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Chapter 22 Electromagnetic induction

 

clip_image002

Above, a magnet is moved into a coil. If the flux through the coil changes (at a steady rate) by DF in time Dt, then an EMF of clip_image004 is induced in each turn. But there are N turns in series. So, the total induced EMF E is as follows:

clip_image006

For example, if the flux changes by 6 Wb in 2 s, and the coil has 100 turns, then the total induced EMF is 300 V.

Continue reading

Chapter 21 Magnetic fields

21.1 Magnetic field B

Magnetic Field B is a field of force that exist around the magnetic body or current carrying conductor[1].

Magnetic field B, also known as magnetic flux density or magnetic field Intensity[2].

clip_image002

we can summarize the result with the following equation:

(Direction) clip_image004

(Magnitude) clip_image006

21.2 Force on a moving charge

clip_image007

If the charge q moving in a magnetic field B with a velocity v. The angle between B and v is q the charge will experience magnetic force as clip_image006[1].

Unit bagi magnetic flux density adalah Tesla,T atau Wbm-2

To find out the direction of the force also can be shown using Fleming’s left-hand rules.

Contoh 1

Satu eletron bergerak dalam medan magnet. Pada satu ketika laju elektron ialah 3.0 ´ 106 ms-1. Magnitud daya magnet yang bertindak pada elektron ialah 5.0 ´ 1013 N. Sudut di antara arah halaju elektron dan arah daya magnet ialah 30°. Kira ketumpatan fluks magnet pada kedudukan elektron medan itu.

[cas elektron = -1.6 ´ 10-19 C]

21.3 Force on a current-carrying conductor

clip_image010

Above, a current-carrying wire is at right-angles to a uniform magnetic field. The field exerts a force on the wire. The direction of the force is given by Fleming’s left-hand rule . The size of the force depends on the current I, the length l in the field, and the strength of the field. This effect can be used to define the magnetic field strength, known as the magnetic flux density, B:

FB = BIl (1)

B is a vector. The SI unit of B is the testa (T). For example, if the magnetic flux density is 2 T, then the force on 2 m of wire carrying a current of 3 A is 2 x 2 x 3 = 12 N.

If a wire is not at right angles to the field, then the above equation becomes

FB=BIl sinq

where q is the angle between the field and the wire. As q becomes less, the force becomes less. When the wire is parallel to the field, sin q = 0, so the force is zero.

Contoh 2

clip_image012

Satu konduktor lurus panjang 2m membawa arus 1.5A dan di letakkan dalam medan magnet. Medan itu mempunyai ketumpatan fluks magnet bermagnitud 3.0 ´ 10-3 T. Sudut antara arah medan dan konduktor ialah 30°. Kirakan magnitud daya magnet yang bertindak pada konduktor dan nyatakan arah daya itu.

21.4 Magnetic fields due to currents

clip_image014

The Biot-Savart law,

On the top, a short length dl of thin wire, carrying a current I, causes a magnetic flux density dB at P. According to the Biot-Savart law

clip_image016

With a suitable constant, this can be turned into an equation:

clip_image018

For a vacuum (and effectively for air), the value of k is 10-7 T m A-1. However, in practice, another constant, m0, is used, and the above equation is rewritten as follows:

clip_image020 (2)

m0clip_image022 is called the permeability of free space. Its value is 4p x 10-7 T m A-1. This is not found by experiment. It is a defined value, linked with the definition of the ampere .

Calculating magnetic flux density Using equation clip_image020[1] and calculus, it

is possible to derive equations for B near wires and inside coils carrying a current I.

clip_image024

B near an infinitely long, thin, straight wire At a distance a from such a wire (as on the top)

clip_image026

Contoh 3

Arus sebanyak 10A mengalir melalui seutas dawai mencancang lurus yang panjang. Kirakan komponen mengufuk medan magnet Bumi. Jika terdapat satu titik neutral 8.0cm dari dawai itu. [m0= 4p ´ 10-7 Hm-1]

clip_image028

B at the centre of a thin coil Or the axis of such a coil, of N turns and radius r (as on the top),

clip_image030

Contoh 4

Arus yang sama magnitud mengalir secara bergilir-gilir melalui seutas dawai tegak lurus yang panjang dan satu gegelung membulat berjejari 10cm dimana satahnya tegak dan selari dengan komponen mengufuk medan magnet pada titik berjarak 10cm di utara atau di selatan dari dawai itu terhadap paduan medan magnet di pusat gegelung itu.

Contoh 5

Satu gegelung membulat diletakkan, dimana satahnya adalah mencancang dan paksinya membuat sudut kecil q dengan meridian magnet itu. Arus dilalukan mengelilingi gegelung supaya medan yang dihasilkan berkecenderungan untuk menentang medan Bumi. Satu magnet mengufuk yang kecil di letakkan di pusat gegelung itu.

