Experimental Problems in Physics

8/16/2019 Experimental Problems in Physics

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3rd

Asian Physics Olympiad

Singapore

Experimental Competition PART B

(The Experimental Competition consists of two parts, Part A and Part B. The

following experiment represents Part B only)

Date: May 10, 2002

Time Available: 2_ hours for Part B

Read the following instructions carefully

1. In this experiment, you are not expected to indicate uncertainties of your

experimental results.

2. Use only the pen provided.

3. Use only the front side of the answer sheets and graph papers.

4. In your answers please use as little text as possible; express yourself

primarily in equations, numbers and figures. If the required result is a

numerical number, underline your final result with a wavy line.5. Write on the blank sheets of paper the results of all your measurements

and whatever else you consider is required for the solution of the question

and that you wish to be marked.

6. It is absolutely essential that you enter in the boxes at the top of each

sheet of paper used, including graph paper, your Name, your Country,

your student number ( Student No.), the part number of the question

(Question Part No.), the progressive number of each sheet (including

graph sheets) ( Page No.) and the total number of sheets that you have

used and wish to be marked for each question (Total pages). If you usesome sheets of paper for notes that you do not wish to be marked, put a

large cross through the whole sheet and do not include them in your

numbering. Do not write anything on the right column which is reserved

for Examiners’ use only.

7. At the end of 2_ hours for this half of the examination, please staple your

answer and graph sheets in the order of their page numbers, before

proceeding to the other half of the examination.


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Objective

To study how the frequency of vibration of a tuning fork varies with an equal

mass clamped on each of its prongs (at a definite point near the prong tip), and

hence to determine the pair of unknown masses X similarly attached to the

prongs.

The Stroboscope

The experiment will make use of a stroboscope (strobe) which is a simple

electronic device consisting of a discharge lamp which can be made to flash for

a short duration with a high intensity at highly regular intervals. The strobe

enables the frequency of a rotating or vibrating object to be measured without

the need for any direct physical contact with the moving object.

Caution: The strobe has a finite life time, specified in maximum number of flashes obtainable. Do not leave it running idly when you are not using it.

Consider a particle rotating with uniform circular motion being illuminated by a

strobe. If the flash frequency is a multiple or sub-multiple of that of the motion,

the particle will appear stationary. It follows that the periodicity of the circular

motion of the particle can be determined by tuning the frequency of the light

flash.

Suppose the frequency of rotation of the particle is x Hz, and that of flashing is y

Hz. Then, in the time interval of 1/ y s between two successive flashes the

particle would have moved through an angle 2π x/ y.

If y/x is an irrational number so that it cannot be expressed as a ratio of two

integers, then the particle would not appear stationary but would appear to rotate

slowly in the forward or backward direction depending on whether y/x is just

slightly smaller or larger than some rational number nearby.

If y/x = q/p where p and q are integers, then the strobe would flash q times for

every p complete cycles. Furthermore, if p and q have no common factors

between them (assumed throughout this write-up), then each flash would show adifferent position of the particle. Thus the particle will exhibit q stationary

positions under the strobe flashlight.

If q becomes too large it might be difficult to count the number of stationary

positions displayed by the rotating particle.

The above theory applied to the rotating particle can be similarly applied to that

of a tuning fork vibrating in simple harmonic motion if we regard the

vibrational motion as equivalent to the motion of the projection of the rotating


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case, because the vibrating object retraces the same path in the opposite

direction every half cycle, there is a chance, though very remote, that an image

in one half of a vibration cycle coincides with that in the next half cycle. It

would result in only one image (but of double the intensity) being recorded,

instead of two. This freak coincidence should be guarded against in an

experimental observation.

Identification of Fundamental Synchronism

Fundamental synchronism is obtained when the lamp flashes once for every

cycle of rotation or vibration of the mechanism under observation, so that the

object appears to stop at one stationary position. However, it will be

appreciated that a similar and indistinguishable result will also occur when the

flash frequency is a sub-multiple (1/2, 1/3, 1/4, etc) of the object movement

frequency. Thus if the object movement frequency is totally unknown, whenadjusting for fundamental synchronism, a safe procedure is to start at a high

flash frequency, when multiple images are obtained, and then slowly reduce the

flash frequency until the first single image appears. This procedure should be

adopted in all measurements to check for fundamental synchronism.

Multiples of Fundamental Frequency

Multiples of fundamental frequency occur when the strobe is flashing at a

higher rate than the cyclical frequency of the object under observation. The

converse when the strobe flashing rate is lower than that of the moving object isreferred to as sub-multiples of fundamental frequency.

If the lamp is flashed at a frequency q times the rotational frequency of the

particle, multiple images can be seen. In such a situation, a rotating particle will

appear as several stationary images spaced equally around the circumference.

Twice this frequency, or q/p=2, will produce two such images at π radians apart,

and three times this frequency, or q/p=3, will yield three images at 2π/3 radians

spacing, etc. The particle rotational frequency is then given by the flash

frequency divided by the number of images seen. In general, if q> p>1, then thestrobe would flash q times for every p cycles of the particle motion, and so there

will still be q stationary positions.

Sub-multiples of Fundamental Frequency

Here, q/p is less than one. If the strobe frequency is exactly 1 /p times that of the

object movement, where p>1, then the object would have moved through p

cycles for every flash, and only one stationary image is seen. If p>q>1, then the


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strobe would have flashed q times for every p full cycles of the object

movement, and the number of stationary images seen would be q.

The Tuning Fork

A tuning fork is designed to vibrate at a fundamental frequency with noharmonics after it is struck. The two prongs of the fork are symmetrical in

every respect so that they move in perfect anti-phase and exert, at any instant,

equal and opposite forces on the central holder. The net force on the holder is

therefore always zero so that the holder does not vibrate, and hence holding it

firmly will not cause any undesirable damping. For the same reason the prongs

of a tuning fork cannot vibrate in like phase as this will result in a finite

oscillatory force on its holder which would cause the vibration to dampen away

very quickly.

It is possible to lower the fundamental frequency of the tuning fork by loading

an equal weight on each arm. The loading on the arms has to be symmetrical in

order to minimise damping of vibration.

For such a loaded tuning fork, the period T of vibration is given by:

T 2 = A(m + B)

where A is a constant depending on the size, shape and mechanical properties of

the tuning fork material and B a constant depending on the effective mass of each vibrating arm.

Items of Apparatus provided:

1. A stroboscope with digital readout.

2. A mini-torch light.

3. A tuning fork with a 31.6g weight loaded symmetrically on each prong

and with the centre-of-mass of the weight coinciding with the point P

marked clearly on each prong.4. Two paper clamps with two detachable levers. The levers are used only

to open the clamps, and they should be removed when you are doing

the experiment.

5. A pair of equal unknown masses X .

6. A series of the following known masses (in pairs): 5g, 10g, 15g, 20g,

25g.

7. Regular graph papers (5 sheets).


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Experimental Steps

Step 1: Fundamental synchronism and measurements of multiple frequencies

(2.7 marks)

(a) Obtain fundamental synchronism between the strobe flash and the

vibrating tuning fork loaded with the original 31.6g mass on each

prong. By dislodging the mass temporarily, check to make sure that

the mass is pre-clamped with its centre-of-mass located at the point P

(which is marked on the prong but hidden by the mass). Record its

fundamental flash frequency.

(b) Keeping the flash frequency above the fundamental frequency, try to

discover as many readings of flash frequencies as possible which

yield observable stationary images of the (31.6g-loaded) tuning fork

frequency. Identify their different q/p values.

(c) Tabulate your data (in the order of increasing q/p) as follow,

keeping q/p as a rational fraction:

Strobe Reading Number of Stationary Images q/p value

Plot a straight-line graph of all the observed strobe flash frequencies

against the corresponding multiple of the tuning fork frequency.

Identify each data point on the graph with its q/p value.


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Step 2: Measurements of sub-multiple frequencies (2.3 marks)

(a) Keeping the strobe frequency below the fundamental frequency of

the (31.6g-loaded) tuning fork, obtain readings of all observable

strobe frequencies which yield stationary images.

