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Chapter 10 Resistance and resistivity

Worksheet Worked examples Practical 1: The I/V characteristics of components Practical 2: The effect of temperature on the resistance of a thermistor Practical 3: Identifying a material from its resistivity End-of-chapter test Marking scheme: Worksheet Marking scheme: End-of-chapter test

Worksheet

Intermediate level

1 2 3 4

Define electrical resistance. State Ohm's law. Write a word equation for the resistance of a length of metal wire in terms of the resistivity of the metal, the length of the wire and its cross-sectional area. A component is connected to a d.c. supply. The supply has negligible internal resistance. At 6.0 V, the current in the component is 0.023 A. When the p.d. is doubled, the current in the component increases to 0.100 A. a b Calculate the resistance of the component at 6.0 V. Does the component obey Ohm's law? Explain your answer. [2] [2] [1] [1] [1]

5

The diagram below shows the I/V characteristics of two components A and B.

I (A) 1.00 A B

0.80

0.60

0.40

0.20

0 0 1 2 3 4 5 6 V (V)

The components are connected in series to a battery. The current in each component is the same and equal to 0.60 A. Calculate the individual resistances of A and B.

[2]

6

A 14m long copper wire of cross-sectional area 4.2 × 10 m is wound into a coil for a loudspeaker. The resistivity of copper is 1.7 × 10­8 m. Calculate the resistance of the wire. [3]

­8 2

Higher level

7

The diagram shows a thermistor connected to a d.c. supply.

6.0 V I (mA)

I S A

60

20 0 0 Time

The supply has negligible internal resistance. When the switch S is closed, the current I in the circuit changes as shown in the graph on the right.

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a b

Explain why the current changes in the manner shown in the graph. Calculate the ratio: minimum resistance of thermistor maximum resistance of thermistor

[2]

[2]

8 The resistance across the ends of a 15 cm long pencil lead is 3.6 . Calculate its

radius given that the pencil lead material has a resistivity of 7.5 × 10­5 m. A piece of metal is shaped into a rectangular block as shown below.

4.0 cm

[3]

9

0.60 cm 0.80 cm

The metal has a resistivity of 4.3 × 10­4 m. a The resistance of the block depends on which pair of faces it is measured between. Calculate the minimum resistance between two opposite faces of the block.

[4]

b

What is the maximum current in the block in a when connected to a 0.050 V supply of negligible internal resistance? [2]

10 A filament lamp is connected to a d.c. supply. The current in the lamp is 2.0 A when

the potential difference across it is 12 V. When operating at 12 V, the filament of the lamp has a cross-sectional area of 4.9 × 10­9 m2 and the resistivity of the filament material is 5.6 × 10­7 m. Calculate the length of the filament in centimetres. [4]

Extension

11 A glass tube contains a conducting liquid of length 5.0 cm. The internal radius of

the tube is 1.4 cm. The resistivity of the liquid is 8.5 × 10­5 m. The liquid is poured onto a horizontal surface and quickly sets in the form of a uniform cylindrical disc of radius 25 cm. Calculate the resistance of this disc across its two opposite larger surfaces. You may assume that the resistivity of the material remains constant. [4] aluminium is one-third that of copper. a For equal length and resistance, calculate the ratio: mass of aluminium mass of copper b [3]

12 The resistivity of aluminium is twice that of copper. However, the density of

Use the Internet to investigate the construction of power cables used for the National Grid. You may be surprised to find that the current-carrying cables are made from aluminium and not copper. Explain why this is so. Total: ­­­ Score: 36 %

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10 Resistance and resistivity

Worked examples

Example 1

A semiconductor diode is connected to a variable d.c. supply of negligible internal resistance. The current in the diode is zero when the p.d. across it is 0.40 V. The current increases to 30 mA when the p.d. across the diode is 0.65 V. Calculate the resistance of the diode at 0.40 V and 0.65 V. Does the diode obey Ohm's law? The resistance R of a component is given by: R= V I

At 0.40 V, the current I is 0 A. Therefore: R= 0.40 = 0 Remember that any number divided by zero is infinite.

(The diode is not conducting; hence its resistance is infinite.) At 0.65 V, the current I is 30 mA. Therefore: R= 0.65 = 22 0.03 Remember to convert the current into amperes. Failure to do this will give the resistance in k.

(The resistance of the conducting diode is quite small.) The resistance of the diode changes with p.d. (or current). Therefore, the current cannot be directly proportional to the p.d. The diode does not obey Ohm's law.

