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Instruction Manualand Experiment Guide

OPTICS SYSTEM 3

N.B.:

Pictures, images and descriptions in this manual may not exactly correspond withthe actual items supplied.

It is also important to note that the experiments in this manual are, only, suggestions.They are not meant to indicate the limitation of the equipment, which can be used in

wide range of experiments, depending on the educational requirement of the teacher.

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GENERAL DESCRIPTION:

The optics kit 3 offers you and your students endless possibilities for exploring the many aspects of modernoptical technology. In a few moments you're ready to conduct experiments utilizing geometric principles andoptics, examine polarization of laser beams, investigate basic and advanced diffractive principles and optics.

LIST OF EXPERIMENTS:

diffraction grating double slit diffraction optical activity passage from interference to diffraction single slit diffraction

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Index of related topics:

D

Diffraction..........................................................................................................................................17Diffraction grating ............................................................................................................................19Double slit diffraction ......................................................................................................................13

I

Interference ......................................................................................................................................17

O

Optical activity..................................................................................................................................23

S

Single slit diffraction ..........................................................................................................................8

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ASSEMBLY INSTRUCTIONS

Assemble the laser on the support stand eventually by using the Polaroid filter or the diverging lens

o The Polaroid filter is solely devoted to mitigating the intensity of the laser beam since thisone is already polarized.

o Cylindrical lens have at least one surface that is formed like a portion of a cylinder. They areused to correct astigmatism in the eye, and, in this case, to stretch a point of light into a line.

Perform your experiments by selecting one from the various slits gratings, polaroids,

Example of double slit interference

Example of a one-dimensional diffraction grating interference

Example of a bidimensional diffraction grating

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Experiment 1.RELATED TOPICS:

Single slit diffraction

The aim of the experiment is to study the single slit diffraction

ITEMS NEEDED:

Laser Slide with single slit

Support for slide Screen Meter Ruler with mm divisions

THEORY:

A plane wave that meets obstacles of a comparable size with its wavelength no longer proceeds linearly butinvades the "blind zone" causing phenomena of light intensity distribution called "diffraction" phenomena.These phenomena can be satisfactory described through the Huyghens - Fresnel's principle. This principlestates, among other things, that all planes reached by a light wave - for instance the points inside a slit -become sources of virtual elementary spherical waves. The observed real wave is the result of the

interference of the elementary waves. This suggests that diffraction and interference are phenomena thatcan be referred to only in a theoretical interpretation.When the diffraction happens at a great distance from the obstacle, it is called Fraunhofer diffraction;whereas, when it occurs at a finite distance from the obstacle, it is called Fresnel dif fraction.The following experiment studies the diffraction figures produced by a narrow slit and collected by a screenparallel to the plane of the slit placed "at infinite".For the teacher's convenience, we give here some theoretical notes, derived from the Huyghens Fresnel'sprinciple, regarding the distribution of the light intensity minima at infinity in the case of a single slit.

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The point Poof the screen in front of the slit receives, in phase, all the pairs of elementary beams comingout in phase from the slit. Therefore that point is the point of maximum light intensity. Let us now look for thepoint P1closer to Poat which there is destructive interference.Remember that destructive and constructive interference can be explained as in the following figure

The elementary beams coming out from the ends of the slit must arrive at P1 with a difference in covered

distance equal to /2. In fact, in that way each wave coming from the first half of the slit (for instance O1) willbe neutralized by a corresponding wave (O2) coming from the second half.

For P1, therefore, the following relationship holds:

sin sin2 2

bor b

= =

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Let us divide the slit in four equal parts and let us choose the difference in covered distance for the beams

coming from the ends of the slit so that it is equal to /2.

For the second point (P2) of destructive interference we have:

sin sin 24 2

bor b

= =

Generalizing the reasoning to the generic point Pof destructive interference we have:

sinb k = with k=1,2,

APPARATUS SETTING:

Align the (turned off) laser, the diaphragm, the adjustable slit and the screen following the diagram of the

following picture.