Perihalkan dan terangkan kesan ke atas magnet itu apabila kekuatan arus bertambah. Gegelung itu mempunyai jejari min 10cm,50 lilitan dan arus sebanyak 0.5A mengalir didalamnya. Jika q=12° dan magnet mengarah ke Timur-Barat.Kirakan komponen mengufuk medan magnet Bumi.

clip_image032

B inside an infinitely long solenoid (coil) The field inside such a solenoid (as on above) is uniform. If n is the number of turn per unit length (per m), then

B=µ0nI

This equation is a reasonable approximation for any solenoid which is at least ten times longer than it is wide.

Contoh 6

Berapa banyakkah lilitan mesti dipunyai oleh satu solenoid jika panjang solenoid itu 2.0m dan arus 5.0A mengalir melaluinya supaya arus menghasilkan ketumpatan fluks magnet bermagnitud 2.0 ´ 10-2 T di pusat solenoid ?

B inside a solenoid with a core The value of B is changed by a core. For example, with a pure iron core, B is increased by a factor of about 1000 (depending on the temperature). The previous equation then becomes:

B = µrµ0 nI

µr, is called the relative permeability of the material. So, for pure iron, µr, is about 1000.

An electromagnet is a solenoid with a core of high µr.

21.5 Force between two current-carrying conductors

Force between two current-carrying wires

clip_image034

X and Y above are two infinitely long, straight wires in a vacuum. The current in X produces a magnetic field, whose flux density is B at Y. As a result, there is a force on Y. F is the force acting on length l.

From equation (3) clip_image036

From equation (1) FB = BI2 l

clip_image038

Note:

• The above equation gives the force of X on Y. Working out the force of Y on X gives exactly the same result.

• If the two currents are in the same direction (as above), then the wires attract each other. If the two currents are in opposite directions, then the wires repel.

Contoh 7

clip_image040

Tiga konduktor yang selari, P, Q dan R berada pada satah yang sama.Jarak antara P dan Q adalah 4cm dan antara Q dan R ialah 2 cm. Arus dalam P dan Q adalah 10A dan 5A masing-masing dan mengalir pada arah yang berlawanan.Daya paduan ke atas Q adalah 5 ´ 10-4N per meter dan bertindak ke arah P. Kirakan magnitud dan arah arus dalam R.

21.6 Definition of ampere

Defining the ampere

The SI unit of current is defined as follows:

One ampere is the current which, flowing through two infinitely long, thin, straight wires placed one metre apart in a vacuum, produces a force of 2 x 10-7 newtons on each metre length of wire.

Using the various factors in equation

clip_image038[1] ,

the above definition can be expressed in the following form (for simplicity, units have been omitted):

If I1=I2=1A,a=1m and l=1m,then F= 2×10-7N. Substituting these in equation above gives µ0 = 4p x 10-7. The above definition is not a practical way of fixing a standard ampere. This is done by measuring the force between two current-carrying coils.

Current Balance

- Measure current by using the principle of the force between two paralell wire using mechanical

clip_image042

Prinsip:

clip_image044

m = jisim pemberat W

b = jarak F ke tuas

c = jarak berat ke tuas

L = jarak XY

apabila sistem tuas seimbang kita dapati

clip_image046

Known that

FB = BIL

B=µ0nI

Maka FB = µ0n I . IL

Then clip_image048

clip_image050

21.7 Torque on a coil

clip_image052

Lets consider the force on PQ and RS that involve in producing the torque.

Force that produce on PQ and RS have the same magnitude but different direction.

so force,F

clip_image054

where

L = height of coil(length of the coil in the magnetic field) and N = no. of turns carrying current,I.

Torque

clip_image056

clip_image058

where

q = angle (below the horizontal from the plan view) between magnetic flux density and the coil.

[coil area,A = Lb]

So the equation for torque will be

clip_image060

When

q = 0°

t = BIAN

Contoh 8

21.8 Determination of ratio e/m and q/m

Using an electron gun

Lets consider the electron gun.

clip_image062

Lets consider the movement of electron in an electro gun

1. The kinetic energy gain by electron = Electric potential energy of electron

2. Electron velocity :

clip_image064

clip_image066

3. Electric force, FE = Magnetic Force, FB , when it’s in equilibrium

clip_image068

clip_image070

4. From equation (2) and (3) we can conclude:

clip_image072

clip_image074

Todetermine any positive charge,q/m using spectrometer.

