348	0,396
0,546	0,517
0,617	0,570
0,839	0,748
1,025	0,885
1,107	0,950
1,750	1,459
2,000	1,642

The refractive index is twice the gradient of the linear graph, $n_{\mathrm{co}} = 2\cdot 0.7710 \approx 1.54$ .

The reason for that is that the travel time for a light pulse

t = \frac{x}{v_{\mathrm{co}}} = \frac{x n_{\mathrm{co}}}{c}

The display will therefore show $y = \frac{1}{2}ct + k \Leftrightarrow y = \frac{1}{2}n_{\mathrm{co}}x + k$ .

The speed of light in the core of the cable is $v_{\mathrm{co}} = \frac{c}{n_{\mathrm{co}}} = \frac{2{,}998\cdot 10^{8}\frac{\mathrm{m}}{\mathrm{s}}}{2\cdot 0.7710} \approx 1.95\cdot 10^{8}\,\frac{\mathrm{m}}{\mathrm{s}}$

Graph titled 'Display y as function of cable length x': Display y [m] from 0,0 to 1,8 against cable length x [m] from 0,0 to 2,0; ten measured points lie on a straight fitted line with y = 0,7710x + 0,1014, R² = 0,9996. — Display y as function of cable length x

1.3 Laser distance meter at an angle from the vertical

$y_1 =\ 312\ \mathrm{mm} \pm 2\ \mathrm{mm},\ y_2 =\ 1273\ \mathrm{mm} \pm 2\ \mathrm{mm}$

1.3b Calculate the angle $\theta_1$ using only these measurements $y_1,\ y_2$ and $H$ (from problem 1.1). Determine the uncertainty $\Delta\theta_1$ . 0.4

\begin{aligned}\theta_1 &= \cos^{-1}\left(\frac{H}{y_2 - y_1}\right)\\ &= \cos^{-1}\left(\frac{907\ \mathrm{mm}}{961\ \mathrm{mm}}\right)\\ &= 19.30^{\circ}\end{aligned}

(see the figure)

Measuring the horizontal part of some triangle is very inaccurate because of the size of the laser dot. No marks will be awarded for that. Using $\delta = 2\ \mathrm{mm}$ as the uncertainty of $y_1,\ y_2$ and $H$ , the uncertainty of $\theta_1$ can be calculated as follows:

\Delta\cos\theta_1 = \Delta\left(\frac{H}{y_2 - y_1}\right)

Using simple derivatives yields

\tan\theta_1\cdot\Delta\theta_1 = \frac{\delta}{H} + \frac{2\delta}{y_2 - y_1}

\Delta\theta_1 = \frac{\left(\frac{\delta}{H} + \frac{2\delta}{y_2 - y_1}\right)}{\tan\theta_1}\cdot\frac{180^{\circ}}{\pi} = \frac{\left(\frac{2}{907} + \frac{4}{961}\right)}{\tan 19{,}30^{\circ}}\cdot\frac{180^{\circ}}{\pi} \approx 1^{\circ}

Otherwise, using min/max method

\Delta\theta_1 = \theta_{1\mathrm{max}} - \theta_1 = \cos^{-1}\left(\frac{H_{\mathrm{min}}}{y_{2\mathrm{max}} - y_{1\mathrm{min}}}\right) = \cos^{-1}\left(\frac{905\ \mathrm{mm}}{965\ \mathrm{mm}}\right) - \cos^{-1}\left(\frac{907\ \mathrm{mm}}{961\ \mathrm{mm}}\right) = 1.0^{\circ}

Alternatively, calculate $\Delta\theta_1$ using $\Delta(y_2 - y_1) = \sqrt{(\Delta y_1)^2 + (\Delta y_2)^2} = \sqrt{2}\delta$ and then

\tan\theta_1\cdot\Delta\theta_1 = \sqrt{\left(\frac{\delta}{H}\right)^2 + \frac{2\delta^2}{(y_2 - y_1)^2}}

Also, accept $\delta = 1\ \mathrm{mm}$ and $\Delta\theta_1 = 0.5^{\circ}$ .