Turn on the laser after having made sure that nobody can look into the light source.

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Centre the slit with the beam (slightly less than half the beam must fall on the fixed margin). Check that the

plane of the slit and the screen are perpendicular with regards to the beam. The light in the region of the

screen must be dim (in a dark room the distance between the slit and the screen can be increased to more

than 10 m.)

PROCEDURE

Use the single slit and turn on the laser. From the qualitative point of view, it is easy to see how the central

bright band is wider than the others (it is double).

The quantitative experiments can vary around the relationship that gives the intensity minima:

sin ( 1, 2, )b k k = = (1)

but sin l = where l is the distance from the screen. We can carry on a further approximation if we

consider a limited number of fringes. In this way sin tan.(within the 5% approximation up to 12 degrees).

In this case we get from (1)

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( 1,2, )b k kx

= = (2)

and by knowing , x, its possible to recover b.

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Double slit diffraction

The aim of the experiment is to study the double slit diffraction.

ITEMS NEEDED:

Laser

Slide with double slit Support for slide Screen Meter Ruler with mm divisions

THEORY

Remember that destructive and constructive interference can be explained as in the following figure

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If the slits are illuminated with a laser light - or with another kind of light, provided that it comes from a point-like source - two secondary sources of coherent light are obtained.The beams coming from those sources cause interference in the whole region in which they aresuperimposed.Let Pbe any point in that region, at distances l

1, and l

2from the slits S

1and S

2that are at a distance dapart,

and let Pbe seen under an angle from the centre of the segment S1S2.

In Pthere will be constructive interference with maximum intensity if, denoting by the light wavelength,

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1 20,1,l l k k = =

but, if the screen is placed at large distance from the slit ( l d ), the above expression becomes

sind k =

but sinl

= where lis the distance from the plane HPshown in figure 1 and = PH so we get for

constructive interference

0,1,d k kl

= =

In a similar way we have that for destructive interference with minimum intensity is instead

( )2 1 0,1,2

d k kl

= + =

The experiment described consists exactly of studying the interference in a plane (screen) placed at a right

angle with the direction of the light beams coming from the double slit. This can substitute Young'sexperiment (light coming from two small holes), very important in the hystory of physics but less convenientfor teaching purposes because of the poor illumination intensity of the figure appearing on the screen.

Because the light intensity of the beams coming from the slits decreases, the experiment must be clarried onin conditions of dim light. If one is in a very large room (at least 10 m a side) darkness must be complete.

APPARATUS SETTING

Allocate the laser turned off, the slide with double slit and the screen as shown in the picture, checking thatthe screen is at right angle with the laser axis.The light in the region of the screen must be dim. (The distance between the slide and the screen can beincreased to about 10 m if the room is completely darkened).

PROCEDURE

After having made sure that nobody can look directly into the laser, turn it on. Centre the slide with thedouble slit in the small light beam, so that it is exactly perpendicular to it (to make negligible the error of d).

Proceed to measure the various quantities. It will be seen that the above relationship is satisfied. The aim ofthe experiment can be that of determining an "unknown" quantity (for instance the light wavelength or thedistance dbetween the slits).

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The measurement implementation is rather simple and the results can be obtained within a few percentagepoints.It is strongly recommended to measure the fringe's width, marking (on the screen) the centres of two darkbands as distant as possible from each other, but still unmistakable.Two other simple and important experiments that can be carried out are the following:

Cover first one and then the other slit to show that the interference figure is only the modulation of

the diffraction figure. Move the laser parallel to itself to show the coherence of light in the whole "thickness" of the beam

(there is always interference). With a traditional light source, which must be of very small size toproduce interference, this cannot be obtained.