(b) Tabulate your readings as in question 1, but in the order of

decreasing q/p, and plot a straight-line graph of all the observed

strobe frequencies against the corresponding sub-multiple of the

(31.6g-loaded) tuning fork frequency. Identify each data point on

the graph with its q/p value.

Step 3: Determination of the pair of unknown masses X (5 marks)

(a)

Remove the 31.6g loading mass from each prong (which would

also reveal the point P marked on the prong) and obtain the

resulting vibrational frequency of the unloaded tuning fork.

(b) Next, obtain the vibrational frequencies of the tuning fork with

each prong loaded with known masses m of 5g, 10g, 15g, 20g and

25g respectively. Ensure that in each case the centre-of-mass of the

load coincides with the point P. Note that the value of m as

labelled on the mass is the total mass of both the mass itself and

that of the given paper clamp (with both its levers removed) used toclamp it.

(c) Tabulate your results using your data obtained in (b) and plot a

graph of T2 against m. Obtain the slope, and the intercept on the

m-axis.

(d) Replace the known loading masses with the unknown masses X ,

and obtain the vibrational frequency under this loading. Deduce X .

Again, note that X also includes the mass of the paper clamp (withboth its levers removed).


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IPhO2002

5

II. OPTICAL BLACK BOX

Description

In this problem, the students have to identify the unknown optical components inside thecubic box. The box is sealed and has only two narrow openings protected by red plastic

covering. The components should be identified by means of optical phenomena observed in

the experiment. Ignore the small thickness effect of the plastic covering layer.

A line going through the centers of the slits is defined as the axis of the box. Apart from the

red plastic coverings, there are three (might be identical or different) elements from the

following list:

• Mirror, either plane or spherical

• Lens, either positive or negative

• Transparent plate having parallel flat surfaces (so called plane-parallel plate)

• Prism

• Diffraction grating.

The transparent components are made of material with a refractive index of 1.47 at the

wavelength used.

Apparatus available:

• A laser pointer with a wavelength of 670 nm. CAUTION: DO NOT LOOK

DIRECTLY INTO THE LASER BEAM.

• An optical rail• A platform for the cube, movable along the optical rail

• A screen which can be attached to the end of the rail, and detached from it for other

measurements.

• A sheet of graph paper which can be pasted on the screen by cellotape.

• A vertical stand equipped with a universal clamp and a test tube with arbitrary scales,

which are also used in the Problem I.

Note that all scales marked on the graph papers and the apparatus for the experiments are

of the same scale unit, but not calibrated in millimeter.


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IPhO2002

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The Problem

Identify each of the three components and give its respective specification:

Possible type of component Specification required

mirror radius of curvature, angle between the mirror axis and

the axis of the box

lens* positive or negative, its focal le ngth, and its position inside the

box

plane-parallel plate thickness, the angle between the plate and the axis of the box

prism apex angle, the angle between one of its deflecting sides and

the axis of the box

diffraction grating* line spacing, direction of the lines, and its position inside the

box

• implies that its plane is at right angle to the axis of the box

Express your final answers for the specification parameters of each component (e.g. focal

length, radius of curvature) in terms of millimeter, micrometer or the scale of graph paper .

You don’t have to determine the accuracy of the results.


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Count ryCou ntr y Student No.Student No. Experiment No.Experiment No. Page No.Page No. Total PagesTotal Pages

ANSWER FORM

1. Write down the types of the optical components inside the box :

no.1. ………………………………………… [0.5 pts ]

no.2.. ………………………………………… [0.5 pts ]

no.3. ………………………………………… [0.5 pts ]

2. The cross section of the box is given in the figure below. Add a sketch in the figure

to show how the three components are positioned inside the box. In your sketch,

denote each component with its code number in answer 1 .

[0.5 pts for each correct position]

axis of the box

direction of the slit direction of the slit


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IPhO2002

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3. Add detailed information with additional sketches regarding arrangement of the optical

components in answer 2, such as the angle, the distance of the component from the slit, and

the orientation or direction of the components. [1.0 pts ]


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4. Summarize the observed data [0.5 pts ], determine the specification of the optical

component no.1 by deriving the appropriate formula with the help of drawing [1.0 pts ],

calculate the specifications in question and enter your answer in the box below [0.5 pts ].

Name of component no.1 Specification


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Count ryCou ntr y Student No.Student No. Experiment NoExperiment No.. Page No.Page No. Total PagesTotal Pages






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4th Asian Physics Olympiad Experimental Competition 25 April 2003

Experimental Competition

II. Cylindrical Bore

Background

There are many techniques to study the object with a bore inside. Mechanical

oscillation method is one of the non-destructive techniques. In this problem, you aregiven a brass cube of uniform density with cylindrical bore inside. You are required to

perform non-destructive mechanical measurements and use these data to plot the

appropriate graph to find the ratio of the radius of the bore to the side of the cube.

The cube of sides a has a cylindrical bore of radius b along the axis of symmetry as shown in Fig. 2.1. This bore is covered by very thin discs of the samematerial. A, B and C represent small holes at the corners of the cube. These holes can beused for suspending the cube in two configurations. Fig. 2.2(a) shows the suspension

using B and C; the other suspension is by using A and B as shown in Fig 2.2(b).

a

C•

•

•

B

A

2b

Fig. 2.1 Geometry of cube with cylindrical bore


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Students may use the following in their derivation of necessary formulae:

For a solid cube of side a

2

6

1 Ma I = about both axes

c.m. = centre of mass

For a solid cylinder of radius b length a

2

2

1mb I Y =

22

4

1

12

1mbma I X +=

Y

X•

•

c.m.•

Fig. 2.2 Two configurations of cube’s suspension

2.2(a)

C•

•

•

B

A

I 1

2.2(b)

I 2

B

A

C

•

•

•

g

Top of stand

X

Y


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Materials and apparatus

1. brass cube

2. stop watch

3. stand

4.

thread5. ruler/ centimeter stick

6. linear graph papers

Experiment

a) Choose only one of the two bifilar suspensions as shown in Fig. 2.2, and derive the

expressions for the moment of inertia and the period of oscillation about the vertical

axis through the centre of mass in terms of abd ,,, and g where is the length of

each thread and d is the separation between threads. (2 points)

b) Perform necessary non-destructive mechanical measurements and use these data to

plot an appropriate graph and then find the value ofa

b. (8 points)

The value of g for Bangkok = 9.78 m/s2.

---------------------------------


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Experimental Competition - Problem No. 2

Black box

APPARATUS AND MATERIALS

1. A double beam oscilloscope.

2. A function generator capable to generate sine, triangle and square waves over the

0.02 Hz to 2 MHz range.3. A "Black box" with two groups of connectors: the ABCD group and A'B'C'D' group.

Besides, there are also two connectors for the standard resistor Rn = 5 kΩ, which is

isolated from the two groups.4. Conductors of negligible resistance.

5. Graph paper.

Warning: You are not allowed to open the black box.

EXPERIMENT

In the black box, there are two groups of passiveelements (that are elements of the types: resistor R,

capacitor C or inductor (induction coil) L). The first

group consists of three elements1 Z , ,

connected in a star circuit as shown in Figure 1. The

elements are led out to the connectors A, B, C and

D, with A - the common connector of the ABCD

group. The second group consists of three elements

1 Z ' , 2 Z ' , 3 Z ' connected in the same manner to

connectors A', B', C' and D', with A'- the common

connector of the A'B'C'D' group (see Figure 2). Figure 1

D

A

Z2

Z3

C

Z1

B

2 Z 3 Z

1. By using the oscilloscope and the function

generator, determine the type and the parameter (that

is resistance of R, capacity of C , inductivity of L) of

each of the elements 1 Z , 2 Z , 3 Z and 1 Z ' , 2 Z ' , 3 Z ' .

[5.0 pts]

Figure 2

D’

A’

Z’2

Z’3

C’

Z’1

B’

2. Connect five points B, C, B', C' and D' together.

We obtain a new black box with terminals DD’A’

(called DD’A’).

a. Draw the electric circuit of this black box. b. Apply a sine wave from the generator to

connectors D and A'.

