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Electrical resistanceIf you connect a lamp to a battery, a current in thelamp causes it to glow. But what determines the sizeof the current? This depends on two factors:

the potential difference or voltage V across thelamp the greater the voltage, the greater thecurrent for a given lamp

the resistance R of the lamp the greaterthe resistance, the smaller the current for agiven voltage.

We will look more carefully at the meaning of voltagein Chapter 11 . For the purpose of this chapter, you justneed to know that we can measure the voltage (alsoknown as potential difference ) across a component by

placing a voltmeter across the component. Now we need to think about the meaning

of electrical resistance . Different lamps havedifferent resistances. It is easy to demonstrate this

by connecting a torch bulb and a car headlamp inseries to a battery. The current is the same in eachcomponent, but the voltage across the torch bulb will

be greater than the voltage across the headlamp. Thetorch bulb has a larger resistance than the headlamp.

The resistance of any component is de ned as theratio of the voltage to the current. As a word equation,this is written as:

resistance = voltagecurrent

or

R = V

I where R is the resistance of the component, V is thevoltage across the component and I is the current inthe component.

You can rearrange the equation above to give:

I = V

R and V = IR

Table 10.1 summarises these quantities and their units.

Quantity Symbol forquantity

Unit Symbolfor unit

current I ampere (amp) A

voltage V volt V

resistance R ohm

Table 10.1 Basic electrical quantities, their symbolsand SI units. Take care to understand the difference

between V (in italics) meaning the quantity voltage

and V meaning the unit volt.

Resistance and resistivity

Calculate the current in a lamp given itsresistance is 15 and the potential differenceacross its ends is 3.0 V.

Step 1 Here we have V = 3.0 V and R = 15 .

Step 2 Substituting in I = V

R gives:

current I =3.015

= 0.20 A

So the current in the lamp is 0.20 A.

Worked example 1

SAQ

1 A car headlamp bulb has a resistance of 36 .Calculate the current in the lampwhen connected to a 12 V battery.

2 You can buy lamps of different brightness to t in light ttings athome ( Figure 10.1 ). A 100 watt lamp glowsmore brightly than a 60 watt lamp. Whichof the lamps has the higherresistance?

Chapter 10

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Chapter 10: Resistance and resistivity

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De ning the ohmThe unit of resistance, the ohm, can be determinedfrom the equation that de nes resistance:

resistance = voltagecurrent

The ohm is equivalent to 1 volt per ampere. That is:

1 = 1 V A 1

Determining resistanceAs we have seen, the equation for resistance is:

R = V I

To determine the resistance of a component, wetherefore need to measure both the voltage V acrossit and the current I through it. To measure the currentwe need an ammeter. To measure the voltage, we needa voltmeter. Figure 10.2a shows how these metersshould be connected to determine the resistance of ametallic conductor, such as a length of wire.

The ammeter is connected in series with theconductor, so that there is the same current in both.

The voltmeter is connected across (in parallel with)the conductor, to measure the voltage across it.

The voltage across the metal conductor can be alteredusing a variable power supply or by having a variableresistor placed in series with the conductor. Thisgives currents at different voltages. The results ofsuch a series of measurements is shown graphicallyin Figure 10.2b .

Figure 10.1 Both of these lamps work fromthe 230 V mains supply, but one has a higherresistance than the other. For SAQ 2 .

The ohm is the resistance of a component whena potential difference of 1 volt is produced perampere of current.

SAQ

3 a Calculate the potential difference across amotor carrying a current of 1.0 A having aresistance of 50 .

b Calculate the potential difference across thesame motor when the current is doubled.Assume its resistanceremains constant.

4 Calculate the resistance of a lamp carrying acurrent of 0.40 A when connectedto a 230 V supply.

V

A

I

I

V

metallic

conductor

a

b

00

Figure 10.2 To determine the resistance of acomponent, you need to measure both current and

potential difference.

