4866.10 - electricity system 1

8/9/2019 4866.10 - Electricity System 1

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Instruction Manualand Experiment Guide

ELECTRICITY SYSTEM 1

4866.10

N.B.:

Pictures, images and descriptions in this manual may not exactly correspond with

the 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

a wide range of experiments, depending on the educational requirements of the

teacher.


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GENERAL DESCRIPTION:

The Electricity kit 1 is dedicated to illustrate the electric components and fundamental circuits laws. It allowsrapid and clear assembly of circuits. All the components of the kit are stored in a foam cushioned plastic

storage case.The kit consists of:

Board in shock resistant plastic, dim. ! " #! cm, with groups of sockets arranged to form a

s$uare, % mm pitch.

&et of components mounted in plug'in housing and linked electrically with two or four plugs for mm

sockets.(imensions:

# plug'in housing: !# " ## " #h mm

(imensions of plug'in housing: !# " !# " #h mm

)lug spacing: % mm.

The international graphic symbol of each component, its rated *alue, the wiring arrangement and the

connection points are printed on the upper side of the plug'in elements

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LIST O E!PERIMENTS:

Tension measurement: the *oltmeter

Electric current measurement: the ammeter

The electrical resistance +irst and &econd hm-s aw

/esistances in series and in parallel )otentiometer

0irchhoff-s laws

The capacitor

harge and discharge of a capacitor

apacitors in series and in parallel

Electric cells

Electric bulbs

/ circuits

/ circuits

/ circuits

The magnetic compass

The electromagnet

ASSEM"L# INSTR$CTIONS:

Some examples of components assembly on the board

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Index o% related topics:B

Battery................................................................................................................................................35

C

Capacitor.....................................................................................................................40, 41, 42, 44, 46

lectric circ!it........................................................................", #, $, 35, 3", 3$, 40, 41, 42, 44, 46, 4#

%

%nd!ctance.....................................................................................................................................63, 65

%nd!ctor...............................................................................................................................................4#

&

&amp b!lb.............................................................................................................................................#

&C' circ!it in series...........................................................................................................................54

&' circ!it............................................................................................................................................4#

(

(a)netic hysteresis............................................................................................................................65

(!t!al*ind!ction................................................................................................................................63

+

+arallel resistance...............................................................................................................................24

+hototransistor....................................................................................................................................3"

+!sh*b!tton sitch...............................................................................................................................$

'

'C circ!it................................................................................................................................42, 44, 46

'elati-e ma)netic permeability..........................................................................................................65

'esistance.........................................................................................................................42, 44, 46, 4#

'esonance...........................................................................................................................................54

SSelf*ind!ction coefficient...................................................................................................................63

Series resistance..................................................................................................................................23

Sitch.........................................................................................................................................#, $, 3$

oltammetric method...................................................................................................................21, 22

/

/heatstone brid)e..............................................................................................................................50

mmeter !sa)e..................................................................................................................................16

i)h*pass filter .................................................................................................................................51

%nd!ctance .........................................................................................................................................61irchhoffs c!rrent la ........................................................................................................2", 31, 33

irchhoffs -olta)e la ..............................................................................................................2$, 33

&o*pass filter ..................................................................................................................................53

(a)net ..............................................................................................................................................5#

(a)netic compass ................................................................................................................5#, 60, 61

(a)netic ener)y and mechanical forces ..........................................................................................65

(a)netic field .......................................................................................................................5#, 60, 61

hms la..........................................................................................................................................25

+otentiometer !sa)e.....................................................................................................................1#, 1$

'C circ!it ....................................................................................................................................51, 53

oltmeter !sa)e.....................................................................................................................10, 12, 14

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Experiment

RELATED TOPICS:

Electric circuit

The purpose of the e"periment is to illustrate a simple electric circuit in ( current

ITEMS NEEDED:

cell holder with batteries

lamp bulb

lamp holder

bridging plugs

T&EOR#:

An electric circuit in ( current is a set of connections with a flow of electric charges without time

discontinuity.The electric circuit is *ery simple and is constituted by a generator of electromoti*e force 2battery3 and a load2lamp3.The goal of the e"perience is to show that the lamp is turned on when the circuit is closed, whilst it is turnedoff when the circuit is broken 2for e"ample by remo*ing a bridging plug3. 4oreo*er, in the case of turning onthe same light intensity must be obser*ed.

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Experiment !

RELATED TOPICS:

Electric circuit

&witch amp bulb

The purpose of the e"periment is to illustrate a simple electric circuit in ( current with a toggle switch.

ITEMS NEEDED:


lamp bulb

lamp holder

bridging plugs

toggle switch

T&EOR#:

By referring to the electric circuit of the pre*ious e"periment it is possible to show how a toggle switch canbreak the circuit in an easy way.As a matter of fact it is possible to turn on5off the lamp. Therefore the circuit has been transformed in asystem controlled by whoe*er is acting on a single point 2the toggle switch3.

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Experiment "

RELATED TOPICS:

Electric circuit

)ush'button switch amp bulb

The purpose of the e"periment is to illustrate a simple electric circuit in ( current with a push'button switch.

ITEMS NEEDED:


lamp bulb

lamp holder

bridging plugs

push'button switch

T&EOR#:

By referring to the electric circuit of the pre*ious e"periment it is possible to show how a push'button switchcan break the circuit in a mechanical way.As a matter of fact it is possible to turn on5off the lamp. Also in this case, as in the preceding e"periment, ispossible to control the turning on5off by simply acting on the push'button switch. 6owe*er the circuit turns on2i.e. the light is on3 7ust during the time inter*al the button is pushed in.

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Experiment #

RELATED TOPICS:

8oltmeter usage

The purpose of the e"periment is to illustrate the *oltmeter usage.

ITEMS NEEDED:


two resistors 1%%

bridging plugs

*oltmeter

T&EOR#:

A multimeter is an electrical instrument capable of measuring *oltage, current, and resistance. There are

two kinds of multimeter: digital and analogue multimeters. Digital multimeters ha*e numerical displays, forindicating the physical $uantity we want to detect 2*oltage, current, or resistance3. Analogue multimetersindicate these $uantities by means of a mo*ing pointer o*er a graduated scale.In this e"periment, we will familiari9e with the measurement of *oltage. In this case we use the multimeter asa *oltmeter. Therefore, the *oltmeter is an instrument for the measure of potential differences between twopoints.The points can be located anywhere in the circuit but normally are placed at the terminals of an acti*ecomponent 2battery, diode, transistor, etc.3 or passi*e component 2resistor, capacitor, inductance3.(ue to this fact the *oltmeter must be connected always in parallel; to the component and ne*er in series;.If we should use the *oltmeter in series; we would obtain a null tension since the points were in contactbefore the insertion of the *oltmeter. In order to illustrate how the *oltmeter works let-s try to reali9e a circuitwith two resistors and one battery.

NOTE:

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&ince e*ery analogical *oltmeter has an inner resistance that is in parallel to the passi*e component 2in ourcase the resistor R13 the circuit is ob*iously modified by the presence of this resistance.If we want that this circuit modification is negligible, the *oltmeter-s inner resistance must be *ery high.

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Experiment $

RELATED TOPICS:

8oltmeter usage


ITEMS NEEDED:


two resistors 1%%

bridging plugs

*oltmeter

T&EOR#:

In this pre*ious e"periment we used two resistors of the same *alue 2e.g. /1


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that is the *oltage is e$ually di*ided into the two resistors. The reader can try to insert a different resistance*alue: a new *alue, in agreement with the pre*ious e$uation, will be shown by the *oltmeter.

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Experiment %

RELATED TOPICS:

8oltmeter usage


ITEMS NEEDED:


two resistors 1%% and one resistor ##%

bridging plugs

*oltmeter

T&EOR#:

In more realistic applications the output *oltage depends upon the resistance of the load it dri*es. If weconsider a load as a resistor, say /, we can obtain the output *oltage at its end as

Vout

=Vin

R2

/ /RL

R1+ R

2 / /R

L

where Vinis the *oltage at the ends of the *oltage generator, while

R2/ /RL=

1

1

RL

+1

R2

.

