current electricity supp notes

8/6/2019 Current Electricity Supp Notes

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CURRENT ELECTRICITY

BASIC ELECTRICAL QUANTITIES.

Current.

An electric current consists of a flow of charge in some conducting material.This flow of charge could be:

a) A flow of electrons, which is the case in metal conductors and cathode ray tubes.b) A flow of ions, which occurs in melts, solutions and gases.c) In special circumstances, a flow of protons can occur.

By convention, a current is regarded as a flow of positive charges, although the type of currentyou are most likely to encounter is a flow of electrons. Current is defined as the rate of flow of

charge:

Current is measured in amperes. If one coulomb of charge passes a point in a conductor everysecond then one ampere of current is flowing.

NB: this is not the definition of the ampere; we will define the ampere later when we deal withelectromagnetism

Potential Difference.

Potential difference is what is commonly (but rather loosely) called voltage'. It is defined as thework done (or energy transferred) in moving a unit charge from one point in a circuit to another.'

The unit of potential difference is the volt. The potential difference between two points in acircuit is one volt when one joule of work is done in moving one coulomb of charge from onepoint to the other.

Resistance.

Charges are not completely free to move; all materials retard orresistthe flow of charge tosome extent. The resistance of a conductor is defined as the ratio of the potential differenceacross the ends of the conductor to the current flowing in the conductor.

Resistance is measured in ohms. ()


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The resistance of a given conductor depends on: The materialthe conductor is made of. The length of the conductor; the longer the conductor the higher the resistance. The thickness of the conductor; the thicker the conductor the lower the resistance. The temperature of the conductor; for a metal conductor, the higher the temperature the

higher the resistance; for semiconductors, the higher the temperature, the lower theresistance.

ELECTRICITY AND MATERIALS

Materials can be classified as conductors, insulators or semi-conductors according to theirelectrical conducting properties.

Any material in which charges are relatively free to move is a conductor. The mostfamiliar example is the metal conductor, in which there are free electrons in a matrix ofpositive ions . Current can also be conducted in aqueous solutions, melts and ionisedgases.

Materials such as plastics and ceramics, where the charges are not free to move areinsulators.

Some materials possess the properties of both conductors and insulators, dependingon the conditions. These are known as semi-conductors. Carbon for example, can bean insulator at room temperature but will conduct a current at higher temperatures. Twoother examples of semi-conductors are silicon and germanium.

If a current is regarded as a flow of charged particles, then it follows that for a material to becapable of carrying a current, two conditions must be met:

firstly there must be charged particles present, (i.e. particles with a netcharge, such aselectrons or ions) and

secondly those particles must be free to move.

This means that a substance can be an insulator even if particles with a net charge are present.

(s)For example - solid table salt - NaCl - will not conduct a current because the ions that arepresent occupy fixed positions in the crystal lattice. If however, the salt is melted or dissolvedin water, the ions become free to move and a current can then flow.

Ohms Law

Ohms Law states that the current flowing in a metal conductor is directly proportional to thepotential difference across its ends provided the resistance of the conductor remains constant.

The resistance of a given conductor is likely to change only with temperature, as the otherdetermining factors will normally remain constant. For metal conductors the higher thetemperature, the higher the resistance. These conductors are called ohmic conductors.


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Some New Electrical Quantities

The current density at a point in a conductor is defined as the current per unit cross-sectionalarea at that point:

At a given temperature, the resistance of a metal conductor is proportional to its length andinversely proportional to its cross sectional area:

The constant of proportionality is called the resistivity of the conductor. The resistivity is

an inherent property of a given material and is temperature dependent - i.e. its value changeswith changes in temperature. For ohmic conductors, the higher the temperature, the higher the

resistivity. Resistivity is defined by the equation: and hence we can also say:

. Resistivity is measured in ohm metres:

The reciprocal of resistivity is called the conductivity i.e.