Answer

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Look at the graph of Figure 10.2b . Such a graphis known as an I V characteristic . The points areslightly scattered, but they clearly lie on a straightline. A line of best t has been drawn. If you extendthis line downwards, you will see that it passesthrough the origin of the graph. In other words, thecurrent I is directly proportional to the voltage V. Thestraight-line graph passing through the origin showsthat the resistance of the conductor remains constant.If you double the current, the voltage will alsodouble. However, its resistance, which is the ratio ofthe voltage to the current, remains the same. Insteadof using:

R = V I

to determine the resistance, for a graph of I againstV which is a straight line passing through the origin,you can also use:

resistance =1

gradient of graph

(Take care! This is only true for an I V graph whichis a straight line through the origin.)

You get results similar to those shown inFigure 10.2b for a commercial resistor . Resistorshave different resistances, hence the gradient of the

I V graph will be different for different resistors.

SAQ

5 Table 10.2 shows the results of an experiment tomeasure the resistance of a carbon resistor whoseresistance is given by the manufactureras 47 10%.a Plot a graph to show the I V characteristic of

this resistor.b Do the points appear to fall on a straight line

which passes through the origin of the graph?c Use the graph to determine the resistance of

the resistor.d Does the value of the resistance fall within

the range given bythe manufacturer?

Voltage V /V Current I /A

2.1 0.040

4.0 0.079

6.3 0.128

7.9 0.19210.0 0.202

12.1 0.250

Table 10.2 Data for SAQ 5 .

Ohms lawFor the metallic conductor whose I V characteristic

is shown in Figure 10.2b , the current through it isdirectly proportional to the voltage across it. This isonly true if the temperature of the conductor does notchange. This means that its resistance is independentof both the current and the voltage. This is becausethe ratio V I is a constant. Any component which

behaves like this is described as an ohmic component ,and we say that it obeys Ohms law . The statement ofOhms law is very precise and you must not confusethis with the equation V = IR.

Ohms lawFor a metallic conductor at constant temperature,the current in the conductor is directly

proportional to the potential difference acrossits ends.

It is easier to see the signi cance of this if weconsider a non-ohmic component. An exampleis a semiconductor diode . This is a componentwhich allows electric current in only one direction.

Nowadays, most diodes are made of semiconductormaterials. One type, the light-emitting diode orLED, gives out light when it conducts.

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120

Figure 10.3 shows the I V characteristic for a light-emitting diode. There are some points you shouldnotice about this graph:

We have included positive and negative valuesof current and voltage. This is because, whenconnected one way round (positively biased), thediode conducts and has a fairly low resistance.Connected the other way round (negatively

biased), it allows only a tiny current through andhas almost in nite resistance.

For positive voltages less than about 2 V, the currentis almost zero and hence the LED has almostin nite resistance. The LED starts to conductsuddenly at its threshold voltage . This depends onthe colour of light it emits, but may be taken to

be about 2 V. The resistance of the LED decreases

dramatically for voltages greater than 2 V.The resistance of a light-emitting diode depends onthe potential difference across it. From this we canconclude that the LED does not obey Ohms law; it isa non-ohmic component .

LEDs have traditionally been used as indicatorlamps to show when an appliance is switched on.

Newer versions, some of which produce white light,are replacing lament lamps, for example in traf clights. This is because, although they are moreexpensive to manufacture, they are more energy-

ef cient and hence cheaper to run, so that the overallcost is less.

SAQ

6 An electrical component allows acurrent of 10 mA through it whena voltage of 2.0 V is applied. When the voltage isincreased to 8.0 V, the current becomes 60 mA.Does the component obey Ohms law? Givenumerical values for the resistanceto justify your answer.

Resistance and temperatureYou should have noted earlier that, for a componentto obey Ohms law, the temperature must remainconstant. You can see why this must be the case byconsidering the characteristics of a lament lamp.Figure 10.4 shows such a lamp; you can clearly

see the wire lament glowing as the current passesthrough it. Figure 10.5 shows the I V characteristicfor a similar lamp.