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Interference Diffraction

The aim of the experiment is to study the passage from interference to diffraction

ITEMS NEEDED:

Laser Diaphragm (optional) Various slides Screen Meter Ruler with mm divisions

THEORY

The conceptual difference between interference and diffraction is not clear-cut: in both cases there is aproblem of wave propagation in special conditions. The theoretical model most used to explain how the twophenomena coincide when diffraction is considered, is the Huyghens-Fresnel's principle. In fact, thatprinciple is based on the interference of infinite elementary waves coming from various parts of the opening.However, it is not necessary that they be only virtual waves, as first thought Huyghens-Fresnel.An interesting correlated treatment of the two phenomena can be found in chapters 28, 29 and 30 ofFeynman's lectures. Because of its length, we cannot repeat it here. We will just mention the logic on which itis based: the properties of the electromagnetic waves entail a set of quantitative conditions for theinterference among waves emitted by 2 or n equal sources; if n is large or infinite, the phenomena areclassically considered "diffraction": the case of the grating and that of a small opening is an opaque screen.The following experiment concerns exactly the phenomenology connected with an increase in the number nof equal sources that cause interference. The arrival point is the diffraction grating.

At the beginning the case of the double slit is presented again. Those slits, to produce a "pure" interferencephenomenon, should be of "infinitesimal" length, or at least less than the wavelength, thus realizingsomething similar to two elementary Huyghens' sources. The interference phenomena are in fact observedalways and only inside the central bright band of the diffraction figure of a slit (if the weak effects in the otherclear bands are neglected). We can therefore understand how the thinning of the slits, which broads thatband, approaches the ideal case of a very small source on 180.What happens with three or more slits is rather intricate but, from a phenomenological point of view, can bedescribed with a limited number of statements:

as the number of slits increases, the bright bands do not move, but their intensity increases and theirwidth decreases;

for a number n>3 of slits, beside those bands there appear other light bands, narrower and less

intense (they are n - 2); the distance among the principal bands is inversely proportional to the distance among the slits; a "coarse" grating (grating 1) with pitch equal to the distance of the few slits in an opaque slit,

produces bright luminous points in the usual positions, but without secondary bands (as the numberof those bands increases, their intensity decreases);

a grating with the same pitch but with slits less wide (grating 2) produces zones less bright in aproportionally wider section;

a grating with narrower pitch (grating 3) gives maxima that are proportionally more distant one fromthe other.

APPARATUS SETTING

Align all the elements of the apparatus according to the diagram (no particular experience is necessary).Make sure that nobody can look directly into the laser and turn it on. Center the fixed diaphragm so that thewhole beam passes through it and the halo and the irregular extension of diffused or reflected light areblocked. Darken partially the room, expecially around the screen (in order to observe all the characteristics of

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the phenomena, darkness should be almost complete).

PROCEDURE

Place the slide with single slit in the slide-holder and notice the width of the central bright band.

Substitute the slide with single slit with the slide with double slit and observe how the interference figure isthe modulation of the preceding diffraction figure.Pass to three slits: some quantities are preserved (total width of the interference figure and distance betweenthe large maxima) and some other characteristics of the phenomenon appear (increase in the light intensity,narrowing of the clear bands and appearance of weak secondary maxima).With four slits the observations are similar to the preceding ones (now, however, the secondary maxima foreach primary maximum are two and they are weaker). With five slits the phenomena observed previouslybecome more evident (the secondary maxima, even weaker, are three). If desired, use also the slide with sixslits, it takes almost the whole beam and therefore its results are not clearly distinguishable from those ofgrating 1.With the grating 1 the secondary maxima disappear and there is a further increase of the light intensity (asthe slits increase, the percentage of energy captured by the slide decreases). With the grating 2 it is possibleto notice the phenomena realted to the halving of the width of the slits, that is the decrease of the light

intensity and expecially the widening of the interference-diffraction figure. With the grating 3 one insteadobserves an increase in the width of the interference figure and expecially a wider spacing of the maxima(due to the decrease of the distance between the slits).