Plot a graph of the ratio of the voltage amplitudes D'A

DA

'

'

U K

U = and the phase shift ϕ

between these voltages as functions of the frequency f of the signal.

c. The graphs possess a particular point at a certain frequency f 0. Determine the value

of the frequency f 0, the ratioD'A

DA

U K

U =

'

'

and the phase shift ϕ at this frequency.

d. Derive the relation between f 0 and the parameters of the elements in the black boxand calculate the values of f 0.. [5.0 pts]

1


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Before attempting to assemble

your apparatus, read the problem text completely!

Please read this first:

1. The time available for the Experimental problem 1 is 2 hours and 45 minutes;

and that for the Experimental problem 2 is 2 hours and 15 minutes.

2. Use only the pen and equipments provided.

3.

Use only the one side of the provided sheets of paper.

4. In addition to blank sheets where you may write freely, there is a set of Answer

sheets where you must summarize the results you have obtained. Numerical

results must be written with as many digits as appropriate; do not forget the units.

5. Please write on the “blank” sheets the results of all your measurements and

whatever else you deem important for the solution of the problem that you wish to

be evaluated during the marking process. However, you should use mainly

equations, numbers, symbols, graphs, figures, and as little text as possible.

6. It is absolutely imperative that you write on top of each sheet: your student code

as shown on your identification tag, and additionally on the “blank” sheets: your

student code, the progressive number of each sheet (Page n. from 1 to N) and the

total number (N) of “blank” sheets that you use and wish to be evaluated (Page

total).

7. The student should start with a new page for each section. It is also useful to write

the number of the section you are answering at the beginning of each such section.

If you use some sheets for notes that you do not wish to be evaluated by the

marking team, just put a large cross through the whole sheet and do not number it.

8. When you have finished, turn in all sheets in proper order (answer sheet first, then

used sheets in order, the unused sheets and problem text at the bottom) and put

them all inside the envelope provided; then leave everything on your desk. You

are not allowed to take anything out of the room.

9.

are not allowed to take anything out of the room.


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Note: The AC voltmeter black port is connected with two spare terminals


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Experimental problem 2

Measurement of liquid electric conductivity 10 points

1. Experimental instructions

In the apparatus of present experiment to measure the conductivity of liquid (i.e.,

water with salt), the sensor deals with ac signal without any contact potential involved

to interfere with the desired experimental results. Meanwhile, since the sensor

(detective windingdoes not directly touch the liquid to be measured, no chemical

reaction would happen during the experiments to damage any part of the apparatus.

Therefore it can be used repeatedly for a long time.

As shown in Fig. 1, the sensor designed for measuring the conductivity of liquid

consists of two circular loops with the same radius, made of soft-iron-based alloy.

Each loop is wound with winding. The numbers of circles of the two windings are

equal to each other. The two alloy loops are aligned along the same axis and

connected closely as one airproof hollow cylinder, as shown in Fig. 2.

Fig. 1 Fig. 2

The sensor is immersed in the liquid to be measured. Winding 11’ is connected to

sine signal generator of frequency about 2.5kHz. The amplitude of its output signal

might drift somewhat. If the drift exceeds certain value, it should be adjusted in time

to keep the output amplitude remain at certain value. Winding 22’ is connected to ac

voltmeter used to measure the induced signal voltage. With the measured magnitude

of the signal voltage, the conductivity of the liquid can be calculated.

2. Experimental principles

The operation principle of the present experimental apparatus can be simply

explained as follows. The ac sine current from the signal generator induces an ac

magnetic field in loop 11’. In turn the magnetic field induces an ac current in the


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conducting liquid. Such induced current induces back a time-varying magnetic field in

loop 22’, which induces an electromotive force in the same loop 22’, being the output

signal of the sensor.

Neglecting the magnetic hysteresis effect, output voltage is a monotonical

function of input voltage . When input voltage and the conductivity of the

liquid are respectively within certain range, a proportional relation holds between

and the ratio of :

(1)

where is the proportionality constant.

In the present apparatus, the liquid container can contain so much liquid to be

measured that the resistance of the liquid outside the cylinder-shaped sensor is

negligible. Therefore the output voltage of the sensor depends mainly on the

‘liquid within the hollow cylinder (referred as “liquid cylinder” hereafter). Thus, it

is possible to use the liquid cylinder to calculate the liquid conductivity. Resistance of

the liquid cylinder is

(2)

where L is the length of the liquid cylinder along its axis, and S is the area of its cross

section. Combination of (1) and (2) leads to

(3)

where or alternatively .

With Eq.(2) and (3) we obtain

(4)

Equation (4) shows that, when using the present sensor to measure the liquid

conductivity, is related to L (length of the hollow cylinder), S (area of its cross section),

, and B as well.

Remark Essentially in the present experiment, in order to obtain the

proportionality constant K and then B accurately, various kinds of liquid with known


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should be required and prepared. Obviously this is not an easy task. Therefore, for

the sake of both convenience and correctness, instead of the various liquids of known

we use externally connected standard resistors. The two ends of the standard

resistor are connected to the two ends of a conducting thread passing through thehollow cylinder of the sensor to form a resistor circuit, as shown in Fig. 3.

Fig. 3

3. Experimental content

1. Draw the experimental circuit diagram for scaling the sensor of the liquid

conductivity (1.0 points) , and complete the connection of the circuit in order to

measure both the input voltage (


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3-2 It can be seen that at some region of less induced current the curve is linear.

Graph this linear part and use the graphical method to obtain the slope B of the

straight part of the curve and its relative uncertainty u(B) or u( B) /B. (1.5 points)

4. With the given axis length of the sensor L=(30.500±0.025) mm and diameter of the

liquid cylinder d =(13.900±0.025)mm, calculate the value of and u(K) or

u(K)/K . (1.0 points)

5Work out the conductivity of the liquid in the container and write the result.

According to the uncertainties of L, d , and B, estimate the uncertainty of the

conductivity. The measurement of the conductivity should be done for six times,

during which the liquid should be stirred for each time.

(1.5 points)

4. Instruments and materials

1. Sensor of liquid conductivity

The sensor has four ports of connection terminals: two terminal ports are

connected to winding 11’ and two terminal ports are connected to the other

winding 22’.

2.

Container filled with the yet-to-be-determined liquid and stirring rod.

3.

The instrument for measuring the liquid conductivity.

On the instrument panel there are:

• Signal generator:

Two ports of connection terminals connected to the signal generator, the

red one for signal output, and the black one for grounding. The amplitude

of output signal can be adjusted by turning the knob.

• ac digital voltmeter.

•

Inserting-type resistor box:

On the panel, there are many ports of connection terminals, between every

two adjacent ports, there is a resistor with relative resistance error of 0.001.

Resistance of these resistors is 0.1,0.2, 0.5,1,2, and 5! respectively.

• Switch 1 2single-pole double throw.

4 Some leads

5. Two pieces of graph paper 20cm"25cmcalculator, recording paper, ruler, and

pen.


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EXPERIMENTAL PROBLEM 2

BIREFRINGENCE OF MICA

In this experiment you will measure the birefringence of mica (a crystal widely used in polarizing optical components).

MATERIAL

In addition to items 1), 2) and 3), you should use,

14) Two polarizing films mounted in slide holders, each with an additional acrylic

support (LABEL J). See photograph for mounting instructions.

15) A thin mica plate mounted in a plastic cylinder with a scale with no numbers;

acrylic support for the cylinder (LABEL K). See photograph for mountinginstructions.

16) Photodetector equipment. A photodetector in a plastic box, connectors and foamsupport. A multimeter to measure the voltage of the photodetector (LABEL L).

See photograph for mounting and connecting instructions.

17) Calculator.

18) White index cards, masking tape, stickers, scissors, triangle squares set.19) Pencils, paper, graph paper.

Polarizer mounted in slide holder with

acrylic support (LABEL J).