In our case, the numerical *alues are

R1=100

R2=100

RL=220 Vin=(1.5+1.5) V=3 V

6ence

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Vout=1.222 V.In the design approach, one could be interested in obtaining a constant output *oltage onto the load. Thedesigner should calculate, on the basis of pre*ious formulas, the desired *alues of resistors / 1 and /# inorder to fulfil the specifications. Try to choose a fi"ed output *oltage and *ary the *alues of the resistors soas to obtain it.

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Experiment &

RELATED TOPICS:

Ammeter usage

The purpose of the e"periment is to illustrate the ammeter usage.

ITEMS NEEDED:


two resistors 1%%

bridging plugs

ammeter

T&EOR#:

The ammeter measures the electrical current that flows in a circuit. =nlike the *oltmeter, the ammeter must be placedbetween points that were connected, this means that there was no difference of potential between them. The ammeteris thus connected in series; because the ammeter breaks the circuit and it is placed in se$uence with respect to theother components of the circuit.As an e"ample we can create a circuit constituted by a battery and two resistances, the ammeter should be placedamong the two resistances.

NOTE:

E*ery analogue ammeter has an internal resistance that, because of the connection in series that breaks the circuit

2in our e"ample it is the connection between the two resistances / 1and /#3, adds itself up into the other componentsof the circuit, always connected in series. This means that the internal resistance can modify the beha*iour of theother components. Therefore to make sure the internal resistance of the ammeter does not affect the circuit, theinternal resistance must be *ery low.

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Experiment '

RELATED TOPICS:

)otentiometer usage

The purpose of the e"periment is to illustrate how a *ariable resistor can be used by an e"ternal user for*arying the *oltage across two points. In this case we show how the *oltage across a lamp can be *aried.

ITEMS NEEDED:


potentiometer 1%% , > ?

lamp holder

lamp bulb

bridging plugs

*oltmeter

T&EOR#:

A potentiometer is an electrical instrument used as a three'terminal *ariable resistor: one is free to mo*eacross the resistor itself, the other two points being its ends. A potentiometer can be used as a continuously*ariable *oltage di*ider with a shaft or slide control for setting the di*ision ratio.In our e"periment, a lamp is directly connected to the *ariable end of the potentiometer. By mo*ing its knob,we correspondingly mo*e the *oltage across the lamp, from % 8 to the ma"imum *oltage pro*ided by the*oltage generator. The *oltage *ariation across the resistor can be obser*ed both with a different lampluminosity and with a different *oltage *alue on the *oltmeter.

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Experiment (

RELATED TOPICS:

)otentiometer usage

The purpose of the e"periment is to illustrate how a *ariable resistor can be used by an e"ternal user for*arying the *oltage across two points. In this case we show how the current flowing along a lamp can be*aried.

ITEMS NEEDED:



lamp holder lamp bulb

connecting leads

bridging plugs

ammeter

T&EOR#:

)otentiometers find their most interesting application as *oltage di*iders, where shaft position determines aspecific *oltage di*ision ratio, as in the pre*ious e"periment has been shown. 6owe*er, there are

applications where we don@t necessarily need a *ariable *oltage di*ider, but merely a *ariable resistor: in thiscase the three'terminal de*ise becomes a two'terminal one. Technically, a *ariable resistor is known as arheotat, but potentiometers can be made to function as rheostats $uite easily. In fact, by simply using thewiper terminal and one of the other terminals, the third terminal can be left unconnected and unused: the

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potentiometer then acts as a rheostat.In our e"periment, a lamp is directly connected to the *ariable end of the potentiometer. 6owe*er, thisterminal is now unconnected at the other end: therefore, a current will flow across the resistor only betweenthe terminal toward the generator and that *ariable. o current will flow in the third terminal. By *arying itsknob we can insert an e$ui*alent *ariable resistance in series with the lamp. As a conse$uence the currentflowing along the lamp will *ary, thus being detectable both by looking at the luminosity of the lamp and by

measuring the current on the ammeter.

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Experiment )

RELATED TOPICS:

8oltammetric method

The purpose of the e"periment is to illustrate a method to measure the *alue of an unknown resistance.

ITEMS NEEDED:

cell holder with batteries 2or dc power supply3

one resistor

bridging plugs

potentiometer

*oltmeter

ammeter

T&EOR#:

The goal of this e"periment is to reali9e a circuit by using two basic instruments as the *oltmeter and theammeter for the determination of an unknown resistance *alue.By using a ( power supply with a *ariable tension % 4 8olt 2where 4 is the ma"imum *oltage pro*idedfrom the generator3 and by reading the potential difference 2V3 at the resistance terminals and the current 2!3that flows into, it is possible to generate a two'column table. In a third column the ratio between Vand ! iscomputable and will pro*ide the *alue in "hmof the unknown resistance R. By using a graph 2V#!)$the slopeof the resulting diagram will pro*ide the same *alue. The reader could co*er the resistor unit with a blacktape and try to calculate its resistance *alue by means of the described method.It should be noted that the ( power supply can be replaced with a proper circuit using a potentiometer, aswe ha*e seen in pre*ious e"periments. +or the sake of simplicity, we ha*e used 7ust this second choice.

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Experiment

RELATED TOPICS:

8oltammetric method

The purpose of the e"periment is to illustrate another method to measure the *alue of a resistance.

ITEMS NEEDED:


one resistor

bridging plugs

potentiometer

*oltmeter

ammeter

T&EOR#:

As in the pre*ious e"periment, the goal is to measure an unknown resistance. This time we place the

*oltmeter before the ammeter, this being a *ery usual situation in the practice. It should be obser*ed that aparticular attention to the inner resistance of the ammeter must be paid. The reader should add this *alue tothat of the unknown resistance.

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Experiment !

RELATED TOPICS:

&eries resistance

The purpose of the e"periment is to illustrate a series resistance circuit.

ITEMS NEEDED:


two resistors

bridging plugs

potentiometer

toggle switch

*oltmeter

ammeter

T&EOR#:

Two resistances R1 and R2 are defined as in erie if they ha*e one terminal in common and if they arecrossed by the same current 2same intensity and sense3. ?hen the resistances are in series the *alue oftotal resistance Ris e$ual to the sum of the *alues of all the resistances, i.e. R= R1+R2.To *erify this relation, the reader can design the circuit illustrated abo*e and then proceed to measure thetotal resistance Rwith the *oltammetric method where the *oltmeter is placed at the terminals of the two inseries resistances.

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Experiment "

RELATED TOPICS:

)arallel resistance

The purpose of the e"periment is to illustrate a parallel resistance circuit.

ITEMS NEEDED:


two resistors

bridging plugs

potentiometer

toggle switch

*oltmeter

ammeter

T&EOR#:

Two resistances R1and R2are defined as being in parallel if they ha*e both terminals connected betweenthe same two sets of electrically common points. This means that they ha*e the same tension drop.If %= 1&R is the conductance of a resistor R obtained when the resistances are in parallel, the totalconductance is e$ual to the sum of all the conductance *alues, that is %= %1+%2or 1&R = 1&R1+1&R2.In order to *erify this, the reader can design the circuit illustrated below and then proceed to measure thetotal *alue of the resistance by means of the *oltammetric method whit the *oltmeter at the end of the circuit.

If for instance R1


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Experiment #

RELATED TOPICS:

hm-s law

The purpose of the e"periment is to illustrate a fundamental physical law: the hm-s law

ITEMS NEEDED:



toggle switch

bridging plug

ammeter

*oltmeter

resistance 1%% , # ? and ##% , # ?

T&EOR#:

The hm-s law is the fundamental law of electricity. It is the historically first relationship between current,*oltage and resistance and was disco*ered by Ceorg &imon hm and published in 1D# 2'he %alvaniiruit !nvetigated *athematiall3. Its principal disco*ery was that the amount of electrical current flowingthrough a metal conductor of a circuit is directly proportional to the *oltage impressed across it, for any gi*entemperature. 6e deri*ed this relationship in a simple mathematical form interrelating all three electricalamounts 2current !, *oltage Vand resistance R3:

V = RI

+rom another point of *iew, this law also indicate how each electrical component shows some degree of

friction to the passage of free electron through it. This sort of opposition to the free motion is 7ust what wedefine as reitane. The hm-s law shows how the amount of current in a circuit depends on the amount of*oltage a*ailable to push electrons through the component, and therefore the amount of resistance in thecircuit to oppose this electron flow. This law can be *erified by means of the circuit abo*e. The reader can

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calculate the *alue of the resistance by applying a *oltage *alue and measuring the corresponding current*alue 2for instance the *alues can be reported in a proper two'column table3. ?e also suggest to change the*alue of the resistance in order to *erify whether the law still holds.

