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Chapter 1Elements of Physics

In this chapter we survey a number of topics in elementary physics that provide a good back-ground for understanding the instrumentation of science. It is impossible to include everything.Though this course is directed to the sciences in general, we shall mostly be using an electricalterminology. We therefore review this terminology beginning with static electricity. We concludeour survey with topics from heat and light.

Static Electricity

ChargeStatic electricity is the physics of electric charge atrest.1 Charge forms a fundamental constituent of twokinds of particle: the electron and the proton. Both carrythe same amount of charge, namely 1.602 x 10 –19

coulomb (C), but opposite sign.2 The proton is theheavier of the two, being approximately 2000 timesthe mass of the electron.

Electric ForceCharges that have the same sign (like charges) repeleach other while unlike charges attract. The magnit-ude of the force F of repulsion/attraction between twocharges q1, q2 separated by a distance r has the form

F = kq1q2

r2 (N). [1-1]

k is a constant whose value depends on the system ofunits used. In the SI (International) system k has thevalue 9.0 x 10 9 N.m2.C–2. This force is a vector and maybe represented by the arrows drawn in Figure 1-1. Theforces act along the line joining the centers of thecharges. If the charges are held at rest, then the forcesare called electrostatic forces. If the charges wereallowed to move then they would tend to move in thedirections shown.

q1

q1 q2

q2

(a)

(b)

Figure 1-1. Electrostatic forces between charges at rest: likecharges (a) and unlike charges (b).

Electric FieldTwo charges at rest exert electrostatic forces on oneother even though separated by a distance of emptyspace. The forces are exerted when the charges are notin actual contact. This action-at-a-distance was diffi-cult for 19th century physicists to accept. As a kind ofcompromise, Michael Faraday described the electricforce in terms of an electric field which enabled theidea of contact to be retained. Charge was imagined togive rise to an electric field in the space surroundingit. The field has a strength and a direction at everypoint and is therefore a vector field. A second chargeat some point in that space is necessarily in contactwith the field there. The field at that point then(somehow) gives rise to the electric force on thecharge. The relationship between the force FE on acharge q and the field strength E is given by

FE = qE (N).

Thus E has units N.C–1. Since charge q can be positiveor negative, the direction of FE can be parallel or anti-parallel to the direction of E (Figure 1-2).

e

p

E

E

E

Figure 1-2. The forces on an electron e and a proton p im-mersed in the same electric field E act in oppositedirections.

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Source of An Electric FieldWe have seen that the source of an electrostatic field isa charge at rest. Michael Faraday developed the con-cept of the electric field. He invented the idea of thelines drawn in Figures 1-3a and b. The lines with thearrows on them he called field lines or flux lines andgave them the unit weber (Wb). He imagined that allspace can be mapped out by an infinite number ofsuch lines (though to keep the figures simple we havedrawn only a few of them). He drew the lines origin-ating on positive charges and terminating on negativecharges. Though having no physical reality, theyenabled him to describe the field in a consistent way.

p

e

(a)

(b)

Figure 1-3a. Representation of the electrostatic field pro-duced by a stationary positive source charge (a) and anegative source charge (b).

The lines enabled Faraday to describe the direction ofthe field and its strength. If we imagine a positive“test” charge placed at any point in this space then itwill actually be “in contact” with one of these imag-inary lines. The direction of the force on the charge isthen given by the direction in which the arrow on theline points. Thus a positive test charge would tend tomove away from a positive source charge andtowards a negative source charge. The magnitude of

the force on the charge, and therefore also the strengthof the field, can be associated with the density of thelines. This relationship between density and fieldstrength is illustrated in Figure 1-4.

(a) (b)

p

Figure 1-4. An attempt to illustrate the relationship betweenthe density of field lines and the strength of a field.

In this figure we can imagine that a positive charge,say a proton p, is the source of an electrostatic field inthe space surrounding it. This field extends every-where in three-dimensional space (Figure 1-3a),though we focus on only that part of the field con-fined to a narrow region to the right of the charge. Ifwe were to orient an area of one square meter perpen-dicular to the field at two arbitrary positions (a) and(b) at different distances from the charge, we wouldcount different numbers of field lines passing throughthe area.

Suppose, for example, that the number of field linespassing through the area at (a) is 4; the field strengththere is therefore 4 Wb.m–2. And suppose that thenumber of field lines drops to only 2 at (b); then thefield strength there is 2 Wb.m–2. It can be shown thatthe magnitude of the electrostatic field produced by astationary point charge q decreases inversely with thesquare of the distance r from q, that is,

E = kq

r2 , …[1-2]

where k is the constant that appears in eq[1-1].This concept of the electric field, though originally

invented by Faraday for convenience, has now beenproven to exist through countless experiments. Theconcepts of field and flux will prove just as powerfulin our description of the magnetic field, as we shallattempt to show later in this chapter.

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Direct Current Electricity

Electric CurrentThus far we have considered the physics of charges atrest. But charges can move. If a net charge moves in adefinite direction an electric current exists. A workingdefinition of current I is charge per second:

I =∆q

∆t. …[1-3]

The unit of current is C.s–1. 1 C.s–1 is given the unit

ampere (A). 3

VoltageMost people take for granted that voltage and currentgo together. But voltage is different from current. Tounderstand the difference between them we examinethe simplest of electric circuits, one comprised of asource of electrical energy and a consumer of elec-trical energy, otherwise called a load (Figure 1-5).

+

RV

conventional current

electron current

Figure 1-5. A circuit consisting of a source of electricalenergy and a load.

We can suppose that our source of electrical energy isa common battery, like the ones used in a flashlight,and the load is a carbon composition resistor (wedescribe what a battery and resistor are made of inChapter 2). The battery, symbolized by the two horiz-ontal lines, is on the left and the resistor, symbolizedby the lightening bolt, is on the right. The “top” orlong side of the battery symbol denotes its positiveterminal while the “bottom” or short side denotes itsnegative terminal. The terminals are given positiveand negative signs because electrons, which have anegative charge emerge from the negative terminal of

a battery and enter the positive terminal. (Rememberlike charges repel, unlike charges attract.)

An Electric Circuit Must Form a Closed LoopThis circuit forms a complete loop. One wire connectsthe positive terminal of the battery with the top of theresistor while a second wire connects the bottom ofthe resistor with the negative terminal of the battery.The circuit must form a complete loop or otherwisethe electrons would not have a path to travel in to getfrom the negative to the positive terminal of thebattery. (Electrons don’t easily move through air!) Theagent which “drives” the electrons around the loop isthe battery voltage. It is said that the battery producesa voltage V (volts) across its terminals which “drives”a current of I (amperes) around the loop. If the batteryis disconnected from the loop then the current dropsto zero. Also if the wires are cut at any point then thecurrent drops to zero. The value of the current is thesame everywhere around the loop—the same amountof current I flows through the wires, through theresistor and through the battery.

The difference between voltage and current shouldbe kept firmly in mind: voltage is a driving force thatappears across something, whereas current, being theflow of electrons, flows through something.4 Figure 1-6 shows the circuit of Figure 1-5 equipped withinstruments (an ammeter and a voltmeter) to enablethe current and voltage we have just defined to bemeasured. The ammeter is connected in series withthe resistor because it displays the current flowingthrough the resistor, whereas the voltmeter isconnected in parallel with the resistor because itdisplays the voltage appearing across the resistor. Wedescribe some examples of these instruments morefully in Appendix A.

V

+

RV

I

Figure 1-6. The circuit of Figure 1-5 equipped with an am-meter (I) and voltmeter (V).

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The Direction of CurrentAt the risk of confusing the reader we now state thatin this course the direction of electric current will betaken to be opposite to the direction of electron flow.In other words, in Figures 1-5 and 1-6 we shall takethe direction of what is called the conventional currentto be clockwise even though we know that theelectrons making up the actual current are movingaround the circuit in a counterclockwise direction.5

Resistance and Ohm’s LawThe size of the current that flows for a given voltageapplied is set by an attribute of the load calledresistance. The resistance R of a load is defined as theratio

R =V

I, …[1-4]

where V is the voltage across the load and I is thecurrent through the load. R has units V.A–1. 1 V.A–1 iscalled an ohm (Ω). If R is a constant, independent of Vand I, then eq[1-4] is known as Ohm’s Law.6

If a battery is the source of electrical energy, thenthe voltage and current are constant (provided thebattery does not run down!). Current I flows continu-ously around the loop in Figures 1-5 and 1-6 in aconstant clockwise direction. For these reasons thiskind of current is called direct current (or DC current).The corresponding voltage is called a DC voltage.

Resistivity and ConductivityWe have just stated that a load has resistance. But wedo not imply that resistance is a fundamentalproperty of matter. In fact, resistance arises from thefundamental property called resistivity. We can defineresistivity as follows. Consider a conductor of length land cross sectional area A. The resistance measuredbetween the ends of the conductor is observed todepend directly on l and inversely on A:

R = ρl

A. …[1-5]

The constant of proportionality ρ is the resistivity. Ithas units (Ω.m).