Thin mica plate mounted in cylinder with

a scale with no numbers, and acrylicsupport (LABEL K).


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A photodetector in a plastic box, connectors and foam support. A multimeter to measure the

voltage of the photodetector (LABEL L). Set the connections as indicated.

DESCRIPTION OF THE PHENOMENON

Light is a transverse electromagnetic wave, with its electric field lying on a plane

perpendicular to the propagation direction and oscillating in time as the light wave travels.

If the direction of the electric field remains in time oscillating along a single line, the wave

is said to be linearly polarized, or simply, polarized. See Figure 2.1.

Figure 2.1 A wave travelling in the y-direction and polarized in the z-direction.

y


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A polarizing film (or simply, a polarizer) is a material with a privileged axis parallel to its

surface, such that, transmitted light emerges polarized along the axis of the polarizer. Call

(+) the privileged axis and (-) the perpendicular one.

Figure 2.2 Unpolarized light normally incident on a polarizer. Transmitted light is polarizedin the (+) direction of the polarizer.

Common transparent materials (such as window glass), transmit light with the same

polarization as the incident one, because its index of refraction does not depend on the

direction and/or polarization of the incident wave. Many crystals, including mica, however,are sensitive to the direction of the electric field of the wave. For propagation perpendicular

to its surface, the mica sheet has two characteristic orthogonal axes, which we will call Axis

1 and Axis 2. This leads to the phenomenon called birefringence.

Figure 2.3 Thin slab of mica with its two axes, Axis 1 (red) and Axis 2 (green).

Let us analyze two simple cases to exemplify the birefringence. Assume that a wave

polarized in the vertical direction is normally incident on one of the surfaces of the thin

slab of mica.


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Case 1) Axis 1 or Axis 2 is parallel to the polarization of the incident wave. The trasmitted

wave passes without changing its polarization state, but the propagation is characterized as if

the material had either an index of refraction n1 or n2 . See Figs. 2.4 and 2.5.

Figure 2.4 Axis 1 is parallel to polarization of incident wave. Index of refraction is n1.

Figure 2.5 Axis 2 is parallel to polarization of incident wave. Index of refraction is n2.

Case 2) Axis 1 makes an angle θ with the direction of polarization of the incident wave.The transmitted light has a more complicated polarization state. This wave, however, can be

seen as the superposition of two waves with different phases, one that has polarization

parallel to the polarization of the incident wave (i.e. "vertical") and another that has

polarization perpendicular to the polarization of the incident wave (i.e. "horizontal").


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Figure 2.6 Axis 1 makes and angle θ with polarization of incident waveCall I

P the intensity of the wave transmitted parallel to the polarization of the incident

wave, and I O the intensity of the wave transmitted perpendicular to polarization of theincident wave. These intensities depend on the angle θ , on the wavelength λ of the lightsource, on the thickness L of the thin plate, and on the absolute value of the difference of

the refractive indices, n1 − n2 . This last quantity is called the birefringence of the material.

The measurement of this quantity is the goal of this problem. Together with polarizers,

birefringent materials are useful for the control of light polarization states.

We point out here that the photodetector measures the intensity of the light incident on it,

independent of its polarization.

The dependence of I P (θ ) and

I O (θ ) on the angle θ is complicated due to other effects notconsidered, such as the absorption of the incident radiation by the mica. One can obtain,

however, approximated but very simple expressions for the normalized intensities IP(θ ) and

IO(θ ) , defined as,

IP(θ ) =

I P(θ )

I P(θ ) + I

O(θ )

(2.1)

and

IO (θ ) =

I O(θ )

I P(θ ) + I O (θ ) (2.2)

It can be shown that the normalized intensities are (approximately) given by,

IP(θ ) =1−

1

21− cos∆φ ( )sin2(2θ ) (2.3)

and


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IO(θ ) =

1

21− cos∆φ ( )sin2(2θ ) (2.4)

where ∆φ is the difference of phases of the parallel and perpendicular transmitted waves.This quantity is given by,

∆φ =2π L

λ n1 − n2 (2.5)

where L is the thickness of the thin plate of mica, λ the wavelength of the incidentradiation and n1 − n2 the birefringence.

EXPERIMENTAL SETUP

Task 2.1 Experimental setup for measuring intensities. Design an experimental setup for

measuring the intensities I P and I

O of the transmitted wave, as a function of the angle θ of

any of the optical axes, as shown in Fig. 2.6. Do this by writing the LABELS of the different

devices on the drawing of the optical table. Use the convention (+) and (-) for the direction

of the polarizers. You can make additional simple drawings to help clarify your design.

Task 2.1 a) Setup for I P (0.5 points).

Task 2.1 b) Setup for I O

(0.5 points).

Laser beam alignment. Align the laser beam in such a way that it is parallel to the table andis incident on the center of the cylinder holding the mica. You may align by using one the

white index cards to follow the path. Small adjustments can be made with the movable

mirror.

Photodetector and the multimeter. The photodetector produces a voltage as light impinges

on it. Measure this voltage with the multimeter provided. The voltage produced is linearly proportional to the intensity of the light. Thus, report the intensities as the voltage produced

by the photodetector. Without any laser beam incident on the photodetector, you can

measure the background light intensity of the detector. This should be less than 1 mV. Do

not correct for this background when you perform the intensity measurements.

WARNING: The laser beam is partially polarized but it is not known in which direction.

Thus, to obtain polarized light with good intensity readings, place a polarizer with either its(+) or (–) axes vertically in such a way that you obtain the maximum transmitted intensity in

the absence of any other optical device.


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MEASURING INTENSITIES

Task 2.2 The scale for angle settings. The cylinder holding the mica has a regular

graduation for settings of the angles. Write down the value in degrees of the smallest interval(i.e. between two black consecutive lines). (0.25 points).

Finding (approximately) the zero of θ and/or the location of the mica axes. To facilitatethe analysis, it is very important that you find the appropriate zero of the angles. We suggest

that, first, you identify the location of one of the mica axes, and call it Axis 1. It is almostsure that this position will not coincide with a graduation line on the cylinder. Thus,

consider the nearest graduation line in the mica cylinder as the provisional origin for the

angles. Call θ the angles measured from such an origin. Below you will be asked to providea more accurate location of the zero of θ .

Task 2.3 Measuring I P and I

O. Measure the intensities I

P and I

O for as many angles θ as

you consider necessary. Report your measurements in Table I. Try to make the

measurements for I P and I

O for the same setting of the cylinder with the mica, that is, for a

fixed angle θ . (3.0 points).

Task 2.4 Finding an appropriate zero for θ . The location of Axis 1 defines the zero of theangle θ . As mentioned above, it is mostly sure that the location of Axis 1 does not coincidewith a graduation line on the mica cylinder. To find the zero of the angles, you may proceed

either graphically or numerically. Recognize that the relationship near a maximum or a

minimum may be approximated by a parabola where:

I (θ ) ≈ aθ 2 + bθ + c

and the minimum or maximum of the parabola is given by,

θm

= −b

2a.

Either of the above choices gives rise to a shift θ of all your values of θ given inTable I of Task 2.3, such that they can now be written as angles θ from the appropriate zero,θ = θ + δθ . Write down the value of the shift δθ in degrees. (1.0 points).

DATA ANALYSIS.

Task 2.5 Choosing the appropriate variables. Choose IP(θ ) or I

O(θ ) to make an analysis

to find the difference of phases ∆φ . Identify the variables that you will use. (0.5 point).


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Task 2.6 Data analysis and the phase difference.

• Use Table II to write down the values of the variables needed for their analysis.

Make sure that you use the corrected values for the angles θ . Include uncertainties.Use graph paper to plot your variables. (1.0 points).

• Perform an analysis of the data needed to obtain the phase difference ∆φ . Report

your results including uncertainties. Write down any equations or formulas used in

the analysis. Plot your results. (1.75 points).

• Calculate the value of the phase difference ∆φ in radians, including its uncertainty.

Find the value of the phase difference in the interval 0,π [ ]. (0.5 points).