1. any positivly charge velocity,

clip_image076

2. The charges follow the circular path, the centripetal force of the charge.

clip_image078

3.Magnetic force of charges

clip_image080

clip_image082

Also clip_image084

clip_image086

clip_image088

clip_image090 for any specific charges.

21.9 Hall effect

1. apabila arus mantap I melalui konduktor/semikonduktor

clip_image092

2. kemudian diletakkan kedalam medan magnet yang kuat dimana I berserenjang dengan B.

3. Medan magnet melencongkan cas positif ke bahagian atas dan cas negatif ke bahagian bawah

clip_image094

4. Pemisahan ini menghasilkan medan elektrik E yang berserenjang dengan arah aliran arus I

- Bila cas bertambah

- Medan elektrik juga bertambah

- Daya gerak elektrik juga meningkat

clip_image096

5. Pemisahan akan berhenti bila daya E, FE=FB

Magnetic force = Electric force

clip_image098

clip_image100 where [clip_image102 n ialah halaju hanyut elektron,n bilangan elektron bebas]

clip_image104

Hall voltage clip_image106

clip_image108


[1] Oxford Dictionary

[2] Fajar Bakti STPM V2 m.s 113

Chapter 15 Direct current circuits

15.1 Internal resistance

15.2 Kirchhoff s laws

15.3 Potential divider

15.4 Potentiometer and Wheatstone bridge

Introduction

Voltage (PD and EMF)

In the circuit below, several cells have been linked in a line to form a battery. The potential difference (PD) across the battery terminals is 12 volts (V). This means that each coulomb (C) of charge will ‘spend’ 12 joules of energy in moving round the circuit from one terminal to the other.

clip_image002

Figure 15- 1

The PD across the bulb is also 12 V. This means that, for each coulomb pushed through it, 12 J of electrical energy is changed into other forms (heat and light energy).

PD may be measured using a voltmeter as shown above.

PD, energy, and charge are linked by this equation:

Energy transformed = charge x PD

For example, if a charge of 2 C moves through a PD of 3 V, the energy transformed is 6 J.

The voltage produced by the chemical reactions inside a battery is called the electromotive force (EMF). When a battery is supplying current, some energy is wasted inside it, which reduces the PD across its terminals. For example, when a torch battery of EMF 3.0 V is supplying current, the PD across its terminals be might be only 2.5 V.

15.1 Internal resistance

(a) explain the effects of internal resistance on the terminal potential difference of a battery in a circuit;

Internal resistance

In reality, when a battery is supplying current, its output PD is less than its EMF. The greater the current, the lower the output PD. This reduced voltage is due to energy dissipation in the battery. In effect, the battery has internal resistance. Mathematically, this can be treated as an additional resistor in the circuit.

clip_image004

Figure 15- 2

The battery above is supplying a current I to an external circuit. The battery has a constant internal resistance r.

From Kirchhoff’s second law :

clip_image006

But clip_image008, so clip_image010

So clip_image012……………(1)

clip_image014

Figure 15- 3

The graph above shows how V varies with I. Unlike earlier graphs, V is on the vertical axis.

Note:

• When I is zero, clip_image016. In other words, when a battery is in open circuit (no external circuit), the PD across its terminals is equal to its EMF

• When R is zero, V is zero. In other words, when the battery is in short circuit (its terminals directly connected), its output PD is zero. In this situation, the battery is delivering the maximum possible current, clip_image018, which is equal to clip_image020. Also, the battery’s entire energy output is being wasted internally as heat.

• As clip_image022, it follows that clip_image024. So the gradient of the graph is numerically equal to the internal resistance of the battery.

If both sides of equation (1) are multiplied by I, the result is clip_image026. Rearranged, this gives the following:

clip_image028

Figure 15- 4

Example 15.1

15.2 Kirchhoff’s law

(b) state and apply Kirchhoff s laws;

Introduction

clip_image032

Figure 15- 5

1. Figure 15-5 shows three typical circuit diagrams that might need to be solved (e.g. given the resistances of all the resistors and the voltages of all the batteries, find all of the currents). Figure 15-5 (a) can be solve easily using Ohm’s Law, but (b) and (c) cannot be solved using the same law. Instead, we must write down Kirchhoff’s laws and solve the equations.

Kirchhoff’s first law (KFL)

clip_image034

Figure 15- 6

Junction is a point where two or more conductor meet together.