+

+ V

I

0

2 V

0

Figure 10.3 The current against voltage ( I V )characteristic for a light-emitting diode. The graph isnot a straight line. An LED does not obey Ohms law.

Figure 10.4 The metal lament in a lamp glows asthe current passes through it. It also feels warm. Thisshows that the lamp produces both heat and light.

I

V

Figure 10.5 The IV characteristic for alament lamp.

Hint

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There are some points you should notice about thegraph in Figure 10.5 :

The line passes through the origin (as for an ohmiccomponent).

For very small currents and voltages, the graph isroughly a straight line.

At higher voltages, the line starts to curve. Thecurrent is a bit less than we would have expectedfrom a straight line. This suggests that the lampsresistance has increased. You can also tell that theresistance has increased because the ratio V I islarger for higher voltages than for low voltages.

The fact that the graph of Figure 10.5 is not a straightline shows that the resistance of the lamp dependson the temperature of its lament. Its resistance mayincrease by a factor as large as ten between when it

is cold and when it is brightest (when its temperaturemay be as high as 1750 C).

SAQ

7 The two graphs in Figure 10.6 show the I V characteristics of a metal wire at two differenttemperatures, 1 and 2.a Calculate the resistance of the wire at each

temperature.b State which is the higher

temperature, 1 or 2.

8 The graph of Figure 10.7 shows the I V characteristics for two electrical components, a

lament lamp and a length of steel wire.a Identify which curve relates to each

component.b State at what voltage both have the same

resistance.c Determine the resistance at the

voltage stated in b .

3

2

1

00 10

I /A

V /V3020

1.5

1.0

0.5

00 5

I /A

V /V1510

1

2

Figure 10.6 I V graphs for a wire at two differenttemperatures. For SAQ 7 .

I /A

V /V0

2

4

0 4 8

3

1

2 6 10

A

B

Figure 10.7 For SAQ 8 .

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ThermistorsThese are components that are designed to have aresistance which changes rapidly with temperature.Thermistors ( therm al res istors ) are made from metaloxides such as those of manganese and nickel. Thereare two distinct types of thermistor.

Negative temperature coef cient (NTC)thermistors the resistance of this type ofthermistor decreases with increasing temperature.Those commonly used in schools and collegesmay have a resistance of many thousands of ohmsat room temperature, falling to a few tens of ohmsat 100 C. You are expected to recall the propertiesof NTC thermistors.

Positive temperature coef cient (PTC) thermistors the resistance of this type of thermistor rises

abruptly at a de nite temperature, usually around100150 C.

SAQ

9 The graph in Figure 10.8 was obtained bymeasuring the resistance R of a particularthermistor as its temperature changed.a Determine its resistance at

i 20 Cii 45 C.

b Determine the temperature when its resistance is i 5000 ii 2000 .

c The sensitivity of the thermistor is de nedas R . This is the gradient of the graph. Usethe graph to estimate the sensitivity ati 20 C

ii 45 C iii 70 C.

The change in their resistance with temperaturegives thermistors many uses.

Water temperature sensors in cars and icesensors on aircraft wings if ice buildsup on the wings, the thermistor sensesthis temperature drop and a small heater is

activated to melt the ice. Baby alarms the baby rests on an air- lled

pad, and as he or she breathes, air from the pad passes over a thermistor, keeping it cool; if the baby stops breathing, the air movement stops,the thermistor warms up and an alarm sounds.

Fire sensors the rise in temperature activatesan alarm.

Overload protection in electric razor sockets if the razor overheats, the thermistorsresistance rises rapidly and cuts off the circuit.

Thermistors at work

/C

R /

k

0

2

4

6

0

5

3

1

10 20 30 40 50 60 70

Figure 10.8 The resistance of an NTC thermistordecreases as the temperature increases. For SAQ 9 .

10 A student connects a circuit with an NTCthermistor, a lament lamp and a battery in series.The lamp glows dimly. The student warms thethermistor with a hair dryer. What change will thestudent notice in the brightnessof the lamp? Explain your answer.