We shall see in Chapter 2 that this relationship fig-ures in the operation of a device called a strain gauge.As well, the resistivity, and hence also the resistance,of a conductor can depend on temperature and the

intensity of the light falling on the conductor.The inverse of resistivity is called conductivity. It is

denoted σ and has the units (Ω.m)–1. A material with asmall resistivity and a large conductivity is called aconductor. Examples are the metals copper, silver andgold. Conversely, a material with a large resistivityand a small conductivity is called an insulator.Examples are rubber, plastic and glass. A materialwith intermediate resistivity and conductivity iscalled a semiconductor. Examples are the elementsgermanium and silicon. We shall have a lot more tosay about conductivity later in this chapter.

Energy and PowerIn the circuits above, there is a continuous transfer ofenergy from battery to resistor. You can prove thisyourself by observing that the battery grows weakerwith time while the resistor heats up. Electrical energyfrom the battery is transferred continuously as heat tothe resistor, which is subsequently transferred to thesurrounding air via conduction, convection and radia-tion. Energy has the unit joule (J). The rate of energyconversion (J.s–1) is called power and has the unit watt(W). 1 J.s–1 = 1 W. The electrical power P can be shownto be the product of the voltage and the current:

P = IV . …[1-6]

Thus 1 W = 1 A.V. If the load obeys Ohm’s Law theneq[1-4] applies and we can rewrite eq[1-6] in theseforms:

P = I2 R =V2

R. …[1-7]

Thus if R is a constant, power is proportional to thesquare of the current and the square of the voltage.

Often we are less interested in the absolute value ofthe power than in the change in the power that resultsfrom some effect. The change may be a decrease or anincrease resulting from attenuation across a resistor inthe former case, or boosting from amplification orfrom some other cause in the latter.

Power Change in dBWe can describe a change in power using a pseudounit called dB (pronounced “dee bee”). Suppose wewish to compare a power level P with a referencepower level Po . We take the logarithm of the ratio ofthe powers and call this a number in bels:7

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bels = log10

P

P0

.

To convert this number to decibels or tenths of bels(dB) just multiply by 10:

dB = 10log10

P

P0

. …[1-8]

A power change of 1/2 is a convenient fraction to takein establishing a reference. If the power is reduced by1

2 , i.e., if P = 1 2Po , then

dB = 10log10

1

2= –3 .

We can interpret this negative number as the “3-dBpoint”—the point where the power is reduced by one-half from the reference power level. In electronics thisis known colloquially as the point at which the poweris “3 dB down”.

We can use this “dB” terminology to describe thechange in a voltage if the device obeys Ohm’s Law. Insuch a device, called a linear device, the electricalpower is proportional to the voltage squared, eq[1-7],allowing us to write eq[1-8] in terms of voltage

dB = 10log10

V2

V02 . …[1-9]

Thus the 3-dB point is also the point at which the volt-age is down by a factor 1/√2 with respect to the refer-ence voltage level. We shall use this terminology inthe following chapters to describe the frequency res-ponse of filters and amplifiers. A graph in which theamplitude is expressed in dB also enables us to distin-guish small features (for a preview of this usage lookahead to Figure A5-2b).

Voltage SourceEnergy sources are classified as being voltage sourcesor current sources. Such sources are also described asbeing ideal or real. A voltage source (Figure 1-7) isdescribed as ideal if it produces a constant voltageacross a load regardless of the current supplied to theload. Voltage sources are sold in a variety of sizes andshapes with the most common type being thechemical cell or battery.

+

V –

+

– V

Figure 1-7. Symbols for ideal voltage sources.

A battery functions as an ideal voltage source so longas the current drawn from it remains small. In prac-tice, a typical cell starts to behave non-ideally as soonas the current drawn from it increases beyond a cer-tain point; at this point the voltage across its terminalsbegins to decrease. This behavior is due to the exis-tence of an effective internal resistance in series withthe battery’s emf (an accompanyment of chemicalaction, lower temperature or age). Thus in this coursewe shall model a real voltage source with the symbolsdrawn in Figure 1-8. A truly ideal voltage source isone whose internal resistance is zero.

–

+ V

I +

–

R i

ε

Figure 1-8. A real voltage source.

Current SourceThe second special case of an energy source is thecurrent source (Figure 1-9). A current source is con-sidered ideal if it delivers a constant current to a loadregardless of the voltage developed across the load. Areal current source is less common than is a real volt-age source. A real current source can be constructedby placing a large resistor in series with a voltagesource or from a circuit employing a transistor or anintegrated circuit (IC).

II

Figure 1-9. Symbols for an ideal current source.

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A real current source, like a real voltage source, doesnot always behave ideally. The current delivered by areal current source is observed to decrease as the volt-age developed across the load increases. This meansthat a real current source behaves electrically like anideal source element in parallel with an internalresistance (Figure 1-10). An ideal current source is onewhose internal resistance is, in principle, infinite.

For various reasons many sensors are designed ascurrent sources. Since sensors are important in thiscourse we devote Chapter 6 entirely to the subject ofsensors.

R

I

V

+

–

Io I = Io – VR i

Figure 1-10. A real current source behaves like an ideal cur-rent source with a resistance in parallel with it.

Resistors in SeriesIn a course on instrumentation it is useful to know afew basics of circuit analysis. Circuit analysis beginswith the issue of resistors in series and in parallel. Forexample, we can show that a circuit consisting of abattery with two resistors in series (Figure 1-11) canbe simplified electrically to one involving a voltagesource and a single resistor. The argument goes asfollows.

V

I

V1 V2

R2R1

Figure 1-11. Two resistors connected in series.

The battery has a voltage V across its terminals anddrives the same current I through both resistors (sincethey are both “inline” and a part of the same single

loop). If the voltages across R1 and R2 are V1 and V2

respectively then

V = V1 + V2 = I R1 + R2( ) ,

= IReq .

Thus the effective resistance of the two resistors is thesum

Req = R1 + R2 . …[1-10]

This expression can be extended to apply to anynumber of resistors in series.

Resistors in ParallelA second example of circuit analysis involves resistorsin parallel (Figure 1-12). We can show that this circuitcan also be simplified electrically to a circuit consist-ing of a voltage source and a single resistor.

V

I

R1

R2

I2

I1

+ –

Figure 1-12. Two resistors connected in parallel.

Both resistors are connected to the same battery andtherefore have the same voltage V across them. Butthe current I supplied by the battery splits up intocomponents I1 and I2, flowing through R1 and R2

respectively. Thus

I = I1 + I2 =V

R1

+V

R2

,

= V1

R1

+1

R2

=

V

Req

.

Thus the effective resistance of the two resistors inparallel is given by the expression

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1

Req

=1

R1

+1

R2

, …[1-11]This expression can be extended to apply to anynumber of resistors in parallel.

Capacitance and CapacitorsNext to an energy source and the resistor the most common element in an electric circuit is thecapacitor. A capacitor can be modelled as a pair of identical parallel metal plates.

Capacitor CharacteristicWe have already seen that the voltage producedacross an ideal resistor is proportional to the currentflowing through the resistor. In contrast, the voltagedeveloped across a capacitor is proportional to thecharge Q stored in the capacitor. Thus the followingrelationship applies:

Q = Cv . …[1-12]

The constant of proportionality, C, is by definition thecapacitor’s capacitance. The value of C depends on thematerials making up the capacitor, in particular on thekind of conductors used and the material called thedielectric, which electrically insulates one plate fromthe other. C has the unit farad (F). 1 F = 1 C.V–1. Onefarad is a large capacitance, the most typical valuesbeing 0.1 µF and 0.01 µF. (The symbol µ stands for“micro” and means 10–6.) Common symbols ofcapacitors are drawn in Figure 1-13.

(a) (b)

(c)

Figure 1-13. Symbols for capacitors: a fixed non-polar type(a), a fixed polarized electrolytic type (b) and a capacitorwhose capacitance can be varied by the user (c).

An ideal capacitor has its own VI characteristic whichcan be found by differentiating eq[1-12] with respectto t:

dQ

dt= i = C

dv

dt. …[1-13]

Here i is the current flowing into the capacitor’s pos-itive terminal. Equation [1-13] shows that i is non-zero

only if dv/dt is non-zero, or when v is changing. Thusin theory a capacitor should not pass a constant or asteady DC current, but should pass a changing DCcurrent or an AC current.

Parallel Plate CapacitorAs we have stated a capacitor can be modelled as apair of identical parallel metal plates. The plates mustbe close enough together, but not touching, for thecapacitance to be significant. We can actually derivean analytical expression for the capacitance of thismodel.

Let us suppose that the plates are of area A withseparation d much less than a dimension of A (so wecan neglect edge effects). We suppose for the momentthat the space between the plates is occupied with avacuum. We charge the capacitor by connecting it to avoltage source (Figure 1-14).