Conductivity is measured in siemensper metre, where 1 siemen = 1


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Superconductivity

The resistivity - and hence, the resistance - of a given metal (ohmic) conductor depends on itstemperature. As the temperature rises, so does the resistivity. Conversely, as the temperaturefalls so the resistivity decreases.For most metals & alloys, the resistivity tends to some constant low value at absolute zero, butfor some - e.g. lead, vanadium & tin - the resistivity becomes effectively zero a few degreesabove absolute zero and the material is then said to be superconducting.An induced current will continue flowing in a superconducting circuit indefinitely in the absenceof a potential difference.

Semiconductors

A semiconductor is a conductor whose resistivity decreases with rising temperature.

The ideal semiconductor is a covalent crystal, incapable of passing a current at absolute zerobecause all of the valence (bonding) electrons are firmly held in the covalent bonds. At highertemperatures some of the valence electrons have sufficient energy to break free from thebonds, and can then move in some net direction under the influence of an electric field. Thematerial is then capable of conducting a current. The higher the temperature, the larger thenumber of liberated electrons and thus the higher the conductivity of the material.

The same effect is produced by impurities and defects in the crystal, and the manufacture ofpractical semiconductors involves the introduction of impurities during crystal growth, a processknown as doping.

Conduction of Current in Metals

The atoms of a metal bond in such a way that a) they are arranged in a regular geometricpattern - i.e. they form a crystal lattice and b) the outer orbitals of the atoms overlap to such anextent that some of the valence (bonding) electrons become free to move through the lattice.Normally these free electrons move randomly through the lattice at fairly high speeds (typicallyin the order of 10 m/s). If a potential difference is established across the ends of the lattice,6

an electric field will then exist in the lattice which will exert a force on the free electrons and

cause them to have a net direction of movement - in the direction of the force. The potentialdifference imposes a drift velocity (typically in the order of 10 m/s) on the free electrons and-3

there is now a netflow of charge.

As they flow through the lattice, the free electrons collide with the positive ions of the lattice.(Each metal atom has effectively lost its free electron(s) and thus is a positive ion.)On collision, some of the kinetic energy which an electron has gained as a result of beingaccelerated by the field is transferred to the ion which results in an increase in the vibrationalenergy of the lattice - and hence its temperature. (This is the origin of the heating effect.)As the amplitudes of vibration of the lattice ions increases, they present larger targets to theelectrons and collisions become more frequent, hence the resistance of the metal increases.


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Calculating drift speed:

Consider a length of conductor in which electriccurrent is flowing:

I= the current flowing in the conductor (in A).

R = the length of the section of conductor (in m).

A = the cross sectional area of the conductor (in m ).2

n = the number of free electrons per unit volume in the conductor (in m ).-3

e = the charge on each electron (in C).v = the average or drift speed of the electrons (m.s ).-1

It follows that:

Volume of section =

Number of free electrons in section =

Total charge in section =

The time required for all of these free electrons to pass a point in the conductor =

Therefore the rate of flow of charge past the point (i.e. the current):

Worked example:

1. Determine the drift speed of the electrons, if 10,0 A current is flowing in a copper wire,of cross sectional area 3,00 x 10 m assuming that 1 electron per atom of copper is-6 2

free to conduct the current. The density of copper is 8,95 g/cm .3

The puzzle here is to obtain a value forn, as one has not been given. The periodic table givesthe relative atomic mass of copper as 63,5 g. Hence we can determine the volume of one mole

of copper: .

From this we can determine the value ofn:

Now:


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BASIC ELECTRIC CIRCUITS (Direct Current)

The most basic electric circuit consists of a source of energy for thecharges, some conductors to carry the current and one or more

components which convert electrical energy to some other form whenthe current passes through them. These components will also, to someextent, resist the flow of charge. The current passes from the source,along the conductors, through the various components and back to thesource again. The charges gain energy in the source and they loseenergy in the components. It is important to realize that energy isconserved in a circuit - i.e. the overall amount of energy gained by thecharges in the source is equal to the overall amount of energy they loseas they go through the various circuit components.In addition, charge is conserved in the circuit. This means that the charges are not created bythe source, nor do they leak out of the circuit and they are not used up as they pass through

components. They simply lose and gain energy as they travel though the circuit.

Circuit components.

A large variety of components exists, some of the more common are:

Sources, which provide energy to the charges moving through the circuit.