Sketch of a table on a hatched floor: an LDM (green) on a stand on the table top sends a red beam at distance y1 to the table top, and a second LDM on a stand near the table edge sends a beam at distance y2 to the floor, at angle θ1 to the dashed vertical; on the right an LDM held against the underside of a hatched block measures the vertical height H to the floor. A separate dashed right triangle has hypotenuse y2 − y1, vertical side H and angle θ1 at the top. — (see the figure)

1.4 Experiment with the optical vessel

1.4a Measure corresponding values of $x$ and $y$ . Set up a table with your measurements. Draw a graph of $y$ as a function of $x$ . 1.6

$x$ [mm]	$y$ [mm]
4	450
17	454
27	457
32	459
39	461
51	466
58	467
66	471
76	473
82	476
90	478
96	480

1.4b Use equations to explain theoretically what the graph is expected to look like. 1.2

The time it takes the light to reach the water surface is

t_1 = \frac{(h - x)/\cos\theta_1}{c}

From the water surface to the bottom the light uses the time

t_2 = \frac{x/\cos\theta_2}{v}

Total travel time forth and back

t = 2t_1 + 2t_2 = 2\frac{(h - x)/\cos\theta_1}{c} + 2\frac{x/\cos\theta_2}{v} = 2\frac{h - x}{c\cos\theta_1} + 2\frac{nx}{c\cos\theta_2}

Hence, the display will show (we simply write $n = \ n_{\mathrm{w}}$ )

y = \tfrac{1}{2}ct + k = \left(\frac{n}{\cos\theta_2} - \frac{1}{\cos\theta_1}\right)x + \frac{h}{\cos\theta_1} + k

which is a linear function of $x$ . Then, using a trigonometric identity and Snell’s law,

\cos\theta_2 = \sqrt{1 - \sin^2\theta_2} = \sqrt{1 - \frac{\sin^2\theta_1}{n^2}}.

From this the gradient $\alpha$ is found to be

\alpha = \frac{n}{\sqrt{1 - \frac{\sin^2\theta_1}{n^2}}} - \frac{1}{\cos\theta_1} = \frac{n^2}{\sqrt{n^2 - \sin^2\theta_1}} - \frac{1}{\cos\theta_1}

1.4c Use the graph to determine the refractive index $n_{\mathrm{w}}$ for water. 1.2

Knowing the gradient $\alpha$ from the graph, the index of refraction $n$ is found by solving this equation. Introducing a practical parameter,

p = \alpha + \frac{1}{\cos\theta_1}

the above equation becomes

p = \frac{n_{\mathrm{w}}^2}{\sqrt{n_{\mathrm{w}}^2 - \sin^2\theta_1}} \Leftrightarrow n_{\mathrm{w}}^4 - p^2 n_{\mathrm{w}}^2 + p^2\sin^2\theta_1 = 0

with the solution

n_{\mathrm{w}} = \sqrt{\frac{p^2 \pm \sqrt{p^4 - 4p^2\sin^2\theta_1}}{2}} = \frac{\sqrt{2}}{2}p\sqrt{1 \pm \sqrt{1 - \left(\frac{2\sin\theta_1}{p}\right)^2}}

From the graph is found $\alpha = 0.3301$ , which leads to $p = 1.388356$ and hence

n_{\mathrm{w}} = 1.34676 \approx 1.347.

All solutions with $n_{\mathrm{w}} \lt 1$ are omitted.

Another and more elegant way of finding $n_{\mathrm{w}}$ is to use Snell’s law in the equation

\alpha = \frac{n_{\mathrm{w}}}{\cos\theta_2} - \frac{1}{\cos\theta_1} = \frac{\sin\theta_1}{\sin\theta_2\cos\theta_2} - \frac{1}{\cos\theta_1} = \frac{2\sin\theta_1}{\sin 2\theta_2} - \frac{1}{\cos\theta_1}

This yields

\sin 2\theta_2 = \frac{2\sin\theta_1}{\alpha + \frac{1}{\cos\theta_1}}

From here $\theta_2$ can be calculated leading to $n_{\mathrm{w}}\ = \frac{\sin\theta_1}{\sin\theta_2}$ . This method also only uses the graph and the angle $\theta_1$ , and measurement of $\theta_2$ is not involved).

The table value for pure water at normal conditions is $n_{\mathrm{w}} = 1.331$ at the wavelength $\lambda = 635\ \mathrm{nm}$ .

The following approximations can be used: For small angles

n_{\mathrm{w}} \approx \frac{\sqrt{2}}{2}p\sqrt{1 + 1 - \frac{1}{2}\left(\frac{2\sin\theta_1}{p}\right)^2} \approx p\sqrt{1 - \left(\frac{\sin\theta_1}{p}\right)^2} \approx p\left(1 - \frac{1}{2}\left(\frac{\sin\theta_1}{p}\right)^2\right)

For very small angles, we get

n_{\mathrm{w}} \approx p \approx\ \alpha + 1

It is much simpler, but not recommendable, to do the experiment with very small $\theta_1 \approx 0$ . Reflections in the water surface will ruin the signal from the bottom.

Graph titled 'Display y as function of the water height x': Display y [mm] from 445 to 485 against water height x [mm] from 0 to 100; twelve measured points lie close to a straight fitted line y = 0,3301x + 448,45, R² = 0,9978. — Display y as function of the water height x

Оригинал в Архива: exp1.pdf · официални решения: exp1_sol.pdf