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SAQ

11 The resistance of a metal wire changes withtemperature. This means that a wire could be usedto sense changes in temperature, in the same waythat a thermistor is used. Suggest one advantage athermistor has over a metal wire for this purpose;suggest one advantage a metalwire has over a thermistor.

ResistivityThe resistance of a particular wire depends on its sizeand shape. A long wire has a greater resistance thana short one, provided it is of the same thickness andmaterial. A thick wire has less resistance than a thinone. You can investigate these relationships using

conducting putty. For a metal in the shape of a wire,its resistance R depends on the following factors:

its length L its cross-sectional area A the material the wire is made from its temperature.At a constant temperature, the resistance is directly

proportional to the length of the wire and inversely proportional to its cross-sectional area. That is:

resistance length

andresistance

1cross-sectional area

Therefore:

resistance length

cross-sectional areaor

R L A

The resistance of a wire also depends on the materialit is made of. Copper is a better conductor than steel,steel is a better conductor than silicon, and so on. Soif we are to determine the resistance R of a particularwire, we need to take into account its length, itscross-sectional area and the material. The relevant

property of the material is its resistivity , for whichthe symbol is (Greek letter rho ).

The word equation for resistance is:

resistance = resistivity length

cross-sectional area

R = L A

We can rearrange this equation to give an equationfor resistivity. The resistivity of a material is de ned

by the following word equation:

resistivity = resistance cross-sectional area

length

= RA L

Values of the resistivities of some typical materialsare shown in Table 10.3 . Notice that the units of

resistivity are ohm metres ( m); this is not the sameas ohms per metre.

Material Resistivity/ m

Material Resistivity/ m

silver 1.60 10 8 mercury 69.0 10

copper 1.69 10 8 graphite 800 10 8

nichrome a 1.30 10 8 germanium 0.65

aluminium 3.21 10 8 silicon 2.3 103

lead 20.8 10 8 Pyrex glass 10 12

manganin b 44.0 10 8 PTFE d 1013 10 16

eureka c 49.0 10 8 quartz 5 1016

Table 10.3 Resistivities of various materials at 20 C. a Nichrome an alloy of nickel, copper and aluminium used in electric

res because it does not oxidise at 1000 C.b Manganin an alloy of 84% copper, 12% manganese and 4% nickel.c Eureka (constantan) an alloy of 60% copper and 40% nickel.d Poly(tetra uoroethene) or Te on.


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SAQ

12 Use the resistivity value quoted inTable 10.3 to calculate the lengthsof 0.50 mm diameter manganin wire neededto make resistance coils withresistances of:

a 1.0 b 5.0 c 10 .

13 1.0 cm 3 of copper is drawn out intothe form of a long wire of cross- sectional area 4.0 10 7 m2. Calculate itsresistance. (Use the resistivityvalue from Table 10.3 .)

14 A 1.0 m length of copper wire has a resistanceof 0.50 .a Calculate the resistance of a 5.0 m length of the

same wire.b What will be the resistance of a 1.0 m length

of copper wire having half thediameter of the original wire?

15 A piece of steel wire has aresistance of 10 . It is stretched totwice its original length. Compare its newresistance with its originalresistance.

Resistivity and temperature

Resistivity, like resistance, depends on temperature.For a metal, resistivity increases with temperature. Aswe saw above, this is because there are more frequentcollisions between the conduction electrons and thevibrating ions of the metal.

For a semiconductor, the picture is different.The resistivity of a semiconductor decreases withtemperature. This is because, as the temperatureincreases, more electrons can break free of theiratoms to become conduction electrons. The numberdensity n of electrons thus increases and so the

material becomes a better conductor. At the sametime, there are more electronion collisions, but thiseffect is small compared with the increase in n.

Find the resistance of a 2.6 m length of eurekawire with cross-sectional area 2.5 10 7 m2.