–

+Q

–Q

+

area A

VE

+ + + + + + +

– – – – – – –

Figure 1-14. A parallel plate capacitor can be charged byconnecting it to an energy source.

The process of charging requires some description forit is really a process of charge separation. We describe itin terms of electron movement. Electrons are attractedinto the positive terminal of the voltage source andaway from the top plate of the capacitor. The top plate

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is therefore left with an excess of positive charge +Q.At the same time electrons are repelled away from thenegative terminal of the voltage source and onto thebottom plate of the capacitor. Thus the bottom plate isleft with an excess of negative charge –Q. Because ofmutual repulsion the charges distribute themselvesuniformly over each plate.8 A uniform electric fieldtherefore arises in the space between the platespointing from positive to negative plate. Analysisshows that the voltage between the plates is given bythe expression

v = V =Qd

εo A, …[1-14]

where εo is a physical constant called the permittivity offree space . From eqs[1-12] and eq[1-14] the capacitanceis given by the expression

C =Q

v=

εo A

d. …[1-15]

Thus the capacitance of a parallel plate capacitor canbe increased by increasing the area of the plates or bydecreasing the separation between the plates. Thecapacitance can also be increased by inserting aninsulating material or dielectric into the gap betweenthe plates. The increase in capacitance results from apolarizing property of the dielectric described by aparameter called the dielectric constant denoted εr. Toinclude the effect of a dielectric we must modify eq[1-15] to read

C = ε r

εo A

d. …[1-16]

The values of εr range over two orders of magnitude,from 1 to about 100. More details on practicalcapacitors are given in Chapter 2.

Energy Stored in a CapacitorThe major usefulness of a capacitor has to do with itscapacity to store an electric charge. To see this wemust first supply the capacitor with charge by con-necting it to an energy source (Figure 1-15). We shallsee that the process of charging an ideal capacitorrequires that the energy source does a certain amountof work.

This circuit is equipped with a switch S that enables

us to begin the process of charging with the capacitorfully discharged. We can suppose that at time t = 0 wemove S to position B to connect the capacitor to theenergy source and thus begin the charging. We sup-pose that at some later time t we observe the capacitorto “contain” a charge Q and to support a voltage Vacross its terminals (as measured, say with avoltmeter).

V C

B

A S

Figure 1-15. A circuit to charge a capacitor.

Our analysis requires a little calculus. We can think ofthe charge as being transferred from the source to thecapacitor in increments or “bundles” of charge dq.The element of work done by the battery dW inplacing an element of charge d q on the capacitor whenthe voltage across the capacitor is v is given by

dW = vdq = Cvdv .

Thus the total work done in charging the capacitorfrom a lower limit of v = 0 volts to an upper limit of v= V volts (the voltage of the source) is given by theintegral

W = Cvdv0

V

∫ ,

=1

2CV 2

, …[1-17a]

=1

2QV =

1

2

Q2

C, …[1-17b,c]

where eqs[1-17b] and c follow by the substitution ofeq[1-12].

This work is not “lost” to the system as heat as itwould be if a resistor were in place of the capacitor.With a little engineering, the capacitor, once charged,could be disconnected from the energy source andconnected to a device (like a resistor in fact) to per-form work (like increasing the resistor’s temperature).The conclusion to be drawn is that the energy

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delivered by the source to the capacitor goes into theproduction of the electric field between the capacitorplates. As long as the electric field exists there, theassociated energy can be regarded as energy stored.

Charging/Discharging a CapacitorIn our treatment of the previous section we ignoredwhat happens during the time when the capacitor isbeing charged or discharged. A circuit to study thistransient state is drawn in Figure 1-16. This circuitincludes a resistance R, which might be the resistanceof a carbon composition resistor or just the effectiveresistance of the capacitor itself.

V C

B

A S

R

Figure 1-16. A capacitor being discharged through a loadresistor R.

We consider the discharge of a capacitor from aninitial state of full charge. The capacitor in Figure 1-16can be fully charged by leaving the switch S inposition B for a sufficiently long time. Then at time t =0, say, we move S to position A to begin the discharge.At some clock time t when the instantaneous voltageacross the capacitor is vC(t) and an instantaneouscurrent i(t) is flowing, the voltage across the capacitorand the voltage across the resistor must be equal, i.e.,

vC(t) – Ri(t) = 0 , …[1-18]

Using eq[1-12] and the fact that i(t) and q(t) are relatedby inversion we have

i(t) = –dq(t)

dt,

(as current i(t) flows charge q(t) decreases) we cansubstitute into eq[1-18] and obtain

dv(t)

dt+

v(t)

RC= 0 . …[1-19]

This expression is a first order differential equation inv(t). You should be able to show by direct substitutionthat a solution is

v(t) = Ve–

tRC , …[1-20]

= Ve–

t

tC

where tC = RC . …[1-21]

Eq[1-20] shows that when the switch is thrown toposition A, the voltage across the capacitor does notdrop to its final value of zero volts instantaneously,but rather, decreases with time exponentially. This isthe same as saying that the capacitor discharges withtime exponentially.

tC, which is a constant (since R and C are constant)is called the time constant; it quantifies how sharplythe voltage decreases. If R is in units of ohms and C inunits of farads then RC has the unit of time inseconds. tC is the time in seconds required for thevoltage to fall from the initial value of V volts to thevalue V/e = 0.368V volts. The bigger the time constantthe more time is required for discharge to take place.

If a resistance were to be included in the chargingcircuit, a similar (complementary) expression wouldbe obtained for vC(t) over the time during which thecapacitor is charging.

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Magnetic FieldIn this course we are concerned with the three field types: gravitation, electric and magnetic. Wehave described the electric field a little already. Though perhaps not consciously, most of usremember a little of magnetic effects from playing with magnets as children. We know that a barmagnet, often made from iron, has a north pole and a south pole. The like poles of such magnetsrepel each other while the unlike poles attract. Magnetic poles must therefore be able to exertforces on one another across empty space, in a manner like electric charges. By this we infer thereexists an entity called a magnetic field.

Basic ObservationsMost of us are aware from an early age of how bar

magnets and the magnetic compass behave. Two bar

magnets will either attract each other or repel each

other, depending on how they are oriented when

brought together. We soon learn that a bar magnet has

a north pole at one end of it and a south pole at the

other. Observations then show that like poles repel

each other while unlike poles attract. If we stand with

a compass at some point on the earth’s surface the

marked end of the compass needle always points in

the general direction of the north geographic pole

(Figure 1-17). In fact, if the marked end of a compass

needle is defined as a north pole then it follows that

the magnetic pole near the north geographic pole is a

south magnetic pole. Because the poles of a magnet

can exert forces on one another across empty space

there must exist an entity called the magnetic field.

Figure 1-17. At any position on the earth’s surface themarked end of a compass needle always points toward themagnetic pole in the northern hemisphere (which is, in fact,a magnetic south pole).

The Test for a Magnetic FieldFor the purpose of understanding, a compass can betaken as an instrument for testing for the existence ofa magnetic field. We can agree that a magnetic fieldexists at some point in space if the needle of a com-pass is forced to take up a special position at thatpoint. We can define the direction of the magneticfield (it is a vector) to be the direction in which thenorth pole of the compass needle points.

Source of a Magnetic FieldJust stating that a bar magnet and the earth have amagnetic field according to our simple test says littleabout the physical source of the field. How this sourcewas discovered can be explained by means of a simpledemonstration.

If a compass is employed as a test instrument and isused to explore the environs of a conductor carryingan electric current, the compass needle is observed totake up special positions with respect to the wire(Figure 1-18). This effect goes away if the current iscut. The conclusion to be drawn is that a magneticfield exists in the region near the conductor havingsomething to do with the current. A close connectionmust exist between electric effects (the current) andmagnetic effects (the action on the compass needle).9

Figure 1-18. Around a current-carrying conductor a com-pass needle always points in a direction perpendicular to,but never towards, the conductor.

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On closer inspection the needle is observed to alwayspoint in a direction perpendicular to the conductorbut never directly towards or directly away from theconductor.

These effects were first studied in detail a centuryago by the English scientist, Michael Faraday. In anattempt to describe them empirically Faraday usedthe concept of the flux line or the field line he hadinvented for the electric field. He showed that themagnetic field produced near a current-carrying con-ductor can be represented graphically in a planeperpendicular to the conductor as a series ofconcentric circles, each indicated by an arrow (Figure1-19). On the left in the figure the current is shownflowing out of the plane of the page while on the rightthe current is shown flowing in the opposite direction.The direction of the current and flux lines can bededuced using a right hand rule. If you place thethumb of your right hand in the direction of thecurrent, and curl your fingers in a gripping actionthen your fingers describe the direction of the field.The strength of the field is denoted by the letter B.

B

X

B

Figure 1-19. Representations of the magnetic field near con-ductors carrying an electric current. The conductor is at thecentre of the circles and perpendicular to the plane of thepage.

The field lines thus drawn are given the same unit asfor the electric field, namely weber (Wb). Here, too, thedensity of the lines can be taken as an indicator of thefield strength. The direction in which the compassneedle points defines the direction of the arrow andthe direction of the field.