The currents at junctions X and Y above illustrate a law which applies to all circuits:

Kirchhoff’s first law

The algebraic sum of currents in a network of conductors meeting at a point is zero

It arises because, in a complete circuit, charge is never gained or lost. The junction rule is based on the conservation of the electric charge . So the total rate of flow of charge is constant. This means that :

clip_image036

Let’s consider:

Current Junction:

clip_image038

Figure 15- 7

Positive Direction:clip_image040

Negative Direction: clip_image042

Used Kirchhoff’s First Law:

clip_image044

Kirchhoff’s second law (KSL)

Energy, work and EMF

1. When we discuss about the KSL we have to represent the EMF in term of magnitude and direction inside the circuit. The EMF device always keeps one of their terminal labeled ‘+’ at higher electric potential than labeled ‘-’. This will present in arrow diagram as:

clip_image046

Figure 15- 8

2. when connected to the circuit, EMF will causes a net flow of positive charge from positive terminal to negative terminal in the same direction as EMF, this flow is part of current. The flows of current through the load (resistor) within the circuit will made the EMF drop this concept name as voltage drop. The direction of the voltage drop oppose the current flow.

clip_image048

Figure 15- 9

clip_image050

Figure 15- 10

The arrangement above is called ‘a circuit’. But, really, there are two complete circuits through the battery. To avoid confusion, these will be called loops.

Lets consider:

Closed Loops

Parallel

Series

clip_image052

Figure 15- 11

Loop 1 and Loop 2 can be form

clip_image054

Figure 15- 12

Loop 1

In the circuit above, charge leaves the battery with electrical potential energy. As the charge flows round a loop, its energy is ‘spent’ – in stages – as heat. The principle that the total energy supplied is equal to the total energy spent (conservation of energy) is expressed by Kirchhoff’s second law.

Kirchhoff’s second law

Round any closed loop of a circuit, the algebraic sum of the EMFs is equal to the algebraic sum of the PDs (i.e. the algebraic sum of all the IRs).

This would means that :

clip_image056

Note:

•From the law, it follows that if sections of a circuit are in parallel, they have the same PD across them.

•’Algebraic’ implies that the direction of the voltage must be considered. There are two rule have to be followed to determine the direction of the voltage.

EMF rule: for a move through ideal EMF device (loop direction) in the direction of EMF arrow, the EMF is ‘+’ (+E) : in the opposite direction is ‘-‘ (-E)

Resistance rule: for a move through a resistance (loop direction) in the direction of current, the voltage drop is ‘-‘ (-IR)

For example, in the circuit below, it’s assume that the current is flow in counter clock wise, the EMF of the right-hand battery is taken as negative (-4 V) because it is opposing the loop direction and the voltage drop is positive because it’s oppose to the loop direction, therefore:

clip_image058

Figure 15- 13

(a) Algebraic sum of EMFs = 18 + (-4) = +14V

(b) Algebraic sum of IRs (Voltage drop) = (2 x 3W) + (2 x 4W) = +14 V

Applying the second Kirchhoff’s law the equation will be :

clip_image060

Resistors in parallel

clip_image062

Figure 15- 14

From Kirchhoff’s second law (applied to the various loops):

E = IR (Loop with total Resistor)

and

E = I1 R1 (Loop with Resistor R1)

and

E = I2 R2 (Loop with Resistor R2)

From Kirchhoff’s first law I = I1 + I2.

So

clip_image064

clip_image066

Resistors in series

If R1 and R2 below have a total resistance of R then R is the single resistance which could replace them.

clip_image068

Figure 15- 15

From Kirchhoff’s first law, all parts of the circuit have the same current /through them because there is only one input and one output

From Kirchhoff’s second law

E= IR and E= IR1 + IR2.

So IR = IR1 + IR2

\ R=R1+R2

For example, if R1 = 3 W and R2 = 6 W, then R = 9 W.

Example 15.2

15.3 Potential divider

(c) explain a potential divider as a source of variable voltage;

(d) explain the uses of shunts and multipliers;

Potential divider

a voltage divider (also known as a potential divider) is a linear circuit that produces an output voltage (Vout) that is a fraction of its input voltage (Vin)

A potential divider or potentiometer like the one below passes on a fraction of the PD supplied to it.

clip_image129

Figure 15- 21

In the input loop above, the total resistance = R1 + R2.

So clip_image131

But Vout = IR2,

so clip_image133

Note:

•The above analysis assumes that no external circuit is connected across R2. If such a circuit is connected, then the output PD is reduced.

In electronics, a potential divider can change the signals from a sensor (such as a heat or Iight.detector) into voltage changes which can be processed electrically. For example, if R2 is a thermistor, then a rise in temperature will cause a fall in R2 and therefore a fall in Vout. Similarly, if R2 is a light­dependent resistor (LDR), then a rise in light level will cause a fall in R2, and therefore a fall in Vout.