Resistors, which retard the flow of current.

Lamps, which convert electrical energy to light.

Heater elements, which convert electrical energy to heat.

Motors, which convert electrical energy to mechanical energy.

Also found, but generally not in simple dc circuits are: Capacitors

Inductors

transistorsThe functions of these last two will not be dealt with at this stage of your course.

And of course, the various components are connected to each other by

Conductors - most of the conductors encountered in circuits are metal conductors,usually copper wire. Other types of conductors are found serving special functions; e.g.electrolytic solutions in cells and semi-conductors in transistors.

Sources.

Charges move through the circuit from where they have high electrical potential energy to wherethey have low electrical potential energy. The source provides the charges with this energy.Various possible sources exist:

Electrochemical cells (usually known simply as `cells'). In these, energy released inchemical reactions is converted to electrical energy. Several cells can be coupledtogether to form a battery.

Generators, alternators, dynamos etc in which mechanical energy is converted toelectrical energy.

Photo-electric cells, in which the energy carried by sunlight is converted to electrical

energy.


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Basic arrangements of the components of a circuit.

Circuit components are arranged in series if the current has to pass through each componentin turn:

If the components are arranged in parallel, the current passes through two or more componentssimultaneously:

CIRCUIT CALCULATIONS.

Work, energy & power in circuits.

When a charge, q, moves through a circuit component,across which there is a potential difference, it either gainsor loses energy.

Because and ,

Also, because ,

and because ,

Power is defined as the rate at which work is done, i.e.

and so:

Consequences of energy and charge conservation in the circuit.

The reading on a voltmeter connected across a component can have two basic meanings: If the component is a source, the reading shows the energy per unit charge that is gained

in the source. If the component has a resistance, the reading shows the energy per unit charge lostin

the component - sometimes called the voltage drop in the component. Because the

resistance of a component is (by definition) given by , the voltage drop across

a component is given byAll circuit components to some extent impede the flow of current - including sources! Sources


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have a resistance, known as the internalresistance, and there is a voltage drop across thatresistance as well. This means that the voltmeter reading across a source can have two possiblevalues: If there is no current flowing in through the source, there can be no voltage drop across

the internal resistance. The voltmeter reading will then show the maximum possibleenergy per unit charge available in the source. We call this the emfor.

If there is a current flowing through the source, the voltmeter reading will show the netenergy per unit charge available in the source - less than the emf, because some work hasto be done getting the charges through the internal resistance of the source.

Hence: , where Vis the voltmeter reading across the source, and ris the inter

resistance. Note that if there is no current flowing, I= zero, and then

Because energy is conserved in a circuit, the total voltage gain in the circuit will be equal to thetotal voltage drop. So if the source is part of a simple circuit, say with just one resistor (other thanthe internal resistance), the voltmeter reading across the source will be equal to the voltmeterreading across the resistor and therefore:

If there are several resistors (or components with resistance), then R would denote the totalexternalresistance of the circuit ( - i.e. excluding the internal resistance of the source).In the equation above, the emfrepresents a rise in potential andIR andIrboth represent dropsin potential and so the sum is indeed zero - i.e. energy is conserved.

Charge is also conserved; i.e. the number of charges in a closed circuit remains constant -charges are neither destroyed, nor created as they pass through the various circuit components.

As a result of this:

If circuit components are in series, the current must be the same though all of them. If circuit components are in parallel, the current is divided between them, so that the total

current flowing is equal to the sum of the currents carried by the individual components.

If several resistors are in series, the total potential difference across all of them together is equalto the sum of the voltages across each of them individually:

i.e.but

andtherefore:

And because the current is the same everywhere:

Therefore:


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If several resistors are in parallel, the potentialdifference across all of them is the same as thatacross each one of them; and the total current is

the sum of the individual currents through eachof them.

i.e.: and

therefore:

Connecting resistors in series increases the total resistance of the circuit, resulting in a smallercurrent for a given potential difference. Connecting resistors in parallel decreases the totalresistance of the circuit resulting in a larger current for a given potential difference.