Fundamental TestAll of the above notwithstanding, there exists a morefundamental physical way of testing for a magneticfield’s existence than by checking to see if a compassneedle deflects. This test is based on the so-called

Lorentz force . If an electric charge q moves withvelocity v through a region where a magnetic field ofstrength B exists then the charge will be subject to amagnetic force given by

Fm = qv × B . …[1-22]

This is a vector cross product, meaning that the vec-tors Fm, v and B are mutually perpendicular.

The meaning of the vector cross product is expand-ed in Figure 1-20 where a positive charge q (a proton)is shown entering a region where a magnetic field ofstrength B exists (pointing out of the plane of thepage). If we suppose for simplicity that the particle’svelocity is initially perpendicular to B, then theLorentz force causes the particle to move in aclockwise direction.

B

pv

Figure 1-20. A proton is shown entering a region from theright in which a magnetic field of strength B exists(pointing out of the plane of the page). The force on theparticle is in the plane of the page perpendicular to v and B,causing the particle to move in a clockwise direction.

Thus we can say that a simpler, more elegant andfundamental test of the existence of a magnetic field ina region of space is if a charge which is moving in thatregion is deflected from its original path by a force ofthe form of eq[1-22]. From eq[1-22] we now have theunit of B in the SI system of units: N.C–1.m–1.s(equivalent to weber.m–2). 1 N.C–1.m–1.s is given thespecial unit Tesla.

Magnetic Field of the EarthMagnetic field has a magnitude and a direction. At apoint on the surface of the Earth the Earth’s magneticfield is described in terms of its strength and declina-tion (angle below the horizon). In the region ofToronto the strength is of the order of 0.3 gauss with adeclination of 2 degrees. We shall return to this topicin Chapter 6.

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Inductance and InductorsIn AC circuits the inductor is the complement of the capacitor. An inductor is modelled as asimple coil of wire. If a current in the coil is made to change with time then an emf is observed toexist between the terminals of the coil. This simple phenomenon, called self induction , is thesource of a number of effects in electricity and electronics. Understanding the inductor involvesnew aspects of the magnetic field.

A Current LoopWe have seen in a previous section that a magneticfield is produced in the space surrounding a current-carrying wire. If conditions are kept the same but thewire is looped to form a coil, then the strength of thefield inside the loop is observed to increase (Figure 1-21a). If a magnetic compass is used to test thesymmetry of the field then that symmetry is found toresemble that of a bar magnet (Figure 1-21b). Faradaydeduced that some connection must somehow existbetween macroscopic magnetism (the bar magnet)and the more elementary physics of a current loop.

N

S

(b)

I

(a)

BBB ~ I

Figure 1-21. The magnetic field lines produced by a currentloop (a) and a bar magnet (b) have a similar symmetry. Thediagram on the left illustrates a microscopic effect, that onthe right a macroscopic effect.

InductionIf the current in the loop is made to change with timethen an emf (voltage) appears between the loop’sterminals. Since this emf results from the changingcurrent and not from some external energy source, itis called an induced emf. The effect is called induction.This emf can be measured with the appropriate ACvoltmeter.

Faraday used his invention of magnetic flux tointerpret induction. He explained that the time rate ofchange of flux through the loop induces an emf in theloop that tends to resist the original change in the flux(or the original change in the current). That is, theinduced emf ε can be described by a mathematical

expression of the form

ε = –dΦB

dt, …[1-23]

where ΦB is the magnetic flux in webers passingthrough the loop. This expression encapsulates twofundamental laws of physics called Faraday’s Law andLenz’s Law. (Lenz was a contemporary of Faraday.)Faraday’s Law states that the emf is proportional tothe time rate of change of the flux and Lenz’s Lawstates that the proportionality has a negative sign.

Since the flux through the loop is proportional tothe current we can rewrite eq[1-23] as

ε = Ldi

dt, …[1-24]

where L, the constant of proportionality, is called theinductance. L is related to the inductor’s geometry andhas the unit henry (H). 1 H = 1 V.s.A–1. One henry is alarge inductance. More typical values are in the mHand µH range. Some symbols of inductors are drawnin Figure 1-22. We shall discuss inductors in moredetail in the section on impedance in this chapter andwhen we get to technical aspects of inductors inChapter 2.

(a) (b)

(c)

Figure 1-22. Symbols used for inductors: a standard induc-tor (a), an inductor whose inductance can be varied by theuser (b) and tapped type (c).

In the next section we discuss the Hall effect, an effectinvolving the magnetic field.

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The Hall EffectThe Hall Effect is an effect that embodies so many physical concepts that the device in which itoccurs forms the quintessential sensor. We confine our attention here to the physics of the effect.Typical devices are described in detail in Chapter 6.

The EffectThe Hall Effect was discovered by E. W. Hall in 1879.It was not until the 1950s, however, that the first prac-tical device was developed employing the effect.

Hall observed that if a semiconductor carrying adirect current is immersed in a magnetic field (Figure1-23), then a voltage develops across the semiconduc-tor in a direction perpendicular to both the currentand the field. This voltage is called the Hall voltage.The sign and magnitude of the voltage provides infor-mation about the magnitude and direction of the fieldand also about the type of majority charge carriermaking up the current flow. (We shall have more oncharge carriers later in this chapter.)

I

F

v

– – – – – – – – – – –

+ + + + + + + + + + +

wB

–q

V

Figure 1-23.. Illustration of the Hall Effect.

The Hall VoltageThe Hall voltage is the quantity that actually getsmeasured in a Hall effect sensor. We can explain howthis voltage arises as follows.

We make three assumptions: (1) that the magneticfield in the figure points into the plane of the page, (2)that an external voltage source induces a(conventional) DC current to flow from left to rightthrough the semiconductor and (3) that the majoritycharge carriers are electrons with charge –q. Theelectrons move to the left with some average (or drift)speed v.

Since the electrons are moving through a magnetic

field, then the Lorentz force, eq[1-22], deflects themupwards towards the top edge of the semiconductor.Thus the top edge becomes more negative than thebottom edge. As the charges build up, the potentialdifference across the semiconductor rises. Eventually,the electric field that results from the collection ofseparated charges balances the effect of the Lorentzforce and the potential difference reaches a maximum.This maximum potential difference denoted VH iscalled the Hall voltage.

Analysis shows that VH has the following form

VH = γIBsinθ , …[1-25]

where γ is called the Hall coefficient and θ is the anglebetween the direction of current and the direction ofmagnetic field.

A number of variables are involved in eq[1-25]. Thismeans that if the numerical values of all of thevariables are known except one then that unknowncan be calculated. For example, let us suppose in theabove example that the direction of I and B are knownbut the identity of the charge carrier is not. If thevoltage of the top of the semiconductor relative to thebottom is found to be negative then it can be inferredthat the charge carriers are electrons. If the voltage ispositive then it can be inferred that the charge carriersare positive entities, or holes (we shall have more tosay about holes later in this chapter).

If we know that the charge carriers are electrons andif we know what the magnitude and direction of thecurrent are then we can calculate from the sign andmagnitude of VH the direction and magnitude of B.Thus a Hall Effect device makes an ideal magneticfield sensor.

This far in this review we have considered some of thephysics that results from the flow of a DC current. Inthe next section we begin our review of the effects thatresult when the current is made to alternate. Thephysics now becomes more mathematical.

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Alternating Current ElectricityWhen a reference is made to a “signal” in the sciences, what is usually meant is an AC signal.“AC” stands for “alternating current”. The major difference between AC and DC is that AC iscurrent that is made to flow for a time in one direction around a circuit loop, and then is reversedand for an equal time is made to flow in the opposite direction—and so on reversing continuous-ly. The audio that drives a speaker in a stereo system is an AC signal, as is the voltage suppliedfrom wall power sockets in the home and laboratory.

AC CircuitsAn AC waveform can be simple or complex. Thesimplest AC signal is a sinusoidal one; it has a simplewaveform that can be studied in detail when display-ed on the screen of an oscilloscope (Figure 1-24 top).The instantaneous voltage v(t) of such a signal swingscontinuously between a positive peak voltage of +Vpeak

and a negative peak voltage of –Vpeak.Just as a DC circuit must be complete in order to

sustain a DC current, so also must an AC circuit becomplete in order to sustain an AC current. Thesource of an AC signal v(t) is called a signal generator .In Figure 1-24 centre and bottom a signal generator isshown in schematic form connected to a load withtwo wires, one “out” and one “return”. In the first halfcycle of the voltage a current i(t) flows clockwisearound the loop. In the second half cycle the currentflow reverses. The current may be changing with timebut at any instant of time and at any point in thecircuit the current has the same instantaneous valueand the same clockwise or counter-clockwisedirection.