Potential dividers are not really suitable for high-power applications because of energy dissipation

Shunt and multiplier

(a) Conversion of Galvanometer to ammeter

clip_image135

Figure 15- 22

Shunts

Shunts is a resistor connected in parallel

Since clip_image137

clip_image139

clip_image141

(b) Conversion of Galvanometer to voltmeter

clip_image143

Figure 15- 23

clip_image145

clip_image147

clip_image149

clip_image151

15.4 Potentiometer and Wheatstone bridge

(e) Explain the working principles of a potentiometer, and its uses;

(f) Explain the working principles of a Wheatstone bridge, and its uses;

(g) Solve problems involving potentiometer and Wheatstone bridge.

Potentiometer

Potentiometer is an instrument that can be used to measure the emf of a source without drawing (considering) any current from the source.

Function : To measure emf a cell

Key Idea : make sure the galvanometer as a null detector

clip_image153

Figure 15- 24

E = lV

where

V = potential difference per unit length of AB.

L = The length of wire

l = length that galvanometer show zero reading

so PD across l,clip_image155

Potentiometer Applications

(a) Measuring a cell’s internal resistance.

clip_image157

Figure 15- 25

clip_image159

clip_image161

clip_image163 or clip_image165

clip_image167

Figure 15- 26

If a graph is plotted,

Gradient, clip_image169

Internal resistance, clip_image171

Intercept, clip_image173

clip_image175

(b) Comparing resistance

clip_image177

Figure 15- 27

clip_image179

clip_image181

clip_image183

Wheatstone bridge

1. A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit.

2. A bridge circuit is a type of electrical circuit in which two circuit branches (usually in parallel with each other) are "bridged" by a third branch connected between the first two branches at some intermediate point along them:

clip_image185

Figure 15- 28. (a) a parallel circuit , (b) a Bridge circuit

3. a ratio between resistance given by clip_image187

Example 15.3

Chapter 14 Electric current

14. Electric Current

14.1 Conduction of electricity

14.2 Drift velocity

14.3 Current density

14.4 Electric conductivity and resistivity

Introduction

Static electricity

if two materials are rubbed together, electrons may be transferred from one to another. As a result, one gains negative charge, while the other is left with an equal positive charge. If the materials are insulators, the transferred charge does not readily flow away. It is sometimes called static electricity.

clip_image002[4]

Figure 14- 1

A charged object will attract an uncharged one. The charged rod has extra electrons. Being uncharged, the foil has equal amounts of – and + charge. The – charges are repelled by the rod and tend to move away, while the + charges are attracted. However, the force of attraction is greater because of the shorter distance.

clip_image004[4]

Figure 14- 2

Charge which collects in one region because of the presence of charge on another object is called induced charge.

14.1 Conduction of electricity

(a) define electric current, and use the equation clip_image006[4]

(b) explain the mechanism of conduction of electricity in metals;

Current

Conventional direction

clip_image008[4]

Figure 14- 3

In the circuit above, chemical reactions in the cell push electrons out of the negative (-) terminal, round the circuit, to the positive (+) terminal. This flow of electrons is called a current.

An arrow in the circuit indicates the direction from the + terminal round to the -. Called the conventional direction, it is the opposite direction to the actual electron flow.

The SI unit of current is the ampere (A).

A current of 1 A is equivalent to a flow of 6 x 1018 electrons per second. However, the ampere is not defined in this way, but in terms of its magnetic effect .

\ One Ampere is the current which, flowing through two infinitely long, thin, straight wires placed one metre apart in a vacuum, produces a force of 2´10-7 N on each metre length of wire

Current may be measured using an ammeter as above.

Conductors and insulators

Current flows easily through metals and carbon. These materials are good conductors because they have free electrons which can drift between their atoms.

Most non-metals are insulators. They do not conduct because all their electrons are tightly held to atoms and not easily moved. Although liquids and gases are usually insulators, they do conduct if they contain ions.

Semiconductors, such as silicon and germanium, are insulators when cold but conductors when warm.

What happen inside the conductor?

Charge

Charge can be calculated using this equation:

Charge,Q = current,I x time,t

The SI unit of charge is the coulomb (C).

Case in point

if a current of 1 A flows for 1 s, the charge passing is 1 C. (This is how the coulomb is defined.) Similarly, if a current of 2 A flows for 3 s, the charge passing is 6 C.

Energy and electrical power

Energy transfer

clip_image010[4]

Figure 14- 4

Above, charge Q passes through a resistor in time t. Work W is done by the charge, so energy W is transformed – the electrons lose electrical potential energy and the lattice gains internal energy (it heats up).

W, Q, and V are linked by this equation:

W= QV

But Q= It,

so W= VIt (1)

Applying V = IR to the above equation gives

W= I2 Rt and clip_image012[4] (2)

For example, if a current of 2 A flows through a 3 W resistor for 5 s, W = 22 x 3 x 5 = 60 J. So the energy dissipated is 60 J. Double the current gives four times the energy dissipation.