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Experiment $

RELATED TOPICS:

0irchhoff-s current law

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The purpose of the e"periment is to show the application of the first 0irchhoff-s law.

ITEMS NEEDED:


three resistors

bridging plugs

potentiometer

toggle switch

connecting leads ammeter

T&EOR#:

The first 0irchhoff-s law or 0irchhoff-s current law establishes that in e*ery node of a circuit in which two ormore branches are connected, the sum of the input current flows is e$ual to the sum of the output currentflows. In other words, the algebraic sum of all currents entering and e"iting a node must e$ual 9ero.In the circuits abo*e illustrated you can see that the node is located where the three branches of the circuitmeet and precisely where the current 2say i13 from resistance R1is then split into two currents 2say i2 and i33into resistance R2and resistance R3, respecti*ely.If you then obser*e the *alue of current read by three ammeters, you will see that whate*er the *alue of inputtension, the *alue of current i1read by the ammeter and connected in series will always be e$ual to the sumof the current flows i2and i3read by the other two ammeters connected in series to the resistances R2andR3, respecti*ely.In practice you can use 7ust one ammeter at a time: to do this, place the ammeter in the i12or R13 branch anduse two bridging plugs 2or connecting leads, when necessary3 in the other two branches, then write down thecurrent *alue you ha*e read. /epeat the procedure for the i2 2or R23 branch and i3 2or R33 branch,respecti*ely, by mo*ing the ammeter and replacing it with a bridging plug 2or connecting leads, whennecessary3. At the end of the e"periment you should ha*e three different current *alues. +inally, try to *erify ifthe 0irchhoff-s law holds.

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Experiment %

RELATED TOPICS:

0irchhoff-s *oltage law

The purpose of the e"periment is to show the application of the second 0irchhoff-s law.

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ITEMS NEEDED:


three resistors

bridging plugs potentiometer

toggle switch

*oltmeter

T&EOR#:

The second 0irchhoff-s law or 0irchhoff-s *oltage law establishes that in e*ery loop of a circuit the sum of thetension drops on each component must e$ual 9ero. In other words, the algebraic sum of all *oltages in aloop must e$ual 9ero.4ake sure that you consider whether the sign of the tension drop is positi*e or negati*e, a positi*e tension

drop meaning that tension changes from a higher *alue to a lower *alue, a negati*e tension drop being thecontrary.In the picture abo*e you can see that both in the electrical diagram and in the assembly chart twoloops are highlighted, one in yellow 2loop 13 and one in green 2loop #3.If we use a *oltmeter we can obser*e that since the resistances R2and R3are in parallel, their tension dropsV2and V3will be the same, therefore it will be easy to *erify 0irchhoff-s second lawF therefore V0= V1+ V2=V1+ V3where V0stands for the supply tension.

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Experiment &

RELATED TOPICS:


The purpose of the e"periment is to show the application of the first 0irchhoff-s law in a more comple" circuit,where se*eral nodes are present.

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ITEMS NEEDED:


se*en resistors

bridging plugs potentiometer

toggle switch

connecting leads

ammeter

T&EOR#:

In this e"periment we would like to *erify the first 0irchhoff-s law or 0irchhoff-s current law for a circuit thatpresents se*eral nodes. In this case, for the sake of simplicity, let us consider the circuit abo*e 2first picture3with two nodes, indicated with ,1and ,2. ?e want to *erify that the total current 2say i13entering the node,1 is e$ual to the sum of the current through the resistances R2$ R3 and R-2say i23-3 and the current through

R52say i53, that is:i1 = i234 + i5

&econdly, we want to *erify that the current entering the node ,22say i53 is e$ual to the sum of the currentsthrough the resistances Rand R/, that is:

i5 = i6 + i7

+or directly measuring the currents flowing through the different branches, we can simply remo*e thecorresponding bridging plug and insert the connecting leads connected to the ammeter, as shown in thesecond picture. This procedure holds for all bridging plugs. Be careful with the correct polarityG nce youha*e read the *alues on the ammeter, you can compare them with those calculated by means of thetheoretical e$uations.It should be noted that for a better comprehension of this e"periment, a power supply instead of the cellholder with batteries is strongly suggested.

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Experiment '

RELATED TOPICS:


0irchhoff-s *oltage law

The purpose of the e"periment is to show the application of the both 0irchhoff-s laws in a more comple"circuit, where se*eral nodes and loops are present.

ITEMS NEEDED:

cell holders with batteries 2or dc power supply3

four resistors

bridging plugs

toggle switch

connecting leads

ammeter

*oltmeter

T&EOR#:

In this e"periment we would like to *erify both the first and second 0irchhoff-s laws. In this circuit theapplication of the *oltage and the current laws will be illustrated for a more realistic circuit in*ol*ing loops andnodes. It should be noted that the circuit uses two cell holders 2you can take the second holder from anotherkit, if a*ailable3 or one cell holder and an e"ternal power supply.?e can indi*iduate three different loops 2say ABE+, A(+ and B(E3 and one node 2B3, as shown in the

picture.In order to apply both 0irchhoff-s laws we must sol*e the following e$uation system:

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i12 (R1+ R2 ) + i4R4 E1= 0 (loop ACDF)i4R4+ E2 i3R3= 0 (loop BCDE)i12= i4+ i3 (node B)

i.e we can obtain the single currents flowing through the corresponding resistances 2be reminded that i is thecurrent flowing into the resistance R3. The first two e$uations are the 0irchhoff-s *oltage law, while the third

e$uation is the 0irchhoff-s current law. It should be noted that we can not use the third loop e$uation 2loopABE+3 because it is a linear combination of the other two e$uations. This is why we use the node e$uation;.4oreo*er, we can measure the current flowing in each node by simply replacing each bridging plug with anammeter, whilst we can detect the *oltage across a resistance by placing a *oltmeter at its terminals.

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Experiment (

RELATED TOPICS:

Electric circuit

Battery

The purpose of the e"periment is to illustrate the difference between batteries connected in series and inparallel

ITEMS NEEDED:


lamp bulb

lamp holder

bridging plugs

T&EOR#:

There is a fundamental difference when you connect batteries in series or in parallel. et us firstly consider asingle 1.> *olt battery connected in parallel to a 1.> *olt lamp. As you can easily *erify, the lamp will normal;bright. If we connect the batteries in parallel they will both gi*e 1.> *olts. The lamp will still normally; brightbut the batteries will last twice as long as the pre*ious case. If we finally connect the batteries in series theywill gi*e H *olts. The lamp will be *ery bright but will blow *ery $uicklyGIt should be noted that some batteries are rechargeable that is an in*erse current can flow through them

without e"ploding; the battery, as could happen for a not rechargeable one. ther than its *oltage a batteryis characteri9ed by its total charge that we can e"press inAmp#hour2Ah3. &ince 1 amp is actually a flow rateof 1 coulomb of electrons per second, in an hour we ha*e H!%% seconds, then 1 amp#hour=300 eond. Inother words it is a measure of the current the battery can pro*ide for an hour. 4ore e"actly, a battery with a

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capacity of 1 amp'hour should be able to continuously supply a current of 1 amp to a load for 1 hour, or #amps for 15# hour, or 15H amp for H hours, and so on, before becoming completely discharged.If we are dealing with a rechargeable battery we must care to use a ma"imum in*erse current e$ual to 151%of its Ah'*alue.In a parallel configuration, two not rechargeable batteries present in practice different electrical properties,and the o*erall beha*iour will depend on the worst one;. If both batteries are rechargeable we can increase

the o*erall Ah'*alue. As a conse$uence if one of them will e"haust before the other, it can be recharged bythe other one.

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Experiment !)