An Analog WaveformThe voltage waveform in Figure 1-24 is an analogwaveform in the sense of being a continuous functionof time. We are therefore justified in representing it bythe continuous function

v(t) = Vpeak sin(ωt) …[1-26]

= Vpeak sin(2πft) ,

where ω = 2πf is the angular frequency in radians persecond (rad.s–1) and f is the linear frequency in cyclesper second (c.s–1). One cycle per second is given thespecial name hertz (Hz). 1 c.s–1 = 1 Hz. It can be seenfrom the measurements performed by the oscilloscope(look closely down the right hand side of Figure 1-24top) that in this example the frequency is 1.000 MHz,the peak-to-peak voltage is 4.28 V and the rms value

over one cycle is 1.51 V. We shall define these terms indue course.

v(t) loadi(t)

(b)

1sthalfcycle

v(t) load

(c)

i(t)2ndhalfcycle

Figure 1-24. A sinusoidal analog AC voltage v(t) taken fromthe screen of an oscilloscope (top). v(t) produced by a signalgenerator results in an AC current i(t) (middle andbottom).

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AverageWe are often interested in the average of an AC voltageor current calculated over one period. The instantan-eous current i(t) averaged over one period is bydefinition given by

I =1

2πi(t)dt

0

2π

∫ .

We can define the average voltage in a similarfashion. You can therefore see that the average of anAC quantity may not be the result you expect. Forexample, the average of a pure sine wave signal(Figure 1-24 top) is zero! However, the average may infact be the measurement you want when the ACsignal contains a DC component. We shall return tothis subject in Chapter 2.

Root-Mean-SquareIt is useful to have a way of comparing a sinewave ACcurrent with a DC current. It is reasonable to define asinewave AC current and a DC current as havingequal effective values if they produce the same heatingin the same resistor. We can find the heating effect ofan AC current by averaging the power losses over onecomplete cycle:

P =1

Ti(t)2 Rdt =

iPeak2 R

Tsin2(ωt)dt

0

T

∫0

T

∫ …[1-27]

=IPeak

2 R

2= Ieff

2 R ,

where Ieff is the “effective value” of the AC current. Ieff

is commonly called the root-mean-square or rms valueand is denoted Irms . Thus

Ieff = Irms =Ipeak

2. …[1-28]

Thus the rms value of a sinewave current (or a sine-wave voltage) is its peak value divided by the squareroot of two. Perhaps surprisingly, the mains, whichsupplies a sinewave voltage of 60 Hz at an amplitudeof 115 volts rms, has a peak voltage of…

Vpeak = 115 2 = 163 volts!

Specifications of devices and circuits are sometimesgiven in terms of rms, other times in peak values. It isimportant to understand the difference between thetwo definitions so that a device is used as intended.

Phase DifferenceEq[1-26] is not the most general form of a sinewavefunction. A more general form is

v(t) = Vpeak sin ωt + φ( ) , …[1-29]

where the additional factor φ is the phase difference. φdetermines the value of v(0), that is, the instantaneousvoltage at a clocktime of 0 seconds. Indeed,

v(0) = Vpeak sinφ ,

so if φ = 0 then v(0)=0. But φ is in general not zero aswe shall see.

The concept of phase difference becomes importantwhen two AC signals coexist at the same point in acircuit or when an AC signal passes through a systemlike a filter. The instantaneous voltage of the signalsmay be zero at different clocktimes. If they have thesame frequency, they might appear individually onthe screen of an oscilloscope as shown in Figure 1-25.Since the signals are zero at different clocktimes theyhave a time difference of, say, ∆t seconds that canactually be measured with a digital oscilloscope. Sincethey have a time difference, they also have a phasedifference.

Figure 1-25. Two signals having the same frequency butdiffering phases pass through zero at different clocktimes.10

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We can calculate the phase difference from measure-ments of the time difference and the period. Since thesignals have the same frequency f they also have thesame period T (which we can measure with cursorsbetween alternate zero crossings on the oscilloscopescreen). Let us suppose the time difference betweenthe signals (as measured between the same zerocrossing of the two signals) is ∆t. Then the followingrelationship applies

∆t

T=

ϕ2π

, …[1-30]

where φ is the phase difference in radians.We have pointed out here a second major difference

between an AC current and a DC current; an AC cur-rent has a phase as well as a magnitude. This fact hasprofound consequences as we shall attempt to showin the next section.

ImpedanceRecall from our discussion of DC circuits earlier inthis chapter that the attribute of a load that limits theflow of a DC current through it is resistance . A similarconcept applies in AC circuits. In an AC circuit theattribute of a load that limits the flow of an ACcurrent is called impedance. But impedance (denotedZ) is a more complicated concept than is resistancebecause of the issue of phase. For the moment, we cancertainly define the magnitude of impedance, |Z|, asthe ratio of peak or rms values of AC voltage andcurrent:

Z =Vpeak

Ipeak

=Vrms

Irms

, …[1-31]

so that |Z| has the unit ohm (Ω), the same unit asresistance.

But this does not mean that we can write ameaningful expression of the form of eq[1-31] usinginstantaneous values of voltage and current, v(t) andi(t). If we try this we end up with a function of timethat is essentially meaningless. 11 The factor whichcomplicates the use of instantaneous values is thephase angle.12

To describe this effect of phase most elegantly we

would need to define the current, voltage and imped-ance as complex numbers, a task that would take usbeyond the intended scope of this basic review. Forthe moment we shall stick with defining the absolutevalues of impedance. For example, we shall write theabsolute values of resistive impedance |ZR|, capacitiveimpedance |ZC|and inductive impedance |ZL| as:

ZR = R

ZC =1

ωC…[1-32a, b, c]

and ZL = ωL

You can see that the resistive impedance is the sameas resistance. The absolute values of the capacitiveand inductive impedance both depend on frequency—capacitive impedance depends inversely on freq-uency while inductive impedance depends directly onfrequency. A graph of eq[1-32b] and [1-32c] for C=1 Fand L=1 H is drawn in Figure 1-26. The data is dis-played here in a log-log graph since frequency canvary over many orders of magnitude.

100 101 102 103 104 105 10610-3

10-1

101

103

Frequency (Hz)

Impe

danc

e

Impedance

Capacitive

Inductive

Figure 1-26. A log-log graph of impedance vs frequency forcapacitive and inductive impedances.

Much of what this course is about concerns the work-ing of sensors. Most sensors are made from semicon-ductor materials, and so the topic of conduction inmaterials is important. We begin our study of thissubject in the next section.

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Conduction of Electricity in a SolidImmediately after this section we will begin our discussion of semiconductors. Before doing sowe need to discuss electrical conduction in solids in greater detail than we have done so far. Wefirst consider conduction in general terms, beginning with conduction in a conductor, and thenbroaden our approach to apply to an arbitrary material. We shall have to be brief.

Modifying Our Model of a ConductorThe model of a conductor we have assumed so far inthis chapter is called in physics the Free Electron GasModel. This name is derived from the fact that theconductor is pictured in the simplest of terms: as amaterial whose internal structure is a regular geomet-rical lattice of stationary positive ions with manybillions of free electrons interspersed amongst them.The electrons are assumed to be completely non-interacting, interacting neither with one another norwith the ions in the lattice. This model enables one todefine current, but it fails to explain the differencesobserved in the conductivities of conductors, semi-conductors and insulators. The model needs to berevised.

As a rather obvious first step we can assume somemeasure of interaction: we can include the collisionsthat the electrons invariably make with the ions asthey are driven along the wire by the internal electricfield (Figure 1-27). These collisions effectively preventthe electrons from accelerating indefinitely to ever-higher speeds. Each electron must now reach an aver-age or drift speed vD at which it moves along the wire.

I

low potential or voltage

high potential

or voltage A

E

Figure 1-27. A section of wire carrying a current I.

This means that in an interval of time dt each electronmoves a distance vDdt along the wire (to the left inFigure 1-27). During this interval the number of elec-trons which pass through an area A are those contain-ed in a volume AvDdt. If the density of electrons in thewire is n then the number of electrons in this volumeis nAvDdt. The current is this number multiplied bythe electronic charge divided by the elapsed time:

I =dq

dt=

nevDAdt

dt= nevDA .

This current is flowing to the right. We can define thevector current density J as the current flowing througha unit area of conductor. Its magnitude is

J =I

A= nev D . …[1-33]

J defined in this way is independent of the conductorgeometry. Ideally, the conductor should support aslarge a current density as is possible in order to deliv-er as much current to a load as is possible without theconductor heating up to an unmanageable hightemperature.

The charge drift velocity can be measured experi-mentally. It is observed to increase linearly with theapplied electric field, i.e.,

vD = µE , …[1-34]

where the constant of proportionality µ is called thecharge mobility. Charge mobility has units m.s–1.J.C–1.In general, µ is large for a good conductor and smallfor a poor one. Substituting eq[1-34] into eq[1-33] andadding the vector notation gives

J = neµE . …[1-35]

Conductivity σ is defined theoretically as the ratio ofthe absolute values of J and E. Thus

σ =J

E= neµ . …[1-36]

Theoretical conductivity has units (Ω.m)–1, consistentwith its experimental counterpart. Resistivity ρ is then

ρ = σ –1. …[1-37]

Thus we now have theoretical expressions for σ and ρ

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which are based on a deeper understanding of a solid.We are now in a position to probe more deeply intothe nature of the charge carriers themselves.