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Diffraction grating

The aim of the experiment is to study the diffraction grating

ITEMS NEEDED:

Laser Various kinds of diffraction grating Screen Meter Ruler with mm divisions

THEORY:

Essentially a diffraction grating is a device with a large number of parallel slits (or reflecting lines).Diffraction gratings with many variable features (fixed or variable spacing, curving of the support and so on)

are used in research and in industry to determine with great precision the wavelengths and the spectra of theradiation sources. The study of the crystal structure through X rays is carried out, enlarging the concept ofgrating, considering the crystals as three dimensions gratings.The characteristic relationship for the diffraction grating is very simple:

sink p = where k = 0,1,2,is an integer linked with an intensity maximum of the diffraction figure produced by thegrating (called "order" or "principal maximum").Instead the complete theory of the grating is rather intricate, therefore we will just make some simpleobservations on the functioning of a grating with slits at constant pitch (our gratings are of that kind).Let us image a plane wave meeting the grating at a right angle. The diffraction figure at a great distance isthe result of the interference of the waves coming from the various slits. Let us divide the grating in pairs of

consecutive slits. Among the elementary waves sent by those slits let us consider those that travel in thedirection that forms an angle with the perpendicular to the grating (see the following figure).

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The waves starting from corresponding pairs of points (for instance from the points O1and O2of the figure)

cover distances that differ of psin. In the direction with a such that the difference in covered distance is a

whole number of the wavelength , the waves will be in phase and will provoke, by constructive interference,maxima of intensity.

With elementary reasonings it is not possible to make any forecasts on what happens for different angles .

However, there should be practically total darkness for each that does not correspond to a maximum (thisis strictly true for p0).

APPARATUS SETTING:

Dispose the laser, the grating and the screen (at about 5 m from the grating) according to the diagram.

Turn on the laser (after having made sure that nobody can look directly into it) and check that the screen andthe grating are at a right angle with the beam.For this experiment the illumination of the room must be only slightly below normal.

PROCEDURE:

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First of all, with the cardboard slightly bent (and tangent to the beams coming out from the grating), movedupward, it is possible to see very well the three-dimensional structure of diffraction.The qualitative study of the diffraction figure produced on the screen revolves around the already mentioned

relationship sink p = that can be verified with very high precision (even in a teaching experiment it

is possible to have errors of less than 1%). The measuring is very simple to perform and only two warnings

are necessary.

The first warning is that the sine of the angle can be mistaken with the tangent and measured by taking theratio between and xin the figure only if a very high precision is not necessary. The second warning is that

if the first maxima of the grating with greates pitch are considered.To quickly find the central maximum (k = 0) it is enough to slightly rotate the grating and to see whichmaximum remains immobile.

EXAMPLE OF EXPERIMENTAL DATA:

By using a motion detector and a light sensor is quick and simple to have a graph of the light intensity vs. thelight sensor position. This is possible by moving the light sensor around the principal maxima while samplingits position with the motion detector.For a distance x= 173 cm and a diffraction grating with p= 0.13 mm we get the following result:

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From the left to the right of the preceding picture we get the position of the principal maxima:0.121 m, 0.129 m, 0.137 m (central peak), 0.146 m, 0.154 m.This means a relative distance of the principal maxima from the central peak of 16 mm, 8 mm, 9 mm, 17 mmor

(0 , [8,9] , [16,17] ,...)mm mm mm = (3)

If we modify the previous formula sink p = we get:

sin k

p

= (4)

or, since sintan, and tan =

9 2

3

(650 10 )(173 10 )0.0087m

(0.13 10 )

x m mk k k

p m

= = =

(5)

with k = 0,1,2,Otherwise said:

(0 , 8.7 , 17.4 ,...)mm mm mm = (6)

in good agreement with the previous experimental data.

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Optical activity

The aim of the experiment is to study the optical activity of dextrose (or normal saccarose). The pictureabove can be obtained with the use of an optical bench.

ITEMS NEEDED:

Laser Slide with polaroid filter Glass tank Water Dextrose (or normal saccarose) Screen

THEORY:

The narrow, intense and monochromatic light beam emitted by a laser allows you to easily perform teachingexperiments on optical activity (Long time ago, the Chemists have used the traditional light sources).Optical activity is a phenomenon connected with the "asymmetry by reflection" of the molecules of manysubstances. Let us consider an asymmetric molecule by reflection (the kind of asymmetry between the twohands), for example, a helical shaped molecule. The polarized light that arrives with the electric vectoroscillating along the helix axis zsets in motion the electrons.