RELATED TOPICS:

Electric circuit )hototransistor

The purpose of the e"periment is to illustrate the different brightness of a lamp due to different batteryconfigurations

ITEMS NEEDED:


lamp bulb

lamp holder

bridging plugs

phototransistor

multimeter

T&EOR#:

In order to illustrate in a $uantitati*e way the different brightness of a lamp due to different batteryconfigurations as shown in the pre*ious e"periment, we can make use of a phototransistor and a multimeter.By referring to the same circuit as abo*e we can place a phototransistor connected to a multimeter in

measuring resistance modality;. This can be done by connecting the black test lead to the 4 7ack andthe red one to the 855f 7ack, the polarity of red being ;. Then it suffices to set the rotary switch at the

desired position and connect test leads across the phototransistor. A phototransistor amplifies the current

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generated by the light striking the acti*e area 2compared to photodiodes, a large output current can beobtained, e*en from a small acti*e area3. The reader can therefore $uantitati*ely *erify the differentresistance due to the different lamp brightness 2batteries in series *ersus batteries in parallel3.

3#


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Experiment !

RELATED TOPICS:

Electric circuit

&witch

The purpose of the e"periment is to illustrate the use of a switch for acti*ating a part of a circuit ande"cluding another one

ITEMS NEEDED:


lamp bulb

lamp holder

bridging plugs

potentiometer

toggle switch

T&EOR#:

The circuit abo*e illustrated shows how to use a switch. A switch can be used to simply open5close a circuit,as shown in a pre*ious e"periment. But can be used in a more smart way to de*iate the current in a differentbranch at a time. This can be *erified by using a lamp: when the switch is the current flows through theupper branch and the lamp will bright at its ma"imumF when it is ++, the current flows through the lowerbranch and the lamp will be less shining. In the latter case, the lamp brightness will depend on the resistanceof the potentiometer 2used here as a rheostat3. ?e want to obser*e that it e"ists a single de*ice for obtainingthe same results called deviator ithwith two positions 2: ormally losed and : ormally pen3.

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Experiment !!

RELATED TOPICS:

Electric circuit

apacitor

The purpose of the e"periment is to illustrate two capacitors in series

ITEMS NEEDED:


bridging plugs

*oltmeter

toggle switch

two capacitor

T&EOR#:

apacitors are passi*e components able to store a gi*en amount of charge depending on its capacity .They are mainly used when a stable constant *oltage is needed in a part of a circuit. As resistors they can beconnected in erieor in parallel. They are in eriewhen they ha*e 7ust one point in common and then theywill ha*e the same charge too, when connected to a *oltage generator. As through resistors in series thesame current 2defined as number of charges per unit of time3 will flow, in static conditions capacitors in seriesha*e the same charge . In order to *erify that, we can read the *oltage by means of a *oltmeter in e*erycapacitors and then multiply the read *alue by the corresponding *alue of capacitance 21and 23:

Q1 =C

1V1 =C

2V2 =Q

2 =Q

In order to *erify the e$ui*alent series capacitance

C= C1C2

C1+ C2a multimeter in capacitance'modality can be used after the battery has been remo*ed from the circuit.

40


41/72

Experiment !"

RELATED TOPICS:

Electric circuit

apacitor

The purpose of the e"periment is to illustrate two capacitors in parallel

ITEMS NEEDED:


bridging plugs

*oltmeter

toggle switch

two capacitor

T&EOR#:

In this e"periment we deal with capacitors in parallel. They are in parallel when they ha*e both ends incommon 2paying attention to the right polarity3 and then they will ha*e the same *oltage too, whenconnected to a *oltage generator. In order to *erify that we can read the *oltage by means of a *oltmeter ine*ery capacitors. The e$ui*alent parallel capacitance is

C = C1 + C2

as one can *erify by means of a multimeter in capacitance'modality to be used after the battery has beenremo*ed from the circuit.

41


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Experiment !#

RELATED TOPICS:

Electric circuit

apacitor /esistance

/ circuit

The purpose of the e"periment is to illustrate the charging phase of a / circuit

ITEMS NEEDED:


bridging plugs

*oltmeter

toggle switch two capacitor

one5three resistor

push button switch

T&EOR#:

et us describe the beha*iour of a *ery important circuit known as R iruit. In our scheme we employ a

resistance /


43/72

flows through the resistance. ow the capacitor is completely charged and its charge is e$ual to J


44/72

Experiment !$

RELATED TOPICS:

Electric circuit

apacitor /esistance

/ circuit

The purpose of the e"periment is to illustrate the discharging phase of a / circuit

ITEMS NEEDED:

bridging plugs

*oltmeter

toggle switch

two capacitor

one5three resistor

T&EOR#:

et us refer to the pre*ious e"periment, that concerning the charging phase of a capacitor. nce thecapacitor is completely charged, switch the generator off, then insert a bridging plug between the firstresistor and the lower bridging plugs. This will inacti*ate the generator 2pay attention to switch it off otherwiseremo*e it directly from the circuit to a*oid a complete discharge of the batteries3.ow the discharging process will start, following the law:

V(t) = Ee t/

nce the process begins, an in*erse current tends to flow from the capacitor through the resistor /.orrespondingly the *oltage across the capacitor tends to decrease as well as the current flowing throughthe resistor. The process will stop when all charges are no longer across the capacitor, the *oltage is e$ualto 9ero and no more current flows through the resistance. ow the capacitor is completely discharged.

44


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It can be obser*ed that a complete discharge will occur only after an infinite time. In practice, we can

con*entionally assume that the capacitor is completely discharged when t


46/72

Experiment !%

RELATED TOPICS:

Electric circuit

apacitor /esistance

/ circuit

The purpose of the e"periment is to illustrate the charging and discharging current in a / circuit

ITEMS NEEDED:

bridging plugs

*oltmeter

toggle switch

two capacitor

potentiometer

lamp bulb

lamp holder

push button switch

T&EOR#:

et us refer to the pre*ious e"periment, the one concerning the charging phase of a capacitor. ?e want toshow the e"istence of the direct 2charging phase3 and in*erse 2discharging phase3 current by using a lamp in

series with the parallel of two capacitors. ?e substitute the resistors with a potentiometer used as a rheostatbecause we need a *ery low resistance in order to appreciate the current flowing through the lamp. Thereader can mo*e the knob of the potentiometer to ad7ust its e$ui*alent resistance. In this way we obtain a*ery low time'constant. In the charging phase a current will flow through the lamp, causing a $uick flash. The

46


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48/72

Experiment !&

RELATED TOPICS:

Electric circuit

Inductor /esistance

/ circuit

The purpose of the e"periment is to illustrate the properties of a / circuit

ITEMS NEEDED:

bridging plugs

*oltmeter

ammeter

toggle switch

one coil potentiometer

T&EOR#:

?hene*er electrons flow through a conductor, a magnetic field will de*elop around it. ?hereas an electricfield flu" between two conductors allows for an accumulation of free electron charge within those conductorsas we ha*e seen in the case of a capacitor, a magnetic field flu" allows for a sort of KinertiaK to accumulate inthe flow of electrons through the conductor. !ndutor are components designed to take ad*antage of thisphenomenon by shaping the length of conducti*e wire in the form of a coil. This shape creates a strongermagnetic field than what would be produced by a straight wire. Therefore, inductors ha*e the e"act opposite

characteristics of capacitors. ?hereas capacitors store energy in an eletri field 2produced by the *oltagebetween two plates3, inductors store energy in a magnetifield 2produced by the current through wire3. Thus,while the stored energy in a capacitor tries to maintain a constant *oltage across its terminals and opposechanges in *oltage, the stored energy in an inductor tries to maintain a constant current through its windings

4#


49/72

and oppose changes in current. A fully discharged inductor 2no magnetic field3, ha*ing 9ero current throughit, will initially act as an open#iruitwhen attached to a source of *oltage 2as it tries to maintain 9ero current3,dropping ma"imum *oltage across its leads. *er time, the inductor@s current rises to the ma"imum *alueallowed by the circuit, and the terminal *oltage decreases correspondingly. nce the inductor@s terminal*oltage has decreased to a minimum 29ero for a KperfectK inductor3, the current will stay at a ma"imum le*el,and it will beha*e essentially as a hort#iruit.