Step 1 Using the equation for resistance, andtaking the value for from Table 10.3 :

resistance = resistivity length

area

R = L A

Step 2 Substituting values:

R = 49.0 10 8 2.6

2.5 10 7

= 5.1

So the wire has a resistance of 5.1 .

Worked example 2


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126

Summary

Resistance is de ned as the ratio of voltage to current. That is: resistance =

voltagecurrent

( R =V I

)

Another term for voltage is potential difference .

The ohm is the resistance of a component when a potential difference of 1 volt is produced per ampere. Ohms law can be stated as: For a metallic conductor at constant temperature, the current in the conductor is directly proportional to

the potential difference (voltage) across its ends.

Ohmic components include a wire at constant temperature and a resistor. Non-ohmic components include a lament lamp and a light-emitting diode. A semiconductor diode allows current in one direction only. A light-emitting diode (LED) emits light

when it conducts.

As the temperature of a metal increases, so does its resistance. A thermistor is a component which shows a rapid change in resistance over a narrow temperature range.

The resistance of an NTC thermistor decreases as its temperature is increased.

The resistivity of a material is de ned as = RA L , where R is the resistance of a wire of that material, A is its cross-sectional area and L is its length. The unit of resistivity is the ohm metre ( m).

Questions1 a A wire has length L, cross-sectional area A and is made of material of resistivity .

Write an equation for the electrical resistance R of the wire in terms of L, A and . [1] b A second wire is made of the same material as the wire in a , has the same

length but twice the diameter. State how the resistance of this wire compareswith the resistance of the wire in a . [2]

c The diagram shows a resistor made by depositing a thin layer of carbon onto a plastic base.

X

A

Y

plastic base

carbon

1.3 10

2

m

The resistance of the carbon layer between X and Y is 2200 . The length ofthe carbon layer is 1.3 10 2 m. The resistivity of carbon is 3.5 105 m.

Show that the cross-sectional area A of the carbon layer is about 2 1010 m2. [2]OCR Physics AS (2822) January 2006 [Total 5]

continued

Glossary

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2 a The electrical resistance of a wire depends upon its temperature and on theresistivity of the material. List two other factors that affect the resistance of a wire. [2]

b The diagram shows an electrical circuit that contains a thin insulated copperwire formed as a bundle.

V

3.0 V

A

The ammeter and the battery have negligible resistance and the voltmeter has anin nite resistance.

The copper wire has length 1.8 m and diameter 0.27 mm. The resistance of the wireis 0.54 .

i Calculate the resistivity of copper. [4] ii State and explain the effect on the ammeter reading and the voltmeter reading

when the temperature of the copper wire bundle is increased. [4]OCR Physics AS (2822) June 2005 [Total 10]

3 a State the difference between the directions of conventional current andelectron ow. [1]

b State Ohms law. [2] c Current against voltage ( I V ) characteristics are shown in Figure 1 for a metallic

conductor at a constant temperature and in Figure 2 for a particular thermistor.

metallic conductor

I

V

Figure 1

continued

thermistor

0

I

V

Figure 2

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128

i Sketch the variation of resistance R with voltage V for: 1 the metallic conductor at constant temperature 2 the thermistor. [3] ii State and explain the change, if any, to the graph of resistance against voltage

for the metallic conductor:

1 when the temperature of the metallic conductor is kept constant at ahigher temperature

2 when the length of the conductor is doubled but the material, temperatureand the cross-sectional area of the conductor remain the same. [4]

OCR Physics AS (2822) January 2005 [Total 10]

4 a State Ohms law. [2] b The I V characteristic for a particular component is shown in the diagram.

V /V0.80.60.40.2 0.2 0

020

40

60

80

100 I /mA

i Name the component with the I V characteristic shown in the diagram. [1] ii Describe, making reference to the diagram, how the resistance of the

component depends on the potential difference V across it. You areadvised to show any calculations. [5]

OCR Physics AS (2822) January 2004 [Total 8]

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