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The electron motion along zgenerates a second wave (coming out) that is linearly polarized according to z.However, since the electrons are compelled to move along the helix, some components of the motion will beperpendicular to the planes passing through z(see figure).A pair of such components would send (for instance in P) waves in phase opposition (that, therefore, wouldannul each other) if there was not the lag due to the fact that only one wave must cover the diameter dof thehelix. It is therefore necessary to correct the wave that comes out and that was initially considered polarizedalong zwith a small field component polarized along the xaxis (see figure).

The wave that will actually come out has the polarization plane rotated of an angle as regards to the planeof the incident wave, even when the linearly polarized light strikes a set of asymmetric molecules arranged atrandom (for instance in a water solution); in fact each molecule appears identical if seen from either side.

APPARATUS SETTING:

Place the various optical elements.Prepare 100 cm

3of highly concentrated solution of dextrose (or saccarose) in water and pour it into the tank.

Take the tank away from the optical bench and turn on the laser (after having made sure that nobody canlook directly into it); place the first polaroid with the axis of easy transmission vertical (when that is the case,the beam is completely intercepted by the second polaroid placed at 0).

PROCEDURE:

After having taken away or lowered the tank, so that the light does not cross the solution, find the position ofthe second polaroid for which the bright spot on the screen disappears. In those conditions the first polaroidgenerates a light beam with horizontal polarization plane and the second intercepts completely that beam.

Now the beam is made to travel for three cm in the solution: the bright spots on the screen reappears. Thesecond polaroid must be rotated until the bright spots disappear again. The angle by which the secondpolaroid has been rotated is equal to the angle of rotation of the light polarization angle (for a dextroseconcentrated solution, an angle of about 20 to the right of the source). To determine with the approximationof 1 or 2 degree the amplitude of that angle, one can take the average between the two positions at whichthe beam reappears when the polaroid is rotated in the two directions.The experiment is repeated making the beam travel for 6 and 12 cm in the solution. Because of the additivityof the effect with regards to the number of molecules met by the light, the rotation angle is found to bedouble and quadruple. If only saccarose is available, it is advisable to start from a distance of 12 cm and topass afterwards to 6 and 3. If the observations are repeated with a concentrated solution of fructose, onefinds that the rotations as regards as the polarization plane are to the left.Another important study that can be performed is studying the effect of the concentration of the solution (forequal covered distances). In all the described experiments it is not difficult to bring the error to below 5%.

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APPENDIXBasics of experimental error theory

We can say that everything we known about the physical world has an inherent uncertainty. In

particular, when we experimentally investigate something there is always an experimental errorand an experimental precision. Since one of the main features of experiments is theirreproducibility, it is very important to deal with this subject in order to be able to explain how good

our results are. This is possible with experimental error theory, a scientific approach to thisproblem.

Let us consider the following example: find the density of a solid rubber cube.

o First trial (with very raw instrument). We can estimate that the mass of the cube isnearly 50 g and the length of a side is nearly 6 cm. So the density would be:

3 0,23148...M M

V L= = = . There are many questions: Where can I stop with

decimal digits to communicate my result? Is it better to have precision on themass measurement or on length measurement? How do we combine our

experimental error on the mass measurement with the experimental error on length

measurement?o Second trial (with more accurate instruments). By using an electronic balance and

a meter stick I find a mass of 60g and a side length of 5,4 cm. So the density wouldbe:

30,381039475...

M M

V L= = = . We still need to answer the questions posed above

but we also have to answer a new question: What makes this trial better than thefirst one?

o Third trial (with much more accurate instruments to measure the side length). Ifwe improve the accuracy of the length measurement, for example by using a vernier

caliper, the problem becomes more involved. This is due to the fact that we do notget the same result if we make more than one measurement. Instead we have a set of

different measurements like (5,455 cm; 5,425 cm; 5,465 cm; ). So we are again

faced with the question: Which one of the measurements (5,455 cm; 5,425 cm;5,465 cm; ) do should I take?