The circuit depicted in the picture is a classical /'circuit. ?hen the switch is first closed, the *oltage acrossthe inductor will immediately 7ump to battery *oltage 2acting as though it were an open'circuit3 and decaydown to 9ero o*er time 2e*entually acting as though it were a short'circuit3. 8oltage across the inductor isdetermined by calculating how much *oltage is being dropped across /, gi*en the current through theinductor, and subtracting that *oltage *alue from the battery to see what is left. ?hen the switch is firstclosed, the current is 9ero, then it increases o*er time until it is e$ual to the battery *oltage di*ided by theseries resistance. This beha*ior is precisely opposite to that of the series resistor'capacitor circuit, wherecurrent started at a ma"imum and capacitor *oltage at 9ero as we ha*e seen in pre*ious e"periments. Try to*erify this effect by *arying the *alue of the resistance / 2for e"ample by using a potentiometer as indicated

in pre*ious e"periments3. As in the /'circuit we can define a time'constant , that in this case is defined as:

=L

R

All considerations about transient and regime state are the dual of those related to the /'circuit case.

4$


50/72

Experiment !'

RELATED TOPICS:

?heatstone bridge

The aim of the e"periment is to realise an accurate method to measure an unknown resistance.

ITEMS NEEDED:

+our resistors

8oltmeter

Ammeter

T&EOR#:

The ?heatstone bridge is a method for an accurate measurement of resistance. It is particularly interesting

because, being a null method, the sensiti*ity the instrument must ha*e is negligible since the *alidity of thedetermination is not affected by the calibration of the instrument.Being gi*en the electric circuit abo*e depicted, if the ammeter signs a null current *alue, it must be L8 < 8A'8B< % so I1M/"< I1M/1and I1M/1< I#M/#, from which it results

Rx=R1R3R

2

50


51/72

Experiment !(RELATED TOPICS:

/ circuit

6igh'pass filter

The aim of the e"periment is to study a high'pass circuit.

ITEMS NEEDED:

potentiometer

capacitor

lamp bulb

lamp holder

function generator

oscilloscope 2optional3

T&EOR#:

?hen we apply a *ariable signal, for instance a sinusoidal signal by means of a fre$uency generator to an/ circuit, an interesting beha*iour can be outlined. In fact, if we consider the circuit depicted abo*e, bysimply *arying the applied fre$uency, a different response of the circuit can be obser*ed. et us concentrateon the lamp and let us start with the lowest fre$uency the generator is able to apply. The lamp will not lightup. ow, let us slowly increase the fre$uency. ?ith raising fre$uency an increase of the intensity of the lightemitted should be obser*ed. In other words, unless a certain fre$uency has been reached, no current willflow through the lampF but abo*e that fre$uency the current will flow. This circuit beha*es as the so'calledhigh#pa ilter. It can be noted that the uto re4uenholds:

12

cfRC

=

et us recall that the *oltage to be applied should be sufficient to feed the lamp we are using in thee"periment. Be sure to use a lamp *oltage that is consistent with the applied one. ?e want to note that

51


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instead of a resistance, a potentiometer has been used in order to reduce the resistance *alue as soon aspossible.

52


53/72

Experiment ")RELATED TOPICS:

/ circuit

ow'pass filter

The aim of the e"periment is to study a low'pass circuit.

ITEMS NEEDED:

potentiometer

capacitor

lamp bulb

lamp holder

function generator

oscilloscope 2optional3

T&EOR#:

?hen one applies a *ariable fre$uency signal to an / circuit connected as abo*e depicted a dualbeha*iour with respect to the pre*ious e"periment can be obser*ed. et us concentrate again on the lampand let us start with the lowest fre$uency the generator is able to apply. The lamp will now light up. Then, letus slowly increase the fre$uency. ?ith increasing fre$uency a decrease of the light intensity should beobser*ed. In other words, until a certain fre$uency has been reached, a current will flow through the lampFabo*e that fre$uency the current will no longer flow. As a conse$uence this circuit beha*es as the so'calledlo#pa ilter. It can be noted that the uto re4uenstill holds:

1

2cf RC=et us furthermore recall that the *oltage to be applied should be sufficient to feed the lamp we are using inthe e"periment. Be sure to use a lamp *oltage that is consistent with the applied one.

53


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Experiment "

RELATED TOPICS:

/ circuit in series

/esonance

The aim of the e"periment is to study an / circuit in series

ITEMS NEEDED:

/esistor

apacitor

oil

amp holder

amp bulbs

+unction generator

scilloscope 2optional3

T&EOR#:

onsider a spring attached to a mass mo*ing hori9ontally on a surface with friction that we rhythmically pullback and forth. After some initial transient period, the mass will mo*e back and forth at the fre$uency of theforce we apply.+or a moment, let-s step back and consider the original undri*en damped oscillator. ?e ha*e two forcesacting on the mass: the spring force 67and the damping force 8 d7&dt.?e now add a periodic dri*ing force which can be written 90o :t.

ewton-s second law now readsmd2x

dt2 = -kx - b

dx

dt+ F

0cos t 4E/CE+/4AT

54


55/72

nce a steady state has been reached, the motion will be sinusoidal with the same fre$uency as the dri*ingforce. The general form of the solution is the familiar cosinusoidal function

x t( ) = Acos t + ( ) 4E/CE+/4ATThe amplitudeAand the phase ;can be found by substituting into and re$uiring that the resulting e$uationbe satisfied for all t. Before proceeding, we rewrite slightly.(i*iding through by m, we ha*e

d2

xdt2

= Kmx b

mdxdt

+ F0m

cost 4E/CE+/4AT

It is con*enient to introduce the following notation, much of which is probably familiar by now.K

m=

0

2

4E/CE+/4AT

b

m= 4E/CE+/4AT

0

= Q 4E/CE+/4AT

F0

K= F0

m02= ALF 4E/CE+/4AT

E$uation now takes the form

d2x

dt2 +

0

Q

dx

dt + 02

x= 02

ALFcos

t 4E/CE+/4AT

ote the difference between :0and :. The symbol N%refers to (


56/72

called a reonane;. The plot is A&AL9*ersus rfor = 0.5$ 1$ 2$ -$and.There are a lot of e"amples ofresonance in physics and engineering.Oou may wish to *erify that the actual peak of the amplitude cur*e is reached not for r = 1but for

r= 11

2Q2

+or large 2small damping3 this makes *ery little differenceF for small *alues of 2large damping3 this shift

becomes significant. +or *alues of smaller than 2 / 2 , there is no ma"imum near the resonance at all. Thiscan all be seen from the preceding figure.+or *ery high fre$uencies 2r>>13,Ais *ery nearlyAL9&r

2independent of . In this case, it is the accelerationterm in that predominates. ?e see then that both for *ery low fre$uencies and for *ery high fre$uencies thedamping is relati*ely unimportant. It is mainly around resonance that the *alue of determines thebeha*iour of the oscillator.The following figure shows the phase of the oscillator in a graph with the same hori9ontal a"is as thepreceding figure and for the same *alues of .

ote that the phase is negati*e, i.e., the oscillator always lags behind the dri*ing force. In the limit of large ,the phase lag abruptly 7umps from 9ero to?as the dri*ing fre$uency passes through :0.If these results were only applicable to a mass on a spring dri*en by a periodic force, they would be hardlyworth studying. 6owe*er, there is an enormous *ariety of problems that all lead to e"cept for the meaning ofthe constant parameters. E"amples include a table on springs set in motion by a *ibrating floor, the

*ibrations of an ammonia molecule dri*en by absorption of infrared radiation, and the charge on a capacitorin an / circuit dri*en by an A *oltage.The correspondence between mehanialand eletrialparameters is shown in the following table:

4echanical &ystem Electrical &ystem

(isplacement x harge '(ri*ing +orce (ri*ing 8oltage (4ass m Inductance L(amping onstant ) /esistance R&pring onstant * /eciprocal apacitance +,C/esonant +re$uency -*,m /esonant +re$uency +,-LC/esonance ?idth . / ),m /esonance ?idth . / R,L

=sing this correspondence, the fundamental e$uation becomes

Ld2q

dt2 +

1

Cq+ R

dq

dt=V

0cos t

and all the other e$uations can be written in terms of the electrical parameters. In particular:

0 1 LQR C

= =

PROCED$RE:

The circuit should be connected to the function generator 2which pro*ides the dri*ing *oltage3 and, ifpossible, to the oscilloscope. alculate the resonant fre$uency for the components and the of the circuit.