Electrons and HolesIf our solid is a conductor then we can explain theexperimental conductivity that is measured almostexactly with our model by assuming that only elec-trons are present as charge carriers. However, whenwe turn our attention to the measured conductivity ofa material of arbitrary type we fail. It had been knownfor a long time that the conductivity of a material ofany type (whether conductor or not) could be writtentheoretically as the sum of the conductivities ofcharge carriers of positive sign as well as negativesign (Figure 1-28).

E

– + vD+vD–

j– j+

Figure 1-28. Within any material the current density vec-tors of negative and positive charge carriers both point inthe direction of the electric field. This means that inprinciple both carriers contribute positively to the totalconductivity.

We can show this by a simple argument. In the lumpof material in Figure 1-28 an electric field is showndirected to the right. Thus negative charge carrierswill move to the left with some drift speed vD– whilepositive charge carriers will move to the right withsome drift speed vD+. From eq[1-34] the vector currentdensity of the negative charge carriers is directedopposite to vD– and therefore in the same direction asvD+. Thus the conductivities of the negative and posi-tive charge carriers add constructively:

σ = n+eµ+ + n–eµ– .

However, the two terms in this result are not usuallyof the same magnitude since the conductivity (ornumber) of one type of charge carrier usually exceedsthe other. In an insulator like glass both n+ and n– arevery small. But in a metal like copper n– is large whilen+ is nearly zero. The carrier that contributes the mostto the conductivity is called the majority charge carrier

while the other is called the minority charge carrier.Semiconductors like germanium and silicon have

conductivities that lay somewhere between the con-ductivities of conductors and insulators. The exper-imental conductivity of a semiconductor can only beexplained theoretically by assuming that positive aswell as negative charge carriers are present in thematerial. The negative charge carriers are, of course,electrons. The positive charge carriers are more diffi-cult to imagine. They are interpreted as regions in thecrystal where there would normally be an electronbut where an electron is “missing”. These regions,lacking an electron, have a net positive charge and aregiven the descriptive name holes.

Intrinsic SemiconductorsSemiconductors are classified as of intrinsic or extrin-sic type depending on the source of their electricalconductivity. The source of the conductivity of anintrinsic semiconductor is heat. An extrinsic semicon-ductor is one whose conductivity is enhanced bydeliberately-introduced impurities. We consider theintrinsic type here.

The drawing in Figure 1-29 is an attempt to repres-ent the structure of a perfect crystal of a semiconduc-tor like silicon (Si) or germanium (Ge). Both elementsreside in the same column of the periodic table alongwith carbon and have a valence of +4, that is, theyhave four electrons in their outer, or bonding, shell.Consequently in crystalline form each atom sharesthese four electrons with its neighbors. The lines inthe figure are intended to represent a single sharedelectron while the circles represent the positive ions ofGe or Si. The figure could be said to more accuratelyrepresent the structure at a temperature of absolutezero when the lattice is completely without vibration-al motion of any kind, in a state of zero energy whenno ions or electrons are out of place.

Figure 1-29. A two-dimensional representation of the struc-ture of a semiconductor crystal at zero K.

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At a non-zero temperature, however, the crystallinelattice will possess small thermal vibrations thatresult from the continuous emission and absorptionof thermal photons. In a semiconductor thisvibrational energy just happens to be of the order ofthe binding energy of the valence electrons, which areonly moderately bound to the atoms. It is thereforepossible for one of these electrons to absorb enoughthermal energy to escape from its parent atom and gowandering about the crystal (Figure 1-30). Left behindin the crystal is a place where there would normallybe an electron. This “vacancy”, which has a netpositive charge, is a hole. Free electrons and holes areproduced in this way in pairs—for each free electronthere is a corresponding hole.

Figure 1-30. A semiconductor crystal at some non-zerotemperature showing a free electron and a hole.

If we apply an electric field to the crystal (say to theright in Figure 1-30) a current will flow. This currentresults from the movement of free electrons to the left.In the neighborhood of the hole it is possible for abound adjacent electron to fall into the hole, thusannihilating the hole or making it “go away”, leavingbehind a new hole where the electron came from. Thishas the effect of making a hole move to the right, soholes, too, take part in conduction. Thus a small con-ductivity arises from heat alone. Since this conduc-tivity is induced by heat it is called natural or intrinsicconductivity. This kind of conductivity is of onlysecondary importance in the functioning of semicon-ductor devices. As we shall show in the next sectionthe more important type is the extrinsic type.

Extrinsic SemiconductorsIn explaining how electrical conduction takes place ina semiconductor device like a diode (in the nextsection), we shall see that extrinsic conductivity ismore important than intrinsic conductivity. Extrinsicconductivity results from a process called doping,whereby a trace amount of a special impurity atom isintroduced into the semiconductor mix while stillmolten. This process results in a so-called N-type or aP-type material depending on the type of impurityintroduced. The word extrinsic refers to the fact thatthis type of conductivity is the result of an externalprocess.

N-Type SemiconductorAn N-type semiconductor is made from a batch ofpure molten material in the following procedure. Letus assume to begin that the material is silicon whosevalence is +4. Trace amounts of a similar sized atom,say phosphorus whose valence is +5, are introducedto the silicon melt to replace some of the siliconatoms. Since the atoms of silicon and phosphorus areof about the same size the lattice is not undulydisturbed by the substitution. Four of the five valenceelectrons of the phosphorus atom pair-bond with theneighboring silicon atoms, leaving the fifth electronunpaired and therefore only loosely bound (Figure 1-31). Thus the energy of this fifth electron is onlyslightly less than the energy of a free electron, so heator an applied electric field can easily supply the elec-tron with sufficient energy to enable it to break freeand take part in conduction (Figure 1-32).

Figure 1-31. A crystal of a P- or donor type semiconductorshowing the donor electron (diagonal line) unpaired andonly loosely bound to the positive impurity ion (in black).

If we apply an electric field to the crystal (Figure 1-32)to the right, then free electrons will move with somedrift speed to the left. Since the positive impurity ions

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cannot move, the contribution of positive charges tothe current is quite negligible. The current is almostwholly made up of electrons, and for that reason thematerial is called an N-type semiconductor (the “N”standing for negative charge carriers).

Figure 1-32. A crystal of an N-type semiconductor showingthe ionized donor ion (black dot) and its free electron (line).

P-Type SemiconductorA P-type semiconductor is made by doping a hostmaterial of germanium or silicon with an impurityhaving a lower valence than the host (for exampleusing aluminum whose valence is +3 in silicon whosevalence is +4). Three of the valence electrons of theimpurity pair-bond with valence electrons of the sur-rounding ions leaving one of the valence electrons ofthe host unpaired (Figure 1-33). Thus a region in thecrystal where an electron would ordinarily be presentis, in fact, lacking an electron. This region has a netpositive charge and is the hole.

Now thermal vibrations might easily induce an

adjacent valence electron to “fall into” the place justdescribed. This electron would, of course, leave itsown vacancy behind. It is possible in this way for the“hole” to migrate through the crystal, which, if noelectric field were present would be a random motion.But if an electric field were applied, say, to the right inFigure 1-33, then adjacent valence electrons would beinduced to move to the left (very short distances, justfar enough to fall into the hole) and the “hole” wouldmove to the right. Thus the hole is in essence not along-lived entity, but a succession of short-lived entit-ies. Short-lived though a single hole may be, thecumulative effect of many billions of such holes is toproduce a long-lived measureable current.

Figure 1-33. The structure of a P-type semiconductor show-ing the impurity (black dot). The unpaired electron of anadjacent host ion (at the left) is indicated by a missing line.

One of the more important semiconductor devices isthe diode, which we are now ready to discuss in thenext section.

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Conduction of Electricity through a PN DiodeThe single most important semiconductor device and historically the first, is the diode. The diodeis used among other things as a rectifier, a device for converting an AC waveform into a pulsatingDC waveform, as a detector, a device for extracting program information from a modulatedwaveform as in radio, and as a source of light. We confine our attention here to the physics ofelectrical conduction through the diode.

ConstructionA diode is fabricated of silicon or germanium as a PNjunction device; it can be thought of as resulting fromthe fusing together of P- and N-type materials(Figures 1-34). We can imagine that immediately afterthe fusion, electrons and holes in both materials moveacross the junction in both directions.

excess ofholes

excess offree electrons

Both materialsare electrically neutral

(a)

N-typeP-type

hhhhhh

hhhhhh

hhhhhh

hhhhhh

eeeeee

eeeeee

eeeeee

eeeeee

hhhhhh

hhhhhh

eeeeee

eeeeee

excess of– charges

excess of+ charges

(b) depletion region – +

P-type N-type

+

+ + +

+ +

------

------

+

+ + +

+ +

materials are nolonger neutral

Only majoritycharge carriersare indicatedhere.