Therefore, the more we analyse the problem the more it gets involved. To search for a possible

solution we can start from the third trial and observe that, generally speaking, when we improve theaccuracy of an instrument we reach a point at which the experimental results are not unique but are

scattered around some value as illustrated in this graph:

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If the number of measurements Nis greater then about 30, the distribution of the experimental data

is bell-shaped and has a value X for which there is a maximum and around which the data arescattered in a nearly symmetrical way. It is also possible to distinguish a value that determines an

interval around Xinto which a significant percentage of the measurements are placed. We need toanswer the questions: Is X the best estimate of our measurement?, How much can we rely on

this value? and What percentage of the measurements are in the interval X-and X+?

To express these questions mathematically, we could try a prototype function that fits our data and

that expresses the probability to get a particular measurement value:

- 3 - 2 - 1 1 2 3

0. 2

0. 4

0. 6

0. 8

this is the graphical representation of the function 2( )

xf x e=

If we want to centre the function around the value Xwe use the expression x-Xin place of x, and if

we want to control the scattering of the measurements around X its possible to divide (x-X)2by

2

2.

The following figure shows f(x)withX=2 and =1;1.5;2

- 2 2 4 6 8 10

0. 2

0. 4

0. 6

0. 8

1

- 2 2 4 6 8 10

0. 2

0. 4

0. 6

0. 8

1

- 2 2 4 6 8 10

0. 2

0. 4

0. 6

0. 8

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Finally, if we want to control the area under the curve we have to multiply it by a normalization

factor A

that would depend on .

Therefore our prototype function is:( )

2

22( )

x X

f x A e

= (1)

where Xis the value for which we have the maximum and determines how the measurements arescattered around X. This is called a Gaussian function or a Normal function, but the underlying

data represent a distribution (still called Gaussian) and not a function. It can be proved that theGaussian distribution is derivable from the binomial distribution assuming that the number of

measurements Nand remains constant.

The physical meaning of all this is that we do not describe a measurement with a single number butrather with a set of values each one with its own probability to appear as an experimental datum.

This probability is governed by the Gaussian distribution. There is an analogy with quantummechanics (for example with the wave packet of a particle) where the interpretation is that if we

make a measure of the position of the particle then the probability to obtain a particular value isgoverned by the Gaussian function and is never a well defined fixed value.

Let us determine the value of A

in (1). We must have a probability of 1 to get a measurement in

the range from -to +(that is, if we perform a measurement we are certain to get some kind ofresult no matter how large or how small that result is):

1( ) 1

2f x dx A

+

= =

To give an interpretation of we can ask what happens if we are only interested in the probability

of finding measurements in the range from X-to X+ instead of the range from -to+:

2

12

1

1( ) 0.68

2

tX

Xf x dx e dt

+ +

=

so , also called then standard deviation (2is called variance), is the amount of uncertainty wehave to allow for, in the most probable value X, if we want to claim a roughly 68% chance of

correctly predicting the result of any single measurement.

To determine X,also called the mean value, we consider a set of Nmeasurements x1, x2, , xN.The probability to get a single result between xiand xi+dxis:

( )2

221

2

ix x

iP e

=

so the probability to get all the results (viewed as independent events) is:

( )2

1

221 2

1...

N

i

i

x x

N NP P P P e

=

=

Since we are speaking about the probability P to get all the results and we can suppose to havealready done our experiment with a set of real results what should be the value of P?

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If we accept the maximum likelihood principle we can make an analogy with entropy and say that Pis proportional to the entropy obtained from our experiment. The value X must be a point of

maximum entropy. By the second principle of thermodynamics we have to maximize P, otherwise

said Xis the value of xthat minimize the exponent: ( )2

1

0N

i

i

dx x

dx =

=

from which it results:

1

1 N

i

i

X xN =

= (2)

that is, the mean value X is the arithmetic mean and describes all the collected data since it is the

value for which the maximum entropy is obtained for our set of data.