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In order to *isually achie*e this phenomenon it-s possible to use the lamp holder and obser*e the lamp bulbbecoming brighter at the resonance, as shown in the circuit./ecord the *oltage across the capacitor and the delay of the peak of this signal relati*e to that of the dri*ingsignal as a function of fre$uency for a number of fre$uencies abo*e and below the resonance. Take se*eraldata points where the gain cur*e changes rapidly near resonance.It should be obser*ed that, as a conse$uence of the abo*e e$uations, if one wants to reduce the resonant

fre$uency, he should increase the *alue of the inductance L. This could be done by inserting theferromagnetic core inside the coil. +urthermore, if a better should be obtained, a resistance in parallel tothe lamp could be inserted.It can be obser*ed that the circuit we ha*e designed acts as a pa#8and ilter: only a band of fre$uency,whose centre is the resonant fre$uency, can e"cite the lamp, the others being stopped.E"actly the same theoretical considerations apply to an / circuit where the inductor and capacitor areplaced in parallel.The only difference is that at the resonant fre$uency we get a current minimuminstead of a ma7imum. Thisphenomenon is called antireonane.It can be obser*ed that the circuit we ha*e designed acts in this case as a top#8and ilter: only outside thecentral band of fre$uency, whose centre is the resonant fre$uency, the lamp can be e"cited, the central bandbeing stopped.

5"


58/72

Experiment "!

RELATED TOPICS:

4agnetic field 4agnetic compass

4agnet

The aim of the e"periment is to illustrate the physical beha*iour of a compass.

ITEMS NEEDED:

compass

cylindrical magnet 2diam. !"H> mm3

T&EOR#:

Before e"plaining how compasses work, let us briefly recall the fundamental laws of magnetism. Theproperty of certain types of minerals to attract the metal iron was already known by the ancient Creeks andmuch earlier in the +ar East. A particular mineral rock, called lodetoneor magnetite, was well'known andemployed in practical applications 2for e"ample, as an aid of na*igation3. It was obser*ed that a piece of thismineral, free to rotate, tends to orient itself in a north'south direction 2by referring to the Earth polarity3.Another surprising property of ob7ects like magnetite is that they possess two different poles, namely northand outh. Already during the 4iddle Ages has been pro*ed that, unlike electric charges, in magnetic ob7ectsone of these poles cannot be isolated by cutting in someway a piece of them. In fact each of the resultingblocks will be a magnetic ob7ect itself possessing its own pair of north and south.

5#


59/72

ike electric charges, poles of the same signs repel one another, whereas opposite poles attract. This forcee"tends itself all o*er the space and can be *isibly mapped by placing the magnet under a sheet andsprinkling *ery small pieces of metals or iron filings on top. The filings will align themsel*es with the magneticfields, gi*ing rise to continuous lines from one pole to the other. The resulting shape is what has been calledmagneti ield.

ur Earth is a sort of big piece of magnetite, thus beha*es as the little lodestone described abo*e andpossesses its own magnetic field 2also known as geomagnetic field3. This phenomenon could be easilyobser*ed if one holds a simple compass: where*er you stand on Earth, it will point toward the orth )ole.The reader can immediately *erify that by using the magnetic compass pro*ided in this kit. )lace it no matterwhere and obser*e that its red arrow will $uickly point toward the orth )ole. This ama9ing property was*ery useful in the past 2and nowadays too3. Imagine for e"ample to be in the middle of the ocean, withoutany idea where the coast is. Pust use your compass: the needle will you show which direction the orth is in.A magneti ompa consists of a small magnet or needle balanced on a pi*ot point without friction. neend of the needle is generally red'coloured to indicate that it points towards north. ow, one could ask why acompass points towards the orth )ole. In order to answer this $uestion, one can imagine as if inside theEarth there e"ists a gigantic bar magnet buried in the core and generated by comple" heat phenomena

in*ol*ing magnetic materials at *ery high temperature. This imaginary magnetic bar has its south endcoinciding with the orth )ole, and *ice *ersa for the &outh )ole. As a conse$uence the north end of thecompass tends to be attracted by the south pole of this bar.It should be noted that the geomagnetic field is *ery weak on the Earth surface 2this is why we need toeliminate as more as we can the friction of the pi*ot point and to use a lightweight magnet3. Thus, e*eryob7ect forming around itself a magnetic field stronger than the geomagnetic one, will redirect the needle ofthe compass with respect to the geographic orth )ole. Try for instance to put the cylindrical magnet close tothe compass needle. A re'orientation of the needle will be obser*ed.

5$


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61/72

Experiment "#

RELATED TOPICS:

4agnetic field Inductance

4agnetic compass

The aim of the e"periment is to illustrate the effect of a coil on a magnetic compass.

ITEMS NEEDED:

push button switch


coil 1% m6

ferromagnetic core 2he"agon steel screw3

paper clip 2not included in the kit3

magnetic compass

T&EOR#:

As we ha*e already seen, a circular magnetic field de*elops around a wire where a current is flowing and ittends to weaken as you mo*e away from the wire. Both phenomena, i.e. the fact that the lines areperpendicular and the farther they get from the centre the weaker they are, can be *erified with a magneticcompass. 4o*e it near the wire in different positions and note how the needle tends to swing.

&ince the magnetic field around a wire has a circular distribution, we could increase its intensity by winding itto a series of circles. Thus, we ha*e created a oil. +or e"ample, if one wraps a wire around a nail, say 1%times, then connects the coil to a battery and brings one end of the nail near the compass, a larger effect onthe compass needle should be obser*ed. This can be *erified in a proper way by obser*ing what happens

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when we use a coil as shown in the picture abo*e. ompare the beha*iour, by using the magnetic compass,with and without the nail 2ferromagnetic core3. Try finally to in*ert the polarity of the batteries and repeat thee"periment. It should be noted that the nail acts as a magnet only when a current is flowing from the battery.In other words we ha*e created an eletromagnet: it has the properties to attract small steel things, but onlywhen a current is impressed. As an e"periment, put a paper clip on the nail, push the button in order to applya current to the coil: the clip will be attracted by the nail. nce the button is released, the current will stop

flowing and the clip will fall down, showing the temporary beha*iour of the nail as a magnet.

62


63/72

Experiment "$

RELATED TOPICS:

Inductance

&elf'induction coefficient 4utual induction

The aim of the e"periment is to calculate the theoretical *alue of the self'induction coefficient of a coil andthen to compare it with the nominal *alue of 1% m6.

ITEMS NEEDED:

oil 1% m6

T&EOR#:

et us consider the following figure representing the section of a ingle laer coil in air wrapped around asupport of null thickness

If L>> R, as it-s known, the induction magnetic induction field inside the coil is parallel to the a"is and nearlyuniform gi*en by

0 0

NB I

L= 4E/CE+/4AT

The flu" of@0through each turn can be found by multiplying@0with the area= R2of the turn. The coil is

formed by ,turns, so the total flu" is @0,2 2

0 0 0 N R

B B NS IL

= = 4E/CE+/4AT

since 05 6 siB L I = we get that the self'induction coefficientLiis2 2

00

si

B N RL

I L

= = 4E/CE+/4AT

If it is not true that L>>R we would ha*e2 2

2 2

0 2 si

N RL R L R

L

= + .

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+or a multi laer coil in air, we can consider each layer as a single layer coil wrapped around the precedingone 2we appro"imate this wrapping as a perfect doubling of the layer as shown in the following figure3. To bee"act, from the point of *iew of the e"ternal coils, the core is constituted by copper and air. But copper isdiamagnetic and so we can neglect the magneti9ation effects.

we ha*e2 1

22 2 2 2

1

0 00 0

2

R Rnumberof layers r

i

i isi

N x N R r i

LL L

= =

+ = =

4E/CE+/4AT

where we ha*e used appro"imate e"pressions for "iand with 2 12R R

r

we indicate the nearest integer to

2 1

2

R R

r

.&ince

,B2rLfrom e$uation we ha*e2 1

220

120

2 4

R R

r

sii

LL R r i

r

=

= + 4E/CE+/4AT

or, by appro"imating 2 12R R

r

with an integer

( ) ( )2 20 2 1 1 1 2 2 2 13

2

24si

L r R R R R R R r R RL

r

+ + + + = 4E/CE+/4AT

which is an intrinsically positi*e constant depending only from geometrical factors.