Task 2.7 Calculating the birefringencen1 −

n2 . You may note that if you add 2

N π to the

phase difference ∆φ , with N any integer, or if you change the sign of the phase, the values

of the intensities are unchanged. However, the value of the birefringence n1 − n2 would

change. Thus, to use the value ∆φ found in Task 2.6 to correctly calculate the birefringence,

you must consider the following:

∆φ =2π L

λ n1 − n2 if L < 82 ×10

−6 m

or

2π − ∆φ =2π L

λ n1 − n2 if L > 82 ×10

−6 m

where the value L of the thickness of the slab of mica you used is written on the cylinder

holding it. This number is given in micrometers (1 micrometer = 10-6

m). Assign 1×10−6mas the uncertainty for L . For the laser wavelength, you may use the value you found in

Problem 1 or the average value between 620 ×10−9 m and 750 ×10−9 m, the reported range

for red in the visible spectrum. Write down the values of L and λ as well as the birefringence n1 − n2 with its uncertainty. Include the formulas that you used to calculate

the uncertainties. (1.0 points).


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27 April 2010 Page 1 of 8

__________________________________________________________________________________________

Question Number 2

Experiment II Understanding Semiconductor Lasers

The purpose of this experiment is to explore the basic characteristics of semiconductor

lasers. We will measure and calculate the fraction of the linear polarization of the

collimated laser beam by using a pair of polarizers and a photoconductor. Finally, we will

determine the maximum value of the power increase per current increment of the

collimated laser.

Safety Caution: Do not look directly into the laser beam, which can damage your eyes!!

◎ Background Description

The photoconductor, the light-sensing device in this experiment, is made ofsemiconductor, which has a band gap of EG = (EC – EV) (see Fig. II-1). When the energy of

the incident photons is larger than that of the band gap, the photons can be absorbed by the

semiconductor to create free electrons and holes. The density of charge carriers, including

electrons and holes, is then increased, and so is the conductivity of the material. In this

experiment, the resistance (the inverse of conductance) is measured by using a multimeter.

EV (Valence Band Edge)

Hole

EG (Energy Gap)

EC (Conduction Band Edge)

Electron

Photon

Fig. II-1 Schematic diagram of an electron-hole pair generated by a

single photon in a semiconductor.


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__________________________________________________________________________________________

Question Number 2

In the semiconductor laser, the light emitting device in this experiment, as the external

source injects electrons and holes into the device, they can combine to emit photons as

shown schematically in the Fig. II-2. Ideally, the combination of one pair of electron and

hole can generate one photon. Realistically, there are also nonradiative processes through

which an electron-hole pair recombines without generating a photon. Thus the number of

photons generated is not equal to the number of electron-hole pairs recombined. The

average fractional number of photons generated by an electron-hole pair is called the

quantum efficiency.

The semiconductor laser can emit a monochromatic, partially polarized and coherent

light beam. The partially polarized light is composed of two parts – linearly polarized and

unpolarized. The light intensity due to the former is denoted by J p and the other by Ju.

When the partially polarized light is incident upon a polarizer, the transmittance of the

linearly polarized part depends on the angle between its polarized direction and the

direction of the polarizer. But for the unpolarized part, a constant portion is allowed to pass

through the polarizer and is independent of the angle.

Hole

Fig. II-2 Schematic diagram of a single photon generated by

an electron-hole air combined in a semiconductor.

EC (Conduction Band Edge)

EV (Valence Band Edge)

Photon

Electron

EG (Energy Gap)


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__________________________________________________________________________________________

Question Number 2

◎ Experiments and Procedures

Exp. II-A Light Response of the Photoconductor

The light source used in this part should be the collimated laser diode (CLD). Use the

circuit diagram shown in Fig. IIA-1 to provide current for the CLD. The symbols used in Fig.

IIA-1 are listed in Table IIA-1.

Fig. IIA-1 Circuit diagram used for the CLD.

Table IIA-1 Symbols used in Fig. IIA-1

Devices Collimated Laser

Diode

5V

dc-power

Variable

Resistors

Ammeter

(Multimeter)

Symbols

Label II-L-# C-D-# C-E-# II-X, II-Y

Operate the CLD with the maximum current. The laser intensity is detected by a

photoconductor (PC). When you shine light on a PC, the conductance increases with the light

intensity. You should minimize the ambient light effect. In this experiment, we actually

measure the resistance, which is the inverse of conductance. The intensity of the laser light

reaching the PC may be varied by using the supplied polarizers or filters. The symbols of

other optical components are given in the Table IIA-2. The partial polarization of the laser

light may be observed by using the experimental setup in the Fig. IIA-2.

A

A


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__________________________________________________________________________________________

Question Number 2

Table IIA-2 Symbols of optical components.

Devices Collimated Laser

Diode

Polarizers Photoconductor Light filter

Symbols

Optional

Label C-C-# II -P -#

II-Q-# II-W-#II-U

Rotating P1, one should observe that the PC resistance varies. Adjust P1 so that the PC

resistance reaches a minimum. If the observed minimum resistance happens in a range, say

°10 or larger of the rotation angle of P1, then the PC is saturated. In this case light filter(s)

should be used to avoid PC saturation near the maximum light intensity.

Fix the P1 position according to the description in previous paragraph. Characterize the

conductance of the PC versus the relative light intensity following the experimental setup

shown in the Fig. IIA-3.

Fig.IIA-2 Experimental setup for the preparation

PCP1FL

P1 P2 PC FL


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__________________________________________________________________________________________

Question Number 2

Exp. II-B The Fraction of Linearly Polarized Laser Light

The light source used in this part should be the collimated laser diode (CLD) with a 15

mA current from the dc power supply. The task in this part is to determine the fraction β of the

laser light that is linearly polarized by using the setup in Fig. IIA-2, which is the same as the

previous section. No error analysis is required in this part.

minmax

minmax)()(

)(

J J

J J

Polarized Linearly J d Unpolarize J

Polarized Linearly J

+

−

=+

= β

J max and J min are the maximum and minimum light intensity detected by PC while rotating P1.

(1) Find the maximum and minimum values of PC resistance ( Rmax and Rmin) by rotating P1

360 °. Transform Rmax and Rmin into the minimum and maximum values of PC conductance

C min and C max. Record the data in the data table. (0.8 points)

(2) Utilizing the conductance versusP

θ graph in Exp. II-A-(2) to determine the relative

intensities J max and J min corresponding to C max and C min. Write down the result on the

answer sheet. (1.6 points)

(3) Calculate β and write down the result on the answer sheet. (0.2 points)

Fig.IIA-2 Experimental setup for the preparation

PCP1FL


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__________________________________________________________________________________________

Question Number 2

Exp. II-C The Differential Quantum Efficiency of the

Collimated Laser Diode

The task of this part is to characterize the relative light intensity versus the current through

the collimated laser diode (CLD) and determine the Differential Quantum Efficiency η,

which will be defined below. Control the current of CLD in the range between 5 mA and 20

mA. Make sure that the PC is not saturated when the current is close to 20 mA. Filters or

polarizers can be used to avoid saturation.

(1) Control the CLD current and measure the corresponding PC resistance values. Record the

data in the data table. Transform your data and plot the PC conductance versus CLD

current on a graph paper. No error analysis is required. (1.3 points)

(2) Based on the graph of step (1), choose a region (∆ I ~ 3 mA) centered around the

maximum slope. By using the conductance versusP

θ graph in Part II-A-(2), transform

and record the data of this region in the table of step (1) into the relative light intensity ( J ).

Plot the relative light intensity ( J ) versus CLD current ( I ) on a graph paper. No erroranalysis is required. (0.8 points)

(3) The maximum radiating power of the CLD is assumed to be exactly 3max =P .0 mW.