Note:

• Equation (1) can be used to calculate the total energy transformation whenever electrical potential energy is changed into other forms (e.g. KE and internal energy in an electric motor). Equations (2) are only valid where all the energy is changed into internal energy. Similar comments apply to the power equations which follow.

As power clip_image014[4], it follows from (1) and (2) that

P= VI P= I2R clip_image016[4]

Example 14.1

14.2 Drift velocity

(c) explain the concept of drift velocity;

(d) derive and use the equation clip_image035[4]

Current and drift velocity

1. we can express current in terms of the drift velocity of the moving charges.

Most electrons are bound to their atoms. However, in a metal, some are free electrons which can move between atoms. When a PD is applied, and a current flows, the free electrons are the charge carriers.

clip_image037[4]

Figure 14- 5

in the wire above, free electrons (each of charge e) are moving with an average speed v. n is the number density of free electrons: the number per unit volume (per m3).

In the wire the number of free electrons = clip_image039[4]

So total charge carried by free electrons = clip_image041[4]

As clip_image043[4]:

\Time,t taken for all the free electrons to pass through clip_image045[4]

As clip_image047[4]

clip_image049[4]

\clip_image051[6]

clip_image053[6] is called the drift speed or drift velocity. Typically, it can be less than a millimeter per second for the current in a wire.

Example 14.2

14.3 Current Density

(e) define electric current density and conductivity;

(f) use the relationship clip_image061[6]

The current density,clip_image063[4] is the current per unit cross-sectional area (per m2).

clip_image065[4] so clip_image067[4]

Note:

• The number density of free electrons is different for different metals. For copper, it is 8 x 1028 m-1.

• When liquids conduct, ions are the charge carriers. The above equations apply, except that e and n must be replaced by the charge and number density of the ions.

Example 14.3

14.4 Electrical conductivity and resistivity

(g) derive and use the equation clip_image076[6]

(h) define resistivity, and use the formula clip_image078[4] ;

(i) show the equivalence between Ohm’s law and the relationship clip_image061[7];

(j) explain the dependence of resistivity on temperature for metals and semiconductors by

using the equation clip_image076[7];

(k) discuss the effects of temperature change on the resistivity of conductors, semiconductors and superconductors.

Conductance

If a PD V is applied across a conductor, and a current I flows, then V = IR.

However, as V is the cause of the current and I s the effect, it is more logical to write this as:

clip_image081[4]

clip_image083[4] is called the conductance.

Resistivity

clip_image085[4]

Figure 14- 6

The resistance R of a conductor depends on its length l and cross-sectional area A:

clip_image087[4]

This can be changed into an equation by means of a constant, clip_image089[4](rho), known as the resistivity of the material:

clip_image091[4]: where l is the length

With this equation, the resistance of a wire can be calculated if its dimensions and resistivity are known.

And clip_image093[4]is the Electrical conductivity,s (sigma)

Example 14.4

14.6 Dependence of resistance on temperature

Lets consider;

1. Current Density

clip_image107[4]

2. Electrical Conductivity

clip_image109[4]

3. Electrical resistivity

clip_image111[4]

then clip_image113[4] but (clip_image115[4])

clip_image117[4]

clip_image119[4]

clip_image121[4]

so clip_image123[4] then

clip_image125[4]

clip_image127[4]

clip_image129[4]constant if temperature constant

E, causes free electron of charges,clip_image053[7], drift opposite the electric field.

clip_image132[4] but clip_image134[4]

clip_image136[4]

clip_image138[4]

so electrical Conductivity

clip_image140[4]

Resistance and temperature

A conducting solid is made up of a lattice of atoms. When a current flows, electrons move through this lattice.

clip_image142[4]

Figure 14- 8

Metals : when free electrons drift through a metal, they make occasional collisions- with the lattice. These collisions are inelastic and transfer energy to the lattice as internal energy. That is why a metal has resistance. If the temperature of a metal rises, the atoms of the lattice vibrate more vigorously. Free electrons collide with the lattice more frequently, which increases the resistance.

Semiconductors (e.g. silicon) At low temperature, the electrons are tightly bound to their atoms. But as the temperature rises, more and more electrons break free and can take part in conduction. This easily outweighs the effects of more vigorous lattice vibrations, so the resistance decreases. At around 100-150 °C, breakdown occurs. There is a sudden fall in resistance – and a huge increase in current. That is why semiconductor devices are easily damaged if they start to overheat.