By substituting the *alues 2L= 20mm$ r=0.11 mm$ R1=5mm$ R2=15mm3 we get LimCto be compared with

the nominal *alue of 101mC.This difference can be essentially e"plained by considering also the mutual induction between turns that

instead we ha*e neglected.

64


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Experiment "%

RELATED TOPICS:

Inductance 4agnetic energy and mechanical forces

/elati*e magnetic permeability

4agnetic hysteresis

The aim of the e"periment is to calculate the theoretical *alue of the sucking force of the coil and then tocompare it with the e"perimental one. In this way it-s possible to obtain an appro"imate *alue for the relati*emagnetic permeability of the ferromagnetic core.

ITEMS NEEDED:

coil 1% m6

ferromagnetic core 2he"agon steel screw3

T&EOR#:

By referring to the pre*ious e"periment we ha*e that the self'induction coefficient Liof the coil is

( ) ( )2 20 2 1 1 1 2 2 2 103

2

24si

L r R R R R R R r R RL kL

r

+ + + + = = 4E/CE+/4AT

where the6 coefficient is determined only by geometrical factors. ?hen the ferromagnetic core is introducedby an amount7 we can imagine the entire system as composed by two subsystems placed side by side: onecoil of length7 with a ferromagnetic core inside and a second coil of length (L#7)with air core.

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In the first coil the self'induction coefficient must be multiplied by r2the relati*e magnetic permeability of theferromagnetic core3 since the magnetic induction field is 0rB B= . 6ere we made the appro"imation of theferromagnetic core *iewed as isotropic and homogeneous.&o, the total self'induction coefficient is

0 0 6

si rL kx k L x = + 4E/CE+/4AT

The current !that flows through the coil-s turns placed at a distance from the coil-s a"is is2 2 3

22 2

yV V S V r V r r V r I

R l yN yL y L

= = = = =

4E/CE+/4AT

where Q is copper-s electric reitivit.By integrating we ha*e that the total current! is

2

1

3 3

2

1

ln

R

R

RV r r V I dy

y L L R

= =

4E/CE+/4AT

&ince we want to obtain the force acting on the ferromagnetic core we ha*e to calculate the magnetic energyof the inductance

21

2M siU L I= 4E/CE+/4AT

By applying e$uations , , and we ha*e

( ) ( )

( ) ( ) ( )

2 2

0 2 1 1 1 2 2 2 1

3

22 2 30 2 1 1 1 2 2 2 1 2

3

1

217

2 24

28 ln

24

r

M

x r R R R R R R r R RU

r

L x r R R R R R R r R R Rr V

r L R

+ + + + = +

+ + + + + 4E/CE+/4AT

The force is gi*en by 2see details later3

.

Mx

I const

Uf

x =

= +

4E/CE+/4AT

or

( ) ( ) ( ) ( )23 3 3 2 2 2

0 2 1 2 1 1 2 2 1

2 2

2

1 2 3lo)

4#

r

x

r R R r R R r R R V Rf

L R

+ + + =

4E/CE+/4ATand since ris positi*e the force is positi*e, hence attracti*e.

NOTE:?e used e$uation because if we start from the totalenergy of the systemE'= E*+ E%, we ha*e to consideralso the work done by the power supply when inserting or e"tracting the ferromagnetic core:

( ) ( )

. . .

M GTx

I const I const I const

U UUf

x x x= = =

= =

4E/CE+/4AT

&ince21 1 1 1 1

2 2M M M

si

U dU dU dt d d L I I

x dt dx dt dt dt

= = = =

and1 1 1

G G G

U dU dU dt dI emf Ix dt dx dt dt = = = = 2at constant !3, we get e$uation

( ) ( )

. .. .

1 1

2

M GT Mx

I const I constI const I const

U UU Udf I

x x x dt x= == =

= = = + = +

4E/CE+/4AT

E!PERIMENTAL PROCED$RE:

If we place the coil in a *ertical plane, with the a"is parallel to the gra*ity acceleration, we can compare the

sucking force with the weight force of the ferromagnetic core. This is possible by regulating the *oltage 8

applied to the coil.By considering the geometrical *alues used in the e"periments abo*e 2L= 20mm$ r=0$11 mm$ R1=5mm$

R2=15mm0 and the physical *alues1F1./710#m$ 0=-710#/&m$ V=5./V3 we get a force that is e"actly

66


67/72

e$ual to the weight force 2 g 0.0 ,3 of the ferromagnetic core.

If we appro"imate the ferromagnetic core ha*ing a characteristic @(C) linear, we obtain a method todetermine the relati*e magnetic permeability.

By e$uating e$uation with %.%D and sol*e for rwe obtain a *alue

650r

Therefore, pay attention to the appro"imation we ha*e made. The characteristics @(C)is in general not linearand so by trying again different applied *oltage you can get different results. This is due to the magnetichysteresis of the ferromagnetic core.

6"


68/72

*PP+NI-

Basics of experimental error theory

/e can say that e-erythin) e 9no abo!t the physical orld has an inherent !ncertainty. %n

partic!lar, hen e experimentally in-esti)ate somethin) there is alays an :experimental error;and an :experimental precision;. Since one of the main feat!res of experiments is their

reprod!cibility, it is -ery important to deal ith this s!b find the density of a solid rubber cube .

o ?irst trial with ery raw instruments. /e can estimate that the mass of the c!be is

nearly 50 ) and the len)th of a side is nearly 6 cm. So the density o!ld be>

3 0,2314#...

M M

V L= = = . =here are many @!estions> :/here can % stop ith decimaldi)its to comm!nicate my res!ltA; :%s it better to ha-e precision on the mass

meas!rement or on len)th meas!rementA; :o do e combine o!r experimental

error on the mass meas!rement ith the experimental error on len)th meas!rementA;

o Second trial with more accurate instruments. By !sin) an electronic balance and

a meter stic9 % find a mass of 60) and a side len)th of 5,4 cm. So the density o!ld

be>

3 0,3#103$4"5...

M M

V L= = = . /e still need to anser the @!estions p!t abo-e b!t e

also ha-e to anser a ne @!estion> :/hat ma9es this trial better than the first oneA;

o =hird trial with much more accurate instruments to measure the side length. %f

e impro-e the acc!racy of the len)th meas!rement, for example by !sin) a -erniercaliper, the problem becomes more in-ol-ed. =his is d!e to the fact that e do not

)et the same res!lt if e ta9e more than one meas!re. %nstead e ha-e a set of

different meas!rements li9e 5,455 cm 5,425 cm 5,465 cm . =h!s e are a)ain

faced ith the @!estion> :/hich one of the meas!rements 5,455 cm 5,425 cm

5,465 cm do sho!ld % ta9eA;

=herefore, the more e analyse the problem the more it )ets in-ol-ed. =o search for a possible

sol!tion e can start from the third trial and obser-e that, )enerally spea9in), hen e impro-e the

acc!racy of an instr!ment e reach a point at hich the experimental res!lts are not fixed b!t are

scattered aro!nd a -al!e as ill!strated in this )raph>

6#


69/72

%f the n!mber of meas!rements Nis )reater then abo!t 30, the distrib!tion of the experimental data

is bell*shaped and has a -al!e -for hich there is a maxim!m and aro!nd hich the data are

scattered in a nearly symmetrical ay. %t is also possible to distin)!ish a -al!e that determines an

inter-al aro!nd -into hich a si)nificant percenta)e of the meas!rements are placed. /e need to

anser the @!estions> :%s -the best estimate of o!r meas!rementA;, :o m!ch can e rely on

this -al!eA; and :/hat percenta)e of the meas!rements are in the inter-al -/ and -0 A;

=o express these @!estions mathematically, e co!ld try a prototype f!nction that fits o!r data and

that expresses the probability to )et a partic!lar meas!rement -al!e>

this is the )raphical representation of the f!nction2

x

f x e=

%f e ant to centre the f!nction aro!nd the -al!e -e !se the expression x/-in place of x, and if

e ant to control the scatterin) of the meas!rements aro!nd -its possible to di-ide 1x/-2!by

!!.