E

V0

hhhhhh

eeeeee

(c)

– +

+

+ + +

+ +

------

------

+

+ + +

+ +

E

depletionregion

V0

Figure 1-34. An attempt to illustrate N- and P-type semi-conductors (a) being fused together to form a junction (b).The result can be thought of as resembling a chargedparallel plate capacitor (c).

In this way a net excess of holes moves to the N-region, a net excess of electrons to the P-region. Thusthe N-region takes on a net positive charge, the P-region a net negative charge. As the charge builds upin the region near the junction an electric field arisesas does a corresponding electric potential difference.Within a few microseconds after fusion, a steady stateevolves in which the net flow of charge in eitherdirection is zero. Thus the junction can be regarded(with caution) as a charged parallel plate capacitor(Figure 1-34c).

As we have stated, the holes and free electrons inboth materials mingle together. This movement inclose proximity results in a random recombination andcancellation of free electrons and holes. Recombina-tion takes place very quickly near the junction (beforethe charges can move very far), thus resulting in anni-hilation of most charge carriers in this region. Thisregion is therefore called the depletion region. Veryinteresting effects now arise if an external potentialdifference is applied across the diode.

Reverse VoltageIf an external voltage source is connected to the diodewith the polarity shown in Figure 1-35a then the Nmaterial is made more positive than before. Thedepletion region widens, the internal electric field Ein

grows stronger and the potential difference across thedepletion region increases. Since Ein is larger thanbefore and since it has the polarity that it does fewelectrons or holes are induced to flow across thejunction. Less likelyhood exists for a current to flowthan when the voltage source was not present. This iscalled the direction of reverse voltage or reverse bias.

However, it should not be assumed that no currentat all flows in this reverse direction. If the voltage inthe reverse direction is increased to a very large value,then the reverse current tends more-or-less to asaturation value Ir. Ir is therefore more-or-less aconstant depending on the particular diode used(more on this below).

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hhhhhh

eeeeee

– +

+

+ + +

+ +

------

------

+

+ + +

+ +

Ein

depletionregiongrows

Vext

N materialP material

V0

(a)

reverse bias

(b)

P N

VextI ≈ 0

Figure 1-35. A representation (a) of the effect of reverse biason a diode in the style of Figure 1-34. The circuit is shownin (b).

Small Forward VoltageIf a source of a small external voltage is connected tothe diode with the polarity shown in Figure 1-36a andb then the situation is quite different from before. Byvirtue of its polarity, the voltage source causes theinternal electric field Ein to decrease. (Recall fromearlier in this chapter that electric field is a vector andinside the diode the two fields point in oppositedirections and therefore subtract.) The depletionregion becomes thinner than before and the reversepotential difference decreases. The likelyhood for acurrent to flow in the direction indicated increases. IfVext remains small, however, this current remainssmall. This decrease in potential across the junctionresults in a very large increase in the flow of holesfrom the P- to the N-region and electrons from the N-to the P-region. Both carriers contribute to a positiveforward current. This current is observed experiment-ally to depend on the applied voltage according to anexponential function of the form:

I = Ir eeV / kT –1( ) …[1-38]

= Ir eV / VT –1( ) ,

where VT = kT/e is a constant at a constant absolutetemperature T. Ir is the reverse saturation currentdescribed above. Eq[1-38] is called the rectifierequation.

hhhhhh

eeeeee

– +

+

+ + +

+ +

------

Ein

depletionregionshrinks

Vext

N materialP material

(a)

V0

forward bias

(b)

P N

VextI

Figure 1-36. A representation (a) of the effect of a small for-ward voltage on a diode. The circuit is shown in (b)

Large Forward VoltageIf the forward voltage is increased further, the reversepotential and the depletion region shrink further. Andif the voltage is increased beyond a certain point,beyond what is called the turn on voltage, the internalelectric field Ein, the reverse potential and thedepletion region vanish altogether. Further, if theforward voltage is increased beyond this point thediode behaves like a linear resistor and passes anappreciable current. The voltage and current are nowrelated by Ohm’s Law:

V = RiI , …[1-39]

where Rf is defined as the diode’s forward resistance.This resistance is more or less constant (and is in factquite small). Thus the PN junction behaves so as topass current preferentially in one direction: from theP- to the N-region. This is precisely what is meant bydiode action.

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Ideal Diodes and Real DiodesThe symbol for a diode is drawn in Figure 1-37a. Thetip of the triangle defines the forward direction andthe line at the tip indicates the N-material end. Forhistor-ical reasons, the P-material end is commonlycalled the anode; the N-material end the cathode.

I

+ –forwarddirection

(a)

(b)

– (c)

P N

V

Figure 1-37. Circuit symbol for a diode (a), the internalregions of a diode (b), and a diode whose cathode is markedby a band (c).

Current versus voltage (IV) graphs for diodes aredrawn in Figure 1-38. An ideal or perfect diode wouldpass current freely (with zero resistance) in the

forward direction (with a vertical IV graph) and blockcurrent totally (with infinite resistance) in the reversedirection.

The IV characteristic of an actual diode, however,differs somewhat from this as is also sketched for Geand Si diodes in the figure. When the forward voltageis relatively small the forward current rises exponen-tially with applied voltage. This is the region in whichthe rectifier equation, eq[1-38], holds. For largerforward voltages the IV characteristic straightens outeventually beoming linear.

It can be seen also that the IV characteristic for Gerises “faster” than does the characteristic for Si. If anasymptote is drawn from the high-current regions ofthese curves back across the voltage axis, differing“turn on” voltages would be obtained: about 0.2 voltsfor Ge and 0.6 volts for Si. These differences play arole in determining how the diodes are used (Chapter2).

Also with reference to the reverse direction of thecharacteristics it can be seen that the reverse currentfor Ge is larger than for Si. This makes Si moresuitable for use as a rectifier than Ge.

Figure 1-38. IV characteristics for the Perfect diode and for typical silicon and germanium diodes.

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The PhotodiodeAs we have stated in the previous section, when adiode is reversed biased by an external voltagesource, a small current flows. This current is enhancedif light is allowed to shine on the PN junction.Ordinarily, a diode is encapsulated in an opaquecovering, but if the covering is made transparent andlight is allowed to fall on the PN junction itself, thenthis reverse current is observed to increase (Figure 1-39). The current that flows is observed to dependdirectly on the light intensity. This is the principle ofoperation of the photodiode.

The intensity given in the figure is in footcandles orfc, a unit we shall define in the next section. Thephotodiode has many technical applications and weshall have more to say about it in Chapters 2 and 6. Figure 1-39. Reverse current chacteristic for a typical

photodiode.

Miscellaneous

We collect here a variety of topics that traditionally belong to the areas of environmental scienceand chemistry as much as to physics. Physical quantities such as pressure, temperature, flow rate,relative humidity, pH and light intensity are routinely measured in the sciences.

Atmospheric PressureThe movement of a weather system into or out of ageographical region is commonly signalled by achange in atmospheric pressure. For the most part arising pressure signals the onset of fairer weatherwhile a dropping pressure indicates imminent stormconditions. The simplest device for measuring atmos-pheric pressure is the mercury barometer (Figure 1-40) an 18th century invention.

h

A

Patm Patm

Figure 1-40. A mercury barometer.

A mercury barometer is in essence a tube that issealed at one end, filled completely with mercury, andthen inverted so that the open end is under thesurface of a pool of mercury. As a result of thisprocedure the pressure in region A of the tube (Figure1-40) is negligibly small. The pressure exerted by theatmosphere just balances the pressure of the mercuryin the tube. It can be shown therefore that the atmos-pheric pressure Patm is given by

Patm =F

A=

ρghA

A= ρgh , …[1-40]

(pascals, Pa) where ρ is the density of mercury, A is thecross sectional area of the tube and h is the height ofthe mercury column. Pressure is expressed in avariety of units (Table 1-4), the unit kPa (kilo Pascals)being the most common in those countries (likeCanada) that have adopted the SI system.

A mercury barometer is useful in learning about theconcept of pressure, but is impractical for computerinterfacing. We shall describe electronic pressure

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sensors which are used in computer interfacing inChapter 6.

Table 1-4. Some Pressure Conversions. 1 kPa is equivalent

to: ` 1000 N.m–2

0.009872 atm

10 millibar

0.29516 in. Hg.

Relative HumidityRelative humidity is a quantity measured more for itsvalue as a comfort indicator as for any other reason. Itis defined in terms of the partial pressure of a gas.

Air is a mixture of the gases nitrogen, oxygen andwater vapor. The total pressure of a mixture of gasesis the sum of the partial pressures of the componentgases. The partial pressure of a gas is the pressure itwould exert if it alone occupied the entire volume atthe same temperature as the mixture. The partialpressure of water vapor in air can vary widely.