The conduction properties of a semiconductor can be changed by doping it with tiny amounts of impurities. For example, a diode can be made by doping a piece of silicon so that a current in one direction increases its resistance while a current in the opposite direction decreases it.

clip_image144[4]

The graphs above are for a typical metal conductor and one type of thermistor. The thermistor contains semiconducting materials

Chapter 13 Capacitor

Chapter 13 Capacitor

13.1 Capacitance

13.2 Parallel plate capacitors

13.3 Dielectrics

13.4 Capacitors in series and in parallel

13.5 Energy stored in a charged capacitor

13.6 Charging and discharging of a capacitor

Introduction

Capacitors store small amounts of electric charge.

clip_image002[6]

Figure 13- 1

A capacitor is a can full of thin metal sheets held very close together but separated by an insulator. You pump electrons in to charge up the plates, and then the capacitor can pump electrons out again to power a circuit. A capacitor is much simpler than a battery, as it can’t produce new electrons — it only stores them. Inside the capacitor, the terminals connect to two metal plates separated by a non-conducting substance, or dielectric (insulator) .In theory, the dielectric can be any non-conductive substance. However, for practical applications, specific materials are used that best suit the capacitor’s function. Mica, ceramic, cellulose, porcelain, Mylar, Teflon and even air are some of the non-conductive materials used. The dielectric dictates what kind of capacitor it is and for what it is best suited. Depending on the size and type of dielectric, some capacitors are better for high frequency uses, while some are better for high voltage applications. Capacitors can be manufactured to serve any purpose, from the smallest plastic capacitor in your calculator, to an ultra capacitor that can power a commuter bus. NASA uses glass capacitors to help wake up the space shuttle’s circuitry and help deploy space probes. Here are some of the various types of capacitors and how they are used.

Air – Often used in radio tuning circuits

Mylar – Most commonly used for timer circuits like clocks, alarms and counters

Glass – Good for high voltage applications

Ceramic – Used for high frequency purposes like antennas, X-ray and MRI machines

Super capacitor – Powers electric and hybrid cars

A battery is a can full of chemicals and metals. An electrochemical reaction produces voltage and current.

13.1 Capacitance

(a) define capacitance;

A capacitor can be charged by connecting a battery across it. The higher the PD V, the greater the charge Q stored. Experiments show that Q a V. Therefore, Q/V is a constant. The capacitance C of a capacitor is defined as follows:

clip_image004[4]

In symbols clip_image006[8]

The higher the capacitance, the more charge is stored for any given PD.

Capacitance is measured in C V-1, known as a farad (F). However, a farad is a very large unit, and the mF (10-6 F) is more commonly used for practical capacitors.

Example 13.1

13.2 Parallel Plate capacitors

(b) describe the mechanism of charging a parallel plate capacitor;

(c) use the formula clip_image012[4] to derive clip_image014[4] for the capacitance of a parallel plate capacitor;

Mechanism of charging a parallel plate capacitor

Electric field near a charged plate

clip_image016[4]

Figure 13- 2

On the top, a metal sphere has a charge Q uniformly distributed over its surface. The electric field E near the surface is given by equation:

clip_image018[4]

But the surface area of the sphere A = 4pR2. So

clip_image020[6] (1)

Conclusion:

This equation also applies to a flat, charged metal plate of surface area A.

The simplest form of capacitor is made up of two parallel metal plates, separated by an air gap.

clip_image022[4]

Figure 13- 3

The capacitor above has been connected to a battery so that the PD across its plates is V. As a result, it is storing a charge Q. (This means that charge Q has been transferred, leaving -Q on one plate and +Q on the other.)

From equation (1) above clip_image020[7]

From equation clip_image024[4]

From these, it follows that clip_image026[4]

But capacitance, clip_image006[9]

So clip_image028[6]

From this equation clip_image028[7]

we can conclude that capacitance in a parallel plate is only depend on area A and distance d

clip_image031[4], so a larger plate area gives a higher C.

clip_image033[4], so a smaller plate separation gives a higher C.

Example 13.2

 

13.3 Dielectrics

(d) define relative permittivity clip_image043[4] (dielectric constant);

(e) describe the effect of a dielectric in a parallel plate capacitor

(f) use the formula clip_image045[4]

Dielectric

clip_image047[4]

Figure 13- 4

1. If the gap between the capacitor plates is filled with a material such as polythene, the capacitance is increased. Any insulating material which has this effect is called a dielectric. In practice, this is achieved by rolling up two long strips of foil with a thin dielectric between them.

2. Placing the dielectric in between foil have three functions:

(a) to make sure the plate do not actually contact with each other, even though the gap is very small.

(b) Increasing the potential different between capacitor plates. Insulting material can withstand larger electric field without experiencing the dielectric breakdown. So the capacitors will store great amount of energy. Dielectric breakdown is a partial ionization that allowed conduction through it.