=he folloin) fi)!re shos f1x2ith-3! and 34.$4!

- - -

?inally, if e ant to control the area !nder the c!r-e e ha-e to m!ltiply it by a normaliDation

factor *

that o!ld depend on .

=herefore o!r prototype f!nction is>

fx= "

e

x #( )222 4E/CE+/4AT

here - is the -al!e for hich e ha-e the maxim!m and determines ho the meas!rements are

scattered aro!nd -. =his is called a :Ea!ssian f!nction; or a :Formal f!nction;, b!t the !nderlyin)

data represent a distrib!tion still called Ea!ssian and not a f!nction. %t can be pro-ed that the

Ea!ssian distrib!tion is deri-able from the binomial distrib!tion ass!min) that the n!mber of

meas!rements Nand remains constant.

=he physical meanin) of all this is that e do not describe a meas!rement ith a sin)le n!mber b!t

rather ith a set of -al!es each one ith its on probability to appear as an experimental dat!m.

=his probability is )o-erned by the :Ea!ssian distrib!tion;. =here is an analo)y ith @!ant!m

mechanics for example ith the a-e pac9et of a particle here the interpretation is that if e

ma9e a meas!re of the position of the particle then the probability to obtain a partic!lar -al!e is)o-erned by the Ea!ssian f!nction and is ne-er a ell defined fixed -al!e.

6$


70/72

&et !s determine the -al!e of *

in . /e m!st ha-e a probability of 1 to )et a meas!rement in the

ran)e from *to Gthat is, if e perform a meas!rement e are certain to )et some 9ind of res!lt

no matter ho lar)e or ho small that res!lt is>1

12

f x dx "

+

= =

=o )i-e an interpretation of e can as9 hat happens if e are only interested in the probability

of findin) meas!rements in the ran)e from -/ to -0 instead of the ran)e from /to0>

2

12

1

1 0.6#

2

t#

#f x dx e dt

+ +

= :

so , also called then :standard de-iation; ! is called :-ariance;, is the amo!nt of !ncertainty e

ha-e to allo for, in the most probable -al!e -, if e ant to claim a ro!)hly 6#H chance of

correctly predictin) the res!lt of any sin)le meas!rement.

=o determine -,also called the :mean -al!e;, e consider a set of Nmeas!rements x, x!, 5, xN.

=he probability to )et a sin)le res!lt beteen xiand xi0dxis>

( )2

221

2

ix x

i$ e

=

so the probability to )et all the res!lts -ieed as independent e-ents is>

( )2

122

1 2

1...

N

i

i

x x

N N$ $ $ $ e

=

=

Since e are spea9in) abo!t the probability P to )et all the res!lts and e can s!ppose to ha-e

already done o!r experiment ith a set of real res!lts hat sho!ld be the -al!e of PA

%f e accept the maxim!m li9elihood principle e can ma9e an analo)y ith entropy and say that P

is proportional to the entropy obtained from o!r experiment. =he -al!e - m!st be a point of

maxim!m entropy. By the second principle of thermodynamics e ha-e to maximiDe P, otherise

said -is the -al!e of xthat minimiDe the exponent> ( )2

1

0N

ii

dx x

dx =

= from hich it res!lts>

1

1 N

i

i

# xN =

= 4E/CE+/4AT

that is, the mean -al!e - is the arithmetic mean and describes all the collected data since it is the

-al!e for hich the maxim!m entropy is obtained for o!r set of data.

=o determine e can proceed in the same ay

( )2

1

22

10

N

i

i

x x

N

de

d

=

=

from hich>

( )2

1

1 N

ii

x xN

=

= 4E/CE+/4AT

B!t hat sho!ld be !se instead of xin e@!ation A %f e !se then e@!ation is sli)htly self*referential

beca!se2

1

1

1 1 ... ...

N

i N i

i

x x x xN N

=

= + + + +

and the i*esim term appears to times. %t is possible to sho

that the correct -al!e of the standard de-iation is>

( )2

1

1

1

N

ii

# xN

=

= 4E/CE+/4AT

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Clearly is not defined for NI1 e are ass!min) N )reater of nearly 30, otherise there are better

distrib!tions to consider.

S!ppose no e ha-e a f!nction 6of se-eral -ariables 5 , , , ...6Q f a b c= and e ant to 9no ho the

experimental error on each -ariable contrib!tes to 6.

/e can say that by -aryin) the -ariables, the @!antity 6-aries of>...

Q Q QQ a b c

a b c

= + + +

and if e identify o!r !ncertainty xith the standard de-iation xe can say that>

...Q a b cQ Q Q

a b c

= + + +

4E/CE+/4AT

the mod!l!s is d!e to the fact that errors co!ld cancel each other and e ant to consider the

maxim!m error.

/e co!ld do better, obtainin) a smaller -al!e, if the -ariables are normal and independent, by

startin) from ( )2

1

1

1

N

Q ii

QN

=

= here , , , ...6i i i iQ f a b c

= is the i*esim -al!e of 6by ta9in) the i*

esim -al!e of each -ariable of o!r set of data, 5 , , ,...6f " B C= is the mean -al!e of 6by ta9in) the

mean -al!e of each -ariable of o!r set of data.

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 ne)lectin)

terms of hi)her order e ha-e>

( ) ( )

2 2

2 22 2

1 1

1 1...

1 1

a b

N N

Q i ii i

Q Qa b

a N b N

= =

= + +

1 4 4 2 4 43 1 4 4 2 4 43

or2 2

2 2 ...Q a b

Q Q

a b

= + + 4E/CE+/4AT

hich is better of since its alays loer.

S!ppose no that the f!nction 6is

1 N

i

i

# xN =

= . By applyin) e@!ation e)et

1 2

2 2

2 2

1 2

...# x x# #

x x

= + +

4E/CE+/4AT

b!t1

1 1N

i

ii i

#x

x x N N=

= = and 1 2 ...x x = = = and so

#

N

= 4E/CE+/4AT

hich is called :standard de-iation of the mean;. nalo)o!sly to the :standard de-iation;, it tells !s

ho )ood is the mean -al!e -and e can ass!me it as the amo!nt of !ncertainty e ha-e to allo

for if e ant to claim a ro!)hly 6#H chance of correctly predictin) the res!lt of any other mean

-al!e it is possible to obtain.

%t is also !sef!l to spea9 abo!t relati-e errorQ

Q

instead of absol!te error Q . =he relati-e error can be

expressed in percenta)e.

?or example let !s ret!rn to the problem of determine the density of a c!be.

Fo, the f!nction 6is the density hich is f!nction of the mass 7 and the side len)th 8> 3M

L= . %f

60M %= and 54L mm= its easy to find that the mean -al!e is4

3 3 3

603,#1 10

54

% %

cm cm

= =

"1


72/72

By applyin) e@!ation e ha-e that the relati-e error is>3 3

3 4

1 1 1 3 1 3M L M L M L

L L M

M L M L M L M L

= + = + = +

.

%f e can s!ppose the precision of the mass meas!rement is 2M % = and the precision of the len)th

meas!rement is 1L mm = e ha-e>

2 3 1 3, 3H 5, 6 H #, $H60 54

= + + =;

this says it is more important to ma9e a caref!l len)th meas!rement than a caref!l mass

meas!rement.

By applyin) e@!ation e )et a better loer estimate of the density error>2 2 2 2

2 2 2 2

$ 2 $ 16,5H

60 54

M L

M L

= + = + = .

=his means that if e ta9e another meas!rement of density theres a probability of nearly 6#H that

the ne -al!e ill lie beteen 4 33,# 0,2 10 %

cm

.

%t is important to note that since the standard de-iation on density is 0,2 x 10 *4)Jcm3e can stop at

the first decimal di)it 3,# x 10*4

.

4866.10 - electricity system 1

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