Extract the maximum slope from the graph in step (2) and transfer it to the value of

max

PG

I

∆≡

∆, which is the maximum ratio of the increased amount of radiating power and

the increase amount of input current. Write down your analysis and the calculated valueG

on the answer sheet. Estimate the error of G. Do not include the error of the Pmax. Write

down your analysis and the calculated value ∆G on the answer sheet. (2.0 points)

(4) The Quantum Efficiency equals the probability of one photon being generated per

electron injected. From a particular bias current of the laser, a small increment of electrons

injected would cause a corresponding increment of photons. The Differential Quantum

Efficiency η is defined as the ratio of the increased number of photons and the increased

number of injected electrons. Determine η of your CLD by using the value of G obtained

in step (3). Write down your analysis and the calculated value η

on the answer sheet.


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__________________________________________________________________________________________

Question Number 2

Estimate the error of η. Write down your analysis and the calculated value ∆η on the

answer sheet. (Laser wavelength = 650 nm. Planck’s constant = 346.63 10 J s−× ⋅ . Light

speed = 8

3.0 10 m s× ) (0.9 points)


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41st International Physics Olympiad – Experimental Competition, July 21st, 2010 1/4

Experimental problem 2

There are two experimental problems. The setup on your table is used for both problems. You have 5hours to complete the entire task (1&2).

Experimental problem 2: Forces between magnets, concepts of

stability and symmetry

Introduction

Electric current I circulating in a loop of area S creates a magnetic moment of magnitude IS m [see Fig. 1(a)]. A permanent magnet can be thought of as a collection of small magnetic moments of

iron (Fe), each of which is analogous to the magnetic moment of a current loop. This (Ampère’s)model of a magnet is illustrated in Fig. 1(b). The total magnetic moment is a sum of all small magnetic

moments, and it points from the south to the northern pole.

(a) (b)

Figure 1. (a) Illustration of a current loop and the produced magnetic field. (b) Ampère’s model of a

magnet as a collection of small current loops.

Forces between magnets

To calculate the force between two magnets is a nontrivial theoretical task. It is known that like poles

of two magnets repel, and unlike poles attract. The force between two current loops depends on the

strengths of the currents in them, their shape, and their mutual distance. If we reverse the current in

one of the loops, the force between them will be of the same magnitude, but of the opposite

direction.

In this problem you experimentally investigate the forces between two magnets, the ring-magnet

and the rod-magnet. We are interested in the geometry where the axes of symmetry of the two

magnets coincide (see Fig. 2). The rod-magnet can move along the z - axis from the left, through the

ring-magnet, and then towards the right (see Fig. 2). Among other tasks, you will be asked to

measure the force between the magnets as a function of z . The origin corresponds to the

case when the centers of the magnets coincide.


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Figure 4. Photograph of the setup, and the way it should be used for measuring the force between

the magnets.

Tasks

1. Determine qualitatively all equilibrium positions between the two magnets, assuming that

the z - axis is positioned horizontally as in Fig. 2 , and draw them in the answer sheet. Label

the equilibrium positions as stable (S)/unstable (U), and denote the like poles by shading, as

indicated for one stable position in the answer sheet. You can do this Task by using your

hands and a wooden stick. (2.5 points)

2.

By using the experimental setup measure the force between the two magnets as a function

of the z - coordinate. Let the positive direction of the z - axis point into the transparent

cylinder (the force is positive if it points in the positive direction). For the configuration when

the magnetic moments are parallel, denote the magnetic force by )( z F

, and when they

are anti-parallel, denote the magnetic force by )( z F

. Important: Neglect the mass of the

rod-magnet (i.e., neglect gravity), and utilize the symmetries of the forces between

magnets to measure different parts of the curves. If you find any symmetry in the forces,

write them in the answer sheets. Write the measurements on the answer sheets; beside

every table schematically draw the configuration of magnets corresponding to each table (an

example is given). (3.0 points)

3.

By using the measurements from Task 2, use the millimeter paper to plot in detail the

functional dependence )( z F

for0

z . Plot schematically the shapes of the curves

)( z F

and )( z F

(along the positive and the negative z - axis). On each schematic graph


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denote the positions of the stable equilibrium points, and sketch the corresponding

configuration of magnets (as in Task 1). (4.0 points)

4.

If we do not neglect the mass of the rod magnet, are there any qualitatively new stable

equilibrium positions created when the z - axis is positioned vertically? If so, plot them on

the answer sheet as in Task 1. (0.5 points)


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Page 1 of 6

Experimental Question 2: An Optical “Black Box”

TV and computer screens have advanced significantly in recent years. Today, most displays consist of a color LCD

filter matrix and a uniform white backlight source. In this experiment, we will study a sample of plastic material which

was considered for use as an ingredient in the backlight illumination of LCD screens.

Equipment

On your desk, you have the following items (see Figure 1):

(1) The sample - a piece of plastic material fixed in a slide frame. The sample is sensitive - do not touch it. To

adjust the sample’s position, use its holder and stand.

(2)

A holder and stand for the slide frame. The stand includes a handle which can be used for fine rotations of the

sample. Do not remove the slide frame from its holder.

(3) A white LED flashlight. The flashlight can be turned on and off using a button at its rear end. Do not confuse

it with the laser (see Figure 2).

(4) A red laser pointer. The laser is marked with a warning label. Do not confuse it with the white flashlight

(see Figure 2). The laser may be turned on and off by moving its black cap back and forth. Don’t try to

remove the cap – it may be dangerous, and you may break the laser. The laser’s battery will weaken after

about an hour - do not keep it turned on longer than necessary. The laser’s wavelength is .

(5) A single stand to be used for the two light sources. At the start of the experiment, the flashlight is fixed to

the stand, while the laser lies on the desk.

(6)

A white screen on wooden legs, covered with millimeter graph paper. There is a hole near the middle of thescreen. You are allowed to make markings on the screen.

(7) A wooden stake that can be moved back and forth on a wooden bench. You are allowed to make markings on

the bench.

(8) A tape measure.

(9) A ruler.

(10) Millimeter graph paper.

(11) A desktop lamp which can be turned on or off for your convenience.

LASER SAFETY:

1. Do not stare into the laser beam!

2. Beware of reflections from metallic surfaces.

3. Do not point the laser at others.

4. Do not try to repair or disassemble the laser.

Call a supervisor if you require assistance.


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Page 2 of 6

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Figure 1: Summary of the equipment.

(1) The plastic sample.

(2) Sample holder and stand.

(3) White LED flashlight.

(4) Red laser.

(5) Stand for light source.

(6) Screen covered with millimeter graph paper.

(7) Wooden stake on bench.

(8) Tape measure.

Figure 2: Close-up of the two light sources.

(3) White

flashlight(4) Red

laser


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Page 3 of 6

Part I – Theory (0.4 points)

a. (0.4 pts.) A light ray is reflected from two mirrors which meet at an angle (Figure 3). Find the angle

between the incoming and outgoing rays. Assume that all light rays lie in the plane perpendicular to the

mirrors’ intersection line.

Part II – Measurements with white light (6.1 points)

Using the white flashlight as your light source, you may observe both the transmission and the reflection properties of

the sample. Figure 4 illustrates the suggested setups for both types of observation. Note: you may observe different

results when illuminating the two sides of the sample.

CAUTION: For viewing transmitted light, you will have to look directly into the flashlight beam through the sample.

Don’t do this with the laser! Also, avoid looking directly into the flashlight itself for long periods of time.

Figure 4: Suggested

observation setups

for white light.

Sample

Reflection:

Screen

White

flashlight

Transmission:

Eye

Sample

White

flashlight

Figure 3: A light ray

reflected from two mirrors.


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Page 5 of 6

Choose the correct option:

A. All the colored patterns result from interference.

B. All the colored patterns result from the dependence of on the wavelength.

C. The patterns depicted in Figure 6 result from interference, while the pattern depicted in Figure 7 results

from dependence of on the wavelength.