Connecting cells in series results in a larger potential difference but at the expense of a largerinternal resistance, as the internal resistances of the individual cells are also in series. If cells areconnected in parallel, the combination has the same potential difference as a single cell but alower internal resistance and will thus succeed in driving a larger current through a given circuit

than one cell on its own.

Connecting ammeters and voltmeters.

Ammeters measure the current flowing in a circuit.

The current must pass throughthe ammeter, it must therefore be connected in series:The ammeter must therefore have as low a resistance as possible or it will increase the total

resistance of the circuit, making any current measurement inaccurate.

Voltmeters measure the potential differencebetween two points in the circuit and are thereforeconnected in parallelwith a circuit component. Ifthe voltmeter were to have too low a resistance, theparallel connection would result in lowering theoverall resistance of the circuit. To reduce theeffect this would have on the accuracy of themeasurement, voltmeters have as high a resistanceas possible.


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Worked example:

12. In the circuit illustrated, the voltmeter, V reads 12 V with

1the switch S open. With S closed, V reads 8 V and the

ammeter reads 2 A.Determine:

2a) The reading on V with the switch closed.b) The internal resistance of the battery.c) The rate at which energy is dissipated as heat in the

battery with the switch closed.d) The value of the unknown resistor R.

(Assume that the resistances of both voltmeters areinfinitely large and that of the ammeter is effectivelyzero.)

The first three parts of the calculation are quite simple:

a) The diagram tells us that we have a 2 ohm resistor and we read that whenthe switch is closed, there will be a 2 ampere current flowing through it:

b) We calculate the internal resistance by dividing the lost volts by the currentflowing:

c) Heating in the battery will be caused by the given 2 ampere current flowingthrough the 2 ohm internal resistance:

d) The value of the unknown resistor presents more of a challenge. We first need todetermine the overall resistance of the circuit:

Next we subtract the internal resistance to obtain the external resistance:

2If we now subtract the resistance of the resistor with the voltmeter V connectedacross it, we get the resistance of the part of the circuit between points A and B:

We now set about simplifying the circuit between points A and B. To do this, wefind the equivalent resistance any known resistors:


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Hence:

Then:

From which:

And then:

We have already calculated the total resistance between A and B to be 2 .

Now we can say that:


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Dealing with more complex circuits: Kirchhoffs Rules.

There are circuits where the circuit simplification methods you learnt at school wont help you.In 1848, Gustav Kirchhoff proposed two simple rules with which you can analyse any dc circuit.

The first of them is known as the junction (or current) rule and is basically a statement of the lawof conservation of charge:

The sum of all the currents entering any junction point is equal to the sum of all thecurrents leaving that junction point.

This means that

The second of them is known as the loop (or voltage) rule and is basically a statement of the lawof conservation of energy:

The sum of the changes in potential around any closed path of a circuit must be zero.

There is a rise in potential - - across the cell and a drop inpotential - V - across the resistor if the current flows asshown. The loop rule - i.e. conservation of energy, says thatplus Vmust be zero. In other words; +IR = 0. If there

were more than one resistor, then +G IR = 0.must have a positive value and eachIR must be negative.If the current goes the other way, the signs must be theopposite way round. It doesnt matter which way round youimagine the current to be flowing as long as you get thesigns the right way round for the direction you have chosen.

Worked Example:

3. You are required to find the values of

2 31I,I andI in the circuit shown:

First, there are three current values in this circuit, and two loops of current as shown below:


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With three unknowns, we need three equations, to solve simultaneously.The equations we need come from Kirchhoffs rules:

The first, we can get from the junction rule: (1)

The other two equations come from the loop rule: + G IR = 0,and we need one for each current loop.

For the left hand loop we get: (2)

And the right hand loop gives us: (3)

Now we set about eliminating unknowns.

First we substitute equation (1) into equation (2):

And also into equation (3):

3By doing this we eliminate one unknown; I. If we now divide equation (5) by two, we get:

Equation (4) and equation (6) now both contain the termwhich we can eliminate by subtracting equation (6) fromequation (4).

1Then we can solve forI:

1We now have a value forI which we can substitute into either equation (4) - or equation (5)

2- and then solve forI :

3Finally, we can substitute these two current values into equation (1) and solve forI :

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