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Example 2

A wire of radius 0.15 mm and length 3.2 m has a resistance of 14 . Calculate the resistance of a wire made from the same material but having five times the length and twice the radius. The resistance R of the wire is given by: R=

l A

The cross-sectional area A of the wire is given by: A = r 2 where r is the radius of the wire. Therefore: R=

l l r 2 r 2

5 = 1.25. The resistance of 22

The resistance of the wire will therefore increase by a factor the new wire is: R = 1.25 × 14 = 17.5 18

Tip

You do not need to use algebra to get the answer. You can first calculate the resistivity of the wire and then use this to calculate the resistance of the wire. The resistivity is given by:

=

RA l l = 3.2 m A = r2 = × (1.5 × 10­4)2 = 7.07 × 10­8 m2

R = 14

=

14 × 7.07 × 10­8 = 3.09 × 10­7 m 3.2

For the new wire: l = 3.2 × 5 = 16m A = r 2 = × (2 × 1.5 × 10­4)2 = 2.83 × 10­7 m2

= 3.09 × 10­7 m

Its resistance R is therefore: R=

l 3.09 × 10­7 × 16 = = 17.5 18 A 2.83 × 10­7

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10 Resistance and resistivity

Practical 1

The I/V characteristics of components

Safety

Always take sensible safety precautions when using mains-operated supplies. Teachers and technicians should follow their school and departmental safety policies and should ensure that the employer's risk assessment has been carried out before undertaking any practical work.

Apparatus

· · · · variable d.c. supply 1 m of 40 swg nichrome wire filament lamp (60 mA, 6 V) silicon diode · · · · digital ammeter digital voltmeter 100 resistor (for diode experiment) connecting leads

Introduction

You can identify a component from its I/V characteristics. In this experiment you will determine the I/V characteristics of a metallic wire kept at a constant temperature, a filament lamp and a semiconductor diode. You will find information on the components mentioned above on pages 90 and 91 of Physics 1.

Procedure

The diagrams show appropriate circuits for investigating the different components. For the diode experiment, it is vital to include a safety resistor.

filament lamp or metallic wire A A diode

100

VITAL SAFETY RESISTOR V V

1 2 3 4 5

Set up the appropriate circuit for the component you are investigating. Change the potential difference across the component from zero to 6.0 V in steps of 0.5 V. Measure the current for each p.d. Record your results in a table. On the same axes, plot a current against voltage graph for each of the components. (You should be able to identify the component from the specific shape of the I/V graph.)

Guidance for teachers

One metre of the nichrome wire has a resistance of about 90 . Using a 1 m length makes it possible to plot the I/V graphs for all the components on the same axes for comparison. If a variable d.c. supply is not available, then a rheostat may be used as a potential divider.

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Practical 2

The effect of temperature on the resistance of a thermistor

Safety

Take care when pouring the boiling water into the beaker. Teachers and technicians should follow their school and departmental safety policies and should ensure that the employer's risk assessment has been carried out before undertaking any practical work.

Apparatus

· · · · · 6.0 V battery 100 ml beaker NTC thermistor plastic bag electric kettle · · · · thermometer digital voltmeter digital ammeter connecting leads

Introduction

In this experiment you will investigate the effect that temperature has on the resistance of a negative temperature coefficient (NTC) thermistor. Their resistive properties are described on page 93 of Physics 1.

Procedure

The circuit shown here may be used to investigate the behaviour of a thermistor.

6.0 V

1 2 3

Put the thermistor in a waterproof plastic bag and place it into a beaker. Pour boiling hot water into the beaker. Stir the water and measure the temperature of the water, the potential difference V and the current I. Record your results in a table. For every 10 °C drop in temperature, measure I and V. Calculate the resistance R of the thermistor at each temperature using the equation: V R= I

A V

thermometer

4 5 6

beaker with hot water

7 8

Repeat the experiment twice and determine the average resistance at each temperature. Plot a graph of resistance R against temperature .

Guidance for teachers

The specifications only require knowledge of NTC thermistors. However, if there is time, selected students could also determine the properties of PTC thermistors.

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10 Resistance and resistivity

Practical 3

Identifying a material from its resistivity

Safety

Do not attempt to measure the current for zero length. This will short out the battery and send a potentially damaging current through the ammeter. Teachers and technicians should follow their school and departmental safety policies and should ensure that the employer's risk assessment has been carried out before undertaking any practical work.

Apparatus

· · · · 6.0 V battery (or d.c. supply) manganin or eureka or nichrome wire ruler micrometer · · · · crocodile clip digital voltmeter digital ammeter connecting leads

Introduction

In this experiment you will determine the resistivity of a metal and identify it by using either a databook or the Internet.

Procedure

The diagram shows an arrangement that may be used to determine the resistivity of a metal. You may use any available wires in the laboratory.

6.0 V

1 Measure the diameter of the wire at different

points. Use the average diameter d to determine the cross-sectional area A of the wire using: d2 A= 4

A crocodile clip

l

2 For a 10 cm long wire, measure the current I and

the potential difference V.

V

resistance wire

3 Record your results in a table. 4 Calculate the resistance R of the wire using:

R= V I

5 Increase the length of the wire in steps of 10 cm and determine

the resistance for each length.