The relative humidity (rH) indicates the amount ofwater vapor in the air. It is defined as the ratio(expressed as a percentage) of the partial pressure ofwater vapor in the air to the equilibrium vaporpressure at a given temperature:

PercentrH =

Partial pressureof water vapor

Equilibrium vapor pressure ofwater at the existing temperature

x 100

… [1-41]

The equilibrium vapor pressure is the pressure atwhich the vapor is in equilibrium with the liquid. Inother words, the equilibrium vapor pressure (or an rHof 100%) cannot be exceeded; if an RH of 100% isapproached close enough the vapor condenses spon-taneously into dew or rain.

Light IntensityWith the rise in importance of optoelectronics, thesubjects of photometry and device responsivity are ofgrowing interest. In photometry (which is not oftentaught in a first year course in physics) the intensity oflight is expressed in a variety of often confusing units.For example, light meters that are commonly used inphotography are sometimes calibrated in foot-candles,other times in a unit called lux.

Light intensity is a measure of the amount of so-called luminous flux which falls on a particular surfacearea. The area in question might be a square meter ora square foot (thus giving rise to different units). It isuseful to consider a generic area A laying flat on theground at noon on a clear summer day when the sunis directly overhead. The rays of light thus fall straightdown (Figure 1-41).

A

E = hf

Figure 1-41. Light flux is described in a manner reminiscentof electric and magnetic flux.

From countless experiments light is now known topossess the attributes of wave and particle, thoughonly the latter attribute will interest us here. Accord-ing to quantum physics each particle, quantum orphoton of light has an energy E given by

E = hf ,

in joules (J) where h is Planck’s constant (6.624 x 10–34

J.s) and f is the frequency of the light (in Hz). If in tseconds n photons strike an area A then the lightpower P is given by

P =nhf

t Watts.

The quantity called Luminous flux is traditionallymeasured in lumens (lm). The conversion between lu-mens and watts is straightforward: 1 lm = 1.496 x 10–10

W. If one lumen of luminous flux falls on an area ofone square foot then the light intensity there is calleda foot-candle. For historical reasons the footcandle (fc)is the unit of intensity most commonly used in photo-graphy. The following conversions apply:

1 lm.ft–2 = 1 fc = 1.609 x 10–9 W.m–2.

There is also an SI equivalent of the foot-candle calledthe lux which is equal to 1 lm.m–2. Thus

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1 fc ≈ 10 lux.

We are now in a position to be able to make the con-versions between the output of a typical light meter(fc or lux) and the unit of most interest in physics,namely W.m–2.

ResponsivityThe subject of responsivity arguably belongs to a sec-tion on technology. We discuss it here because of itsgeneral nature.

Not all devices, elements or sensors respond to elec-tromagnetic radiation equally and in the same way.The responsivity factor R of an element is defined bythe expression

I = RP , …[1-42]

where I is the measured photocurrent (A) induced in asensor by the incident light, and P is the optical power

(W) incident on the sensor. R depends more-or-lessstrongly on the wavelength.

Relative responsivity is a responsivity curve nor-malized to the maximum responsivity and expressedin %. The relative responsivities of various elementsversus wavelength are sketched in Figure 1-42. Thisfigure extends from the very short wavelength region(far ultraviolet) to the long wavelength region (infra-red). The range of wavelengths in the electromagneticspectrum of most interest to us in this course lay inthe visible range, extending from about 4000 to 7500Å, or in terms of color from violet to deep red (theregion is shown shaded in the figure).

It can be shown that germanium and silicon have afairly wide responsivity, with germanium peaking inthe far infrared. Of the three elements shown in thefigure, selenium has a responsivity most like thehuman eye, making it of wide use in sensors. Selen-ium and germanium detectors are discussed in moredetail in Chapters 2 and 6.

Figure 1-42. The responsivity of various elements: the human eye (dark), selenium, germanium and silicon.

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Practice Problems

1. What must be the power rating of a 100 Ω resistorif 100 V is to be applied to it? For a 100 kΩ resis-tor?

2. In a section of a circuit the power is reduced by afactor of 1/4. What is the reduction in dB?

3. In a section of a circuit the voltage is reduced by afactor of 1/4. What is the reduction in dB?

4. A power supply delivers 20.0 V at a current of0.50 A. What is the power output? How muchenergy is delivered in 5.0 s?

5. The 12V power supply drawn in the followingfigure is used to charge a single cell nickel-cad-mium battery. Answer the following questions:

+

–

12V

PowerSupply

R

Ni-Cadrechargeablebattery

(a) If the battery typically develops 1.3V while beingcharged what size resistor is needed to produce a50 mA current?

(b) Assuming that the battery develops 1.5 V whenfully charged, calculate the current.

(c) How much power is being dissipated as heat inthe battery after it is fully charged?

(d) How much power is being dissipated in theresistor R?

6. If a battery develops 1.5V across its terminalswhen unloaded but only 1.3V when connected toa 100 Ω load, what is the battery’s internal imped-ance?

7. A voltmeter whose resistance is 1000 Ω is used tomeasure the voltage of a worn-out flashlight bat-tery. If the reading is 0.9 V, what is the battery’sinternal resistance?

8. A 100 Ω 1/4 W resistor is connected to the outputof a power supply capable of producing a max-

imum of 20V. At high output levels is it likely thatth resistor will burn out? Explain why or why not.

9. Two A cells of 1.5 V are connected in series andthen to a load resistor of 200 Ω. Calculate thepower dissipated in the resistor.

10. Two A cells of 1.5 V are connected in parallel andthen to a load resistor of 200 Ω. Calculate thepower dissipated in the resistor.

11. Calculate the voltage drop across R2 in thefollowing circuit.

I

10 V

R1 = 100 Ω R2 = 200 Ω

12. Calculate the current through R1 in the followingcircuit.

I I2

I1

+ –

R1 = 100 Ω

R2 = 200 Ω

10 V

13. A D cell of 1.5 V is connected to a 0.1 µF capacitor.How much work is done in charging thecapacitor?

14. An electric current flowing through a 10 Hinductor is made to change by 0.5 A in 1 ms.Calculate the emf induced across the inductor.

15. In the following circuit V = 10 V, R = 10 Ω, and C= 1 F. If the capacitor was initially charged to avoltage of 10 V how much time is required for thevoltage across the capacitor to drop to 3.68 volts?Sketch a diagram of the voltage decay.

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V C

B

A S

R

16. At some point on the earth’s surface the magneticfield strength is measured to be 0.3 gauss with adeclination of 5.0 degrees. Calculate the horizon-tal and vertical components of the field.

17. What height of a mercury column in meters issupported by an atmospheric pressure of 101 kPa?

18. At high noon on a summer day the solar insola-tion is observed to be 1000 W.m–2. What would bethe maximum power input to a silicon solar arrayof dimensions 6 cm x 12 cm?

19.

EndNotes for Chapter 11 To fully explain what charge is to the satisfaction of the professional particle physicist would take us into recent,advanced areas of physics and many pages of abstruce mathematics. It is far simpler for us to think of charge as being simplypositive or begative.2 The signs are quite arbitrary. Legend has it that Benjamin Franklin was the one to coin the terms negative and positive forthose charged bodies which were subsequently found through later experiments to possess an excess of electrons and adeficiency of electrons, respectively.3 The unit of current, the ampere, is defined more strictly in physics than is implied here. However, the difference is notimportant in a course in instrumentation.4 Our use of the phrases “voltage across” and “current through” are deceptively simple but extremely important for acorrect understanding of electricity. The idea of voltage being a driving force has historical roots; it was originally called anelectromotive force, or emf for short.5 The reason for using the direction of conventional current instead of the direction of the actual electron current is partlyhistorical and partly consistency. Most textbooks in electricity and magnetism targeted to the university level use the conven-tional current direction. Some high school and community college textbooks focussed on technology use the electron currentconvention. You would need a second year course in electricity and magnetism to appreciate the rationale for the conventionalcurrent direction.6 A resistor does not so much “resist” the flow of current as “limit” the flow of current. The use in physics and electronics ofthe word “resistor” is unfortunate; it would have been better had a resistor been first called a “limiter” in the 19th century.7 This seemingly peculiar definition was proposed by Alexander Graham Bell, the inventor of the telephone. Bell, in hiswork with the deaf and his long-time research into the science of hearing discovered that the human ear responds to changesin sound intensity not linearly as might be expected, but logarithmically.8 This fact of uniform distribution of static charges on the outside of a conductor is discussed in a full scale study of staticelectricity.9 The intimate connection referred to takes up a large area of study in physics called electricity and magnetism or moreformally, electromagnetic theory.10 This picture, like many others in this book, was transferred from a Tek TDS210 digital oscilloscope (described inAppendix A) using the program Oscar (described in Appendix C).11 This would be like trying to divide two vectors, which is equally meaningless.12 AC voltage, current and impedance are analogous to vectors in mechanics. Just as a vector has a magnitude and adirection, an AC voltage, current and impedance have a magnitude and a phase angle. Indeed, AC quantities are given asimilar name: phasors.