(c) Increasing the capacitance of the capacitor. From the experiment its shows that, the ratio of the capacitance with dielectric clip_image049[4] and capacitance without inserting dielectricclip_image051[4]is constant, the constant named as dielectric constant,clip_image053[8].

clip_image055[4]

The total charge in the capacitor is always the same with or without the dielectric.

clip_image057[4]

Figure 13- 5

3. The dielectric constant,clip_image053[9] also called the relative permittivityclip_image060[4] , indicates how easily a material can become polarized by imposition of an electric field on an insulator (dielectric).

4. The capacitance of the parallel plate capacitor when inserted with dielectric is

clip_image062[4] or clip_image064[4]

The dielectric constantclip_image053[10], of the dielectric is the factor by which the capacitance is increased.

5. In electrolytic capacitors, the dielectric is formed by the chemical action of a current. This gives a very thin dielectric, and a very high capacitance. But the capacitor must always be used with the same plate positive, or the chemical action is reversed. Capacitors have a maximum working voltage above which the dielectric breaks down and starts to conduct.

Example 13.3

13.4 Capacitor in series and parallel

(g) Derive and use the formulae for effective capacitance of capacitors in series and in parallel;

Capacitors in series

clip_image077[4]

Figure 13- 6

if C1, and C2 have a total capacitance of C, then C is the single capacitance which could replace them.

Two capacitors in series store only the same charge Q as a single capacitor. So, clip_image079[4] and clip_image081[4] . But clip_image083[4]

So clip_image085[4]

Or clip_image087[4]

Example 13.4

Capacitors in parallel

clip_image101[4]

Figure 13- 7

if C1, and C2 have a total capacitance of C, then C is the single capacitance which could replace them.

Capacitors in parallel each have the same PD across them.

So ,clip_image103[4]

And clip_image105[4] and clip_image107[4]

Together, the capacitors act like a single capacitor with a larger plate area.

So clip_image109[4]

clip_image111[4]

and clip_image113[4]

Example 13.5

13.5 Energy stored by a capacitor

(h) use the formulae clip_image127[4](derivations are not required);

Work must be done to charge up a capacitor. Electrical potential energy is stored as a result.

If a charge of 2 C is moved through a steady PD of 10 V, then, using equation

Electrical potential, clip_image129[4],

\work done W = QV = 2 x 10 = 20 J.

So the stored energy is 20 J, numerically, this is the area under the graph below.

clip_image131[4]

Figure 13- 9

Energy stored in Capacitor

When a capacitor is being charged, Q and V are related as in the graph below. As before, the energy stored is numerically equal to the area under the graph, which is clip_image133[4]. As clip_image006[10] , this can be expressed in three ways:

Energy stored, clip_image135[4]

clip_image137[4]

Figure 13- 10

Example 13.6

13.6 Charging and discharging

(i) describe the charging and discharging process of a capacitor through a resistor;

(j) define the time constant, and use the formula clip_image149[6];

(k) derive and use the formulae clip_image151[4],clip_image153[4]and clip_image155[6] for charging a capacitor through a resistor;

(l) derive and use the formulae clip_image157[4],clip_image159[4], and clip_image155[7] for discharging a capacitor through a resistor;

(m) solve problems involving charging and discharging of a capacitor through a resistor.

Discharge of a capacitor

clip_image162[4]

Figure 13- 11

The capacitor above is charged from a battery and then discharged through a resistance R.

clip_image164[4]

Figure 13- 12

Graph A shows how, during discharge, the charge Q decreases with time t, according to the following equation:

clip_image166[4] where e = 2.713

RC is called the time constant clip_image168[6]. (it equals the time which the charge would take to fall to zero if the initial rate of loss of charge were maintained.)

clip_image149[7]

Increasing R or C gives a higher time constant, and therefore a slower discharge.

The gradient of the graph at any time t is equal to the current at that time.

clip_image170[4]

Figure 13- 13

Graph B shows how the current decreases with time. The area under the graph is numerically equal to the charge lost.

Note:

•Each graph is an exponential decay curve, with the same characteristics as a radioactive decay curve. A half-life can be calculated in the same way.

Charging a capacitor

The capacitor below is charged through a resistance R. The graph shows how the charge builds up.

clip_image172[4]

Figure 13- 14

The equation for the charging process is clip_image174[4]

The charge reaches a maximum value of clip_image176[4] which is equal to VC.

The charging current starts at a maximum value of V/R and falls to a lower value in the same way as it does when the capacitor discharges.

Voltage-time graphs

Since the voltage across a capacitor is proportional to the charge on it the variations of voltage with time are the same shape as the charge-time graphs. The equations for calculating the voltage at any time are similar to those for charge, substituting V for Q

Example 13.7

 

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