D. The patterns depicted in Figure 6 result from the dependence of on the wavelength, while the pattern

depicted in Figure 7 results from interference.

e. (1.4 pts.) With the white light set up as in part (d), measure the deflection angle of violet light (at the far

blue end of the spectrum) for the dominant peak depicted in Figure 7. The deflection angle is defined in Figure

8. Record all intermediate measurements. Provide error estimates.

f. (1.4 pts.) Illumination of the sample at different angles of incidence results in different deflection angles for the

dominant transmitted peaks. Measure the minimal deflection angle of the dominant peak for transmitted

violet light (there is only one such minimal angle). Record all intermediate measurements. Provide error

estimates.

g. (0.8 pts.) Using the angle from part (c), express the refraction index of the sample in terms of either or

. You may use the reversibility of light propagation and the fact that there is only one minimal angle

.

h. (0.7 pts.) Find the refraction index of the sample for violet light and its error estimate.

Part III - Laser measurements (3.5 points)

Remove the flashlight from the light-source stand, and replace it with the laser. You can use the white screen to view

both transmission and reflection patterns, as illustrated in Figure 9. The laser has a limited battery life - do not keep it

turned on longer than necessary. When aligning the components, it may help to rotate the laser around its axis.

WARNING: Do not look directly into the laser beam or its reflections! Do not look at the laser light throughthe sample – use the provided screen.

Figure 7: Bright patternfarther to the right from

the light source.

R G B

Sample Figure 8: The

deflection angle .


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Page 6 of 6

ReflectionTransmission

Screen

Sample

LaserScreen

Sample

Laser

Observe the alternating pattern of bright and dim fringes on the screen as you slightly rotate the sample. The dimming

of some of the fringes is due to destructive interference between different regions of each “tooth” on the sample.

i. (1 pt.) Use one of the setups in Figure 9, with the sample illuminated perpendicularly by the laser beam.

Record the deflection angles of the observed fringes as a function of the fringe number. Define the center

of the pattern as . Use the provided table on the answer form. Record all intermediate measurements.

Provide error estimations.

j. (1.5 pts.) Using a linear graph, find the spacing between two adjacent “teeth” of the sample. Error bars on the

graph are not required. Provide error estimation for .

k. (1 pt.) Using the formula you derived in part (g), find the refraction index of the sample for the laser’s red

wavelength. Record any additional measurements. Provide error estimates. WARNING: Do not look through

the sample! Use the provided screen.

Figure 9: Suggested

observation setups

for laser light.


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Experimental Competition: 14 July 2011

Problem 2 Page 1 of 2

2. Mechanical Blackbox: a cylinder with a ball inside

A small massive particle (ball) of mass m is fixed at distance z below the top of a long hollow

cylinder of mass M . A series of holes are drilled perpendicularly to the central axis of the cylinder.These holes are for pivoting so that the cylinder will hang in a vertical plane.

Students are required to perform necessary nondestructive measurements to determine the numerical

values of the following with their error estimates:

i. position of centre of mass of cylinder with ball inside.

Also provide a schematic drawing of the experimental set-up for measuring the centre ofmass. [1.0 points]

ii. distance z [3.5 points]

iii.

ratioM

m . [3.5 points]

iv.

the acceleration due to gravity, g . [2.0 points]

Equipment: a cylinder with holes plus a ball inside, a base plate with a thin pin, a pin cap, a ruler, a

stopwatch, thread, a pencil and adhesive tape.

CM x is the distance from the top of the cylinder to the

centre of mass.

R is the distance from the pivoting point to the centre of

mass.

Thin pin

for pivoting

Base plate

to be clampedto a table top

pivot

M

O

CM

CM x

L

R

z

m


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Experimental Competition: 14 July 2011

Problem 2 Page 2 of 2

Caution: The thin pin is sharp. When it is not in use, it should be protected with a pin cap for safety.

Useful information:

1. For such a physical pendulum,2

2

2CM

d M m R I g M m R

dt , where

CM I is

the moment of inertia of the cylinder with a ball about the centre of mass and is the angular

displacement.

2.

For a long hollow cylinder of length L and mass M , the moment of inertia about the centre of

mass with the rotational axis perpendicular to the cylinder can be approximated by

2

1

3 2

LM .

3. The parallel axis theorem:2

centre of massI I x M , where x is the distance from the rotation

point to the centre of mass, and M is the total mass of the object.

4. The ball can be treated as a point mass and it is located on the central axis of the cylinder.

5. Assume that the cylinder is uniform and the mass of the end-caps is negligible.

Cylinder with holes

plus ball inside

Base plateStopwatch

Adhesivetape

Thread(for balancing)

Ruler

Pin cap


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Solar cells E2

Page 1 of 9

2.0 Introduction

Equipment used for this experiment is displayed in Fig. 2.1.

Figure 2.1 Equipment used for experiment E2.

List of equipment (see Fig. 2.1):

A: Solar cell

B: Solar cell

C: Box with slots for the mounting of light source, solar cells, etc.

D: LED-light source in holder

E: Power supply for light source D

F: Variable resistor

G: Holder for mounting single solar cell in the box C

H: Circular aperture for use in the box C

I: Holder for mounting two solar cells in the box C


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Solar cells E2

Page 2 of 9

J: Shielding plate for use in the box C

K : Digital multimeter

L: Digital multimeter

M: Wires with mini crocodile clips

N: Optical vessel (large cuvette)

O: Measuring tape

P: Scissors

Q: Tape

R : Water for filling the optical vessel N

S: Paper napkin for drying off excess water

T: Plastic cup for water from the optical vessel N (not shown in Fig. 2.1)

U: Plastic pipette (not shown in Fig. 2.1)

V: Lid for the box C (not shown in Fig. 2.1)

Data sheet: table of fundamental constants

Speed of light in vacuum Elementary charge Boltzmann’s constant

A solar cell transforms part of the electromagnetic energy in the incident light to electric energy by

separating charges inside the solar cell. In this way an electric current can be generated. Experiment

E2 intents to examine solar cells with the use of the supplied equipment. This equipment consists of

a box with holders for light source and solar cells along with various plates and a lid. The variable

resistor should be mounted in the box, see Fig. 2.2. One of the three terminals on the resistor has

been removed, since only the two remaining terminals are to be used. Also supplied are wires withmini crocodile clips and two solar cells (labeled with a serial number and the letter A or B) with

terminals on the back. The two solar cells are similar but can be slightly different. The two

multimeters have been equipped with terminals for designated use as ammeter and voltmeter,

respectively, see Fig. 2.3. Finally, the experiment will make use of an optical vessel together with

some drinking water from the bottle.


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Figure 2.2 (a) Box with light source and resistor for mounting. (b) The resistor mounted in the box. Notice

that the small pin on the resistor fits in the hole to the right of the shaft.

Figure 2.3 Multimeters equipped with terminals for use as ammeter (left) and voltmeter (right),

respectively. The instrument is turned on by pressing “POWER” in the top left corner. The instrument turns

off automatically after a certain idle time. It can measure direct current and voltage as well as alternatingcurrent and voltage . The internal resistance in the voltmeter is 10 MΩ regardless of the measuring range.The potential difference over the ammeter is 200 mV at full reading, regardless of the measuring range. In

case of overflow the display will show “l”, and you need to select a higher measuring range. The “HOLD”

button (top right corner) should not be pushed, except if you want to freeze a measurement.


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WARNING: Do not use the mul timeter as an ohmmeter on the solar cells since the measur ing

cur rent can damage them. When changing the measur ing range on the mul timeters, please tur n

the dial with caution. I t can be unstable and may break. Check whether there is a number under

the decimal point when measur ing –

i f the dial is not ful ly in place, the mul timeter wil l notmeasur e, even i f there are digi ts in the display.

Notice: Do not change the voltage on the power supply. It must be 12 V throughout the

experiment. (The power supply for the light source should be connected to the outlet (230 V ~) at

your table.)

Notice: Uncertainty considerations are only expected when explicitly mentioned.

Notice: All measured and calculated values must be given in SI units.

Notice: For all measurements of currents and voltages in this experiment, the LED-light

source is supposed to be on.


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2.1 The dependence of the solar cell current on the distance to the light source

For this question you will measure the current, , generated by the solar cell when in a circuit withthe ammeter, and determine how it depends on the distance, , to the light source. The light is

produced inside the individual light diodes and is therefore to be measured as shown in Fig. 2.4.

Figure 2.4 Top view of setup for question 2.1. Note the aperture a immediately in front of the solar cell A.