To determine

we can proceed in the same way

( )2

1

221

0

N

i

i

x x

N

de

d

=

=

from which:

( )2

1

1 N

i

i

x xN

=

= (3)

But what should be use instead of xin equation (3)? If we use (2) then equation (3) is slightly self-

referential because

2

1

1

1 1( ... ... )

N

i N i

i

x x xN N

=

= + + + +

and the i-esim term appears two

times. It is possible to show that the correct value of the standard deviation is:

( )

2

1

1

1

N

ii

X xN

=

=

(4)

Clearly is not defined for N=1 (we are assuming N greater of nearly 30, otherwise there are betterdistributions to consider).

Suppose now we have a function Qof several variables ( , , ,...)Q f a b c= and we want to know how

the experimental error on each variable contributes to Q.

We can say that by varying the variables, the quantity Qvaries of:

...Q Q Q

Q a b ca b c

= + + +

and if we identify our uncertainty xwith the standard deviation xwe can say that:

...Q a b cQ Q Q

a b c = + + +

(5)

the modulus is due to the fact that errors could cancel each other and we want to consider the

maximum error.We could do better, obtaining a smaller value, if the variables are normal and independent, by

starting from (4) ( )2

1

1

1

N

Q i

i

QN

=

=

where ( , , ,...)i i i iQ f a b c= is the i-esim value of Q by

taking the i-esim value of each variable of our set of data, ( , , ,...)f A B C= is the mean value of Q

by taking the mean value of each variable of our set of data.

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Since ( ) ( ) ( ) ( )2 2 2

2 2 2 2... ...i i i i i i

i i i i

Q Q Q QQ Q a b a b

a b a b

= = + + + +

(by

neglecting terms of higher order) we have:

( ) ( )

2 2

2 22 2

1 1

1 1...

1 1

a b

N N

Q i i

i i

Q Qa b

a N b N

= =

= + +

or2 2

2 2 ...Q a bQ Q

a b

= + +

(6)

which is better of (5) since its always lower.

Suppose now that the function Qis just the arithmetic mean1

1 N

i

i

X xN =

= . By applying equation (6)

we get

1 2

2 2

2 2

1 2

...X x xX X

x x

= + +

(7)

but1

1 1N

i

ii i

Xx

x x N N=

= =

and1 2

...x x = = = and so

XN

= (8)

which is called standard deviation of the mean. Analogously to the standard deviation, it tells

us how good is the mean value Xand we can assume it as the amount of uncertainty we have toallow for if we want to claim a roughly 68% chance of correctly predicting the result of any other

mean value it is possible to obtain.

It is also useful to speak about relative errorQ

Q

instead of absolute error

Q . The relative error can

be expressed in percentage.

For example let us return to the problem of determine the density of a cube.

Now, the function Qis the density which is function of the mass M and the side length L:3L

= .

If 60g= and 54L mm= its easy to find that the mean value is 43 3 3

60 3,81 1054

g gcm cm

= =

By applying equation (5) we have that the relative error is:3 3

3 4

1 1 1 3 1 3L M L M L

L L M

L M L M L M L

= + = + = +

.

If we can suppose the precision of the mass measurement is 2M

g = and the precision of the

length measurement is 1L

mm = we have:

2 3 13,3% 5,6% 8,9%

60 54

= + + =

(this says it is more important to make a careful length measurement than a careful massmeasurement).

By applying equation (6) we get a better (lower) estimate of the density error:

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2 2 2 2

2 2 2 2

9 2 9 16,5%

60 54

M L

M L

= + = + = .

This means that if we take another measurement of density theres a probability of nearly 68% that

the new value will lie between 43

(3,8 0, 2) 10 g

cm

.

It is important to note that since the standard deviation on density is 0,2 x 10-4

g/cm3we can stop at

the first decimal digit 3,8 x 10-4.

4864.30 - optics system 3

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