The distance is measured from inside the light diode to the surface of the solar cell.

Do not change the measuring range on the ammeter in this experiment: the internal resistance of the

ammeter depends on the measuring range and affects the current that can be drawn from the solar

cell. State the serial numbers of the light source and of solar cell A on your answer sheet. Mount the

light source in the U-shaped holder (the light source has a tight fit in the holder, so be patient when

mounting it. Mount solar cell A in the single holder and place it together with the circular aperture

immediately in front of the solar cell. The current as a function of the distance to the light sourcecan, when is not too small, be approximated by

where and are constants.

2.1a Measure I as a function of r , and set up a table of your measurements. 1.0

2.1b Determine the values of I a and a by the use of a suitable graphical method. 1.0


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2.2 Characteristic of the solar cell

Remove the circular aperture. Mount the variable resistor in the box as shown on Fig. 2.2. Place the

light source in slot number 0, furthest away from the resistor. Mount solar cell A in the single

holder without the circular aperture in slot number 10. Build a circuit as shown in Fig. 2.5, so thatyou can measure the characteristic of the solar cell, i.e. the terminal voltage U of the solar cell as a

function of the current I in the circuit consisting of solar cell, resistor and ammeter.

Figure 2.5 Electrical diagram for measuring the characteristic in question 2.2.

2.2a Make a table of corresponding measurements of U and I . 0.6

2.2b Graph voltage as function of current 0.8

2.3 Theoretical characteristic for the solar cell

For the solar cells in this experiment, the current as function of the voltage is given by the equation

(( ) ) where the parameters , and are constant at a given illumination. We take the temperature to

be . The fundamental constants and are the elementary charge and Boltzmann’sconstant, respectively.

2.3a Use the graph from question 2.2b to determine . 0.4The parameter can be assumed to lie in the interval from 1 to 4. For some values of the potentialdifference , the formula can be approximated by

( )

2.3bEstimate the range of values of for which the mentioned approximation is good.Determine graphically the values of and for your solar cell. 1.2


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2.6 Couplings of the solar cells

The two solar cells can be connected in series in two different ways as shown in Fig. 2.7. There are

also two different ways to connect them in parallel (not shown in the figure).

Figure 2.7 Two ways to connect the solar cells in series for question 2.6. The two ways to connect them in

parallel are not shown.

2.6

Determine which of the four arrangements of the two solar cells yields the highest

possible power in the external circuit when one of the solar cells is shielded with the

shielding plate (J in Fig. 2.1). Hint: You can estimate the maximum power quite well by

calculating it from the maximum voltage and maximum current measured from each

configuration.

Draw the corresponding electrical diagram.

1.0

2.7 The effect of the optical vessel (large cuvette) on the solar cell current

Mount the light source in the box and place solar cell A in the single holder with the circular

aperture immediately in front, so that there is approximately 50 mm between the solar cell and the

light source. Place the empty optical vessel immediately in front of the circular aperture as shown in

Fig. 2.8.


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Figure 2.8 Experimental set-up for question 2.7.

2.7aMeasure the current I , now as a function of the height, h, of water in the vessel, see Fig.

2.8. Make a table of the measurements and draw a graph.1.0

2.7b Explain with only sketches and symbols why the graph looks the way it does. 1.0

Mount the light source in the box and place solar cell A in the single holder so that the distance

between the solar cell and the light source is maximal. Place the circular aperture immediately in

front of the solar cell.

2.7c

For this set-up do the following:

- Measure the distance between the light source and the solar cell and the current .- Place the empty vessel immediately in front of the circular aperture and measure the

current .- Fill up the vessel with water, almost to the top, and measure the current .

0.6

2.7dUse your measurements from 2.7c to find a value for the refractive index for water.Illustrate your method with suitable sketches and equations. You may include additional

measurements.

1.6


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11

Experimental problem: “Semiconductor element”

In this experiment a semiconductor element ( ), an adjustable resistor (up to 140 Ω),

a fixed resistor (300 Ω), a 9-V-direct voltage source, cables and two multimeters are at disposal.

It is not allowed to use the multimeters as ohmmeters.a) Determine the current-voltage-characteristics of the semiconductor element taking into

account the fact that the maximum load permitted is 250 mW. Write down your data in

tabular form and plot your data. Before your measurements consider how an overload of the

semiconductor element can surely be avoided and note down your thoughts. Sketch the

circuit diagram of the chosen setup and discuss the systematic errors of the circuit.

b) Calculate the resistance (dynamic resistance) of the semiconductor element for a current of

25 mA.

c)

Determine the dependence of output voltage U 2 from the input voltage U 1 by using the

circuit described below. Write down your data in tabular form and plot your data.

The input voltage U 1 varies between 0 V and 9 V. The semiconductor element is to be

placed in the circuit in such a manner, that U 2 is as high as possible. Describe the entire

circuit diagram in the protocol and discuss the results of the measurements.

d) How does the output voltage U 2 change, when the input voltage is raised from 7 V to 9 V?

Explain qualitatively the ratio ∆U 1 / ∆U 2.

e) What type of semiconductor element is used in the experiment? What is a practical

application of the circuit shown above?

Hints: The multimeters can be used as voltmeter or as ammeter. The precision class of these

instruments is 2.5% and they have the following features:

measuring range 50 µA 300 µA 3 mA 30 mA 300 mA 0,3 V 1 V 3 V 10 V

internal resistance 2 k Ω 1 k Ω 100 Ω 10 Ω 1 Ω 6 k Ω 20 k Ω 60 k Ω 200 k Ω


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11

Experimental Problems

Problem 4: Lens experiment

The apparatus consists of a symmetric biconvex lens, a plane mirror, water, a meter stick,

an optical object (pencil), a supporting base and a right angle clamp. Only these parts may

be used in the experiment.

a) Determine the focal length of the lens with a maximum error of ± 1 %.

b) Determine the index of refraction of the glass from which the lens is made.

The index of refraction of water is nw = 1.33. The focal length of a thin lens is given by

( )1 2

1 1 1n 1f r r

⎛ ⎞= − ⋅ −⎜ ⎟

⎝ ⎠,

where n is the index of refraction of the lens material and r 1 and r 2 are the curvature

radii of the refracting surfaces. For a symmetric biconvex lens we have r 1 = - r 2 = r, for a

symmetric biconcave lens r 1 = - r 2 = - r .

Solution of problem 4:

a)

For the determination of f L , place the lens on the mirror and with the clamp fix the pencil to the supporting base.

Lens and mirror are then moved around until the

vertically downward looking eye sees the pencil and its

image side by side.

In order to have object and image in focus at the same

time, they must be placed at an equal distance to the eye.

In this case object distance and image distance are the

same and the magnification factor is 1 .

It may be proved quite accurately, whether magnification 1 has in fact been obtained, if

one concentrates on parallatical shifts between object and image when moving the eye:

only when the distances are equal do the pencil-tips point at each other all the time.

The light rays pass the lens twice because they are reflected by the mirror. Therefore

the optical mapping under consideration corresponds to a mapping with two lenses

placed directly one after another:


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1.2 Experimental competition

Exercise A

Follow the acceleration and the deceleration of a brass disk, driven by an

AC electric motor. From the measured times of half turns, plot the angle,

angular velocity and angular acceleration of the disk as functions of time.

Determine the torque and power of the motor as functions of angular

velocity.

Instrumentation

1. AC motor with switch and brass disk

2. Induction sensor

3. Multichannel stop-watch (computer)

Instruction

The induction sensor senses the iron pegs, mounted on the disk, when

they are closer than 0.5 mm and sends a signal to the stop-watch. Thestop-watch is programmed on a computer so that it registers the time

at which the sensor senses the approaching peg and stores it in mem-

ory. You run the stop-watch by giving it simple numerical commands,

i. e. pressing one of the following numbers:

5 – MEASURE.

The measurement does not start immediately. The stop-watch waits

until you specify the number of measurements, that is, the number

of successive detections of the pegs:

3

Experimental Problems in Physics

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