6 Plot a graph of resistance R of the wire against its length l. 7 Draw a straight line of best fit through the data points. 8 Find the gradient of the line, which is equal to ,

A where is the resistivity of the metal.

R

gradient = 0 0

A

9 Calculate the resistivity using the relationship:

= gradient × A

l

10 Use the Internet or a science databook to identify the metal.

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End-of-chapter test

Answer all questions.

1 2

Name a non-ohmic component. The graph shows the I/V characteristics of two types of filament lamps X and Y. a b c Does lamp X obey Ohm's law? Explain your answer. [1] Calculate the resistance of lamp X at 2.0 V. [3]

I (mA) 100 Y

[1]

X

80

60

Describe how the resistance of lamp Y depends on the current. [1]

40

20

0 0 1 2 3 4 5 6 V (V)

3

A negative temperature coefficient (NTC) thermistor and its connecting leads are coated with a high-resistivity plastic material. The thermistor is placed in a beaker containing hot water. The temperature of the water is kept constant at 80 °C. The I/V characteristic of the thermistor is shown below.

I (mA) 100 80 °C 80

60

40

20

0 0 2 4 6 8 10 12 V (V)

a b

Calculate the resistance of the thermistor. State and explain the change, if any, to the shape of the I/V graph when the temperature is lowered and maintained at 30 °C.

[2] [2] [2]

4 5

A wire is made of a material of resistivity . Write an equation for the resistance R of a wire of length l and diameter d. According to a databook, a manganin wire of radius 0.15 mm has a resistance of 5.33 per metre of length. a b Calculate the resistivity of manganin. Explain how your answer to a would change if the manganin wire had twice the radius. Total: ­­­ Score: 18

[4] [2] %

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10 Resistance and resistivity

Marking scheme

Worksheet

1 2 3 4

Resistance = potential difference [1] current The current in a metallic conductor kept at a constant temperature is directly proportional to the potential difference across its ends. [1] Resistance = a b R= resistivity × length [1] cross-sectional area R = 260 [1]

V 6.0 = [1]; I 0.023

At double the p.d., the resistance is R= 12 = 120 [1] 0.100

The resistance is not constant. Therefore, the current cannot be directly proportional to the voltage. The component is non-ohmic. [1]

5 6 7

RA = R= a

1.0 = 1.7 [1]; 0.60

RB =

3.5 = 5.8 [1] 0.60

1.7 × 10­8 × 14 l [1]; R = 5.67 [1] [1]; R = A 4.2 × 10­8 As current flows in the thermistor, its temperature increases. This causes a decrease in the resistance of the thermistor and therefore an increase in the current (the p.d. is constant at 6.0 V). [1] Eventually, the temperature reaches a maximum and therefore the resistance of the thermistor is lower but constant. The current is therefore also constant. [1]

b

Rmin =

6.0 6.0 = 100 , Rmax = = 300 [1] 0.060 0.020 Rmin 100 1 = = 0.33 [1] Rmax 300 3

Ratio =

8

R=

l l 7.5 × 10­5 × 0.15 [1]; A = = = 3.13 × 10­6 m­2 3.1 × 10­6 m­2 [1] A R 3.6

so r= 3.13 × 10­6 = 9.97 × 10­4 m 1.0 × 10­3 m [1]

A = r2

9

a

Minimum resistance shortest length and largest cross-sectional area [1] l = 0.60 cm, R= A = 4.0 × 0.80 = 3.2 cm2 = 3.2 × 10­4 m2 [1]

l 4.3 × 10­4 × 6.0 × 10­3 = [1]; R = 8.1 × 10­3 [1] A 3.2 × 10­4

I = 6.2 A [1]

b

I=

V 0.050 = [1]; R 8.1 × 10­3

10 R = =

l=

V 12 = 6.0 [1] I 2.0 l= 6.0 × 4.9 × 10­9 = 0.053 m [1]; 5.6 × 10­7 l = 5.3 cm [1]

RA [1];

11 Volume remains constant (r 2h = constant) [1]

252 × l = 1.42 × 5.0 (l = new `length'), so R= l = 1.57 × 10­2 cm = 1.57 × 10­4 m [1]

l 8.5 × 10­5 × 1.57 × 10­4 = [1]; R = 6.8 × 10­8 [1] A × (25 × 10­2)2

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10 Resistance and resistivity

99

12 a Mass = density × volume (M = DV) [1]

Resistance remains the same, and equal length l, therefore:

Cu × l Al × l = ACu AAl

Ratio =

so

AAl Al = = 2 [1] ACu Cu

DAl AAl MAl DAl × AAl × l 1 2 = = = × = 0.67 [1] MCu DCu × ACu × l DCu ACu 3 1

( )( )

The mass (and hence weight) of aluminium overhead cables is 67% that of copper for equal length and resistance. b Internet research. Cables are typically aluminium with steel reinforcement.

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10 Resistance and resistivity

Marking scheme

End-of-chapter test

1 2

Semiconductor diode or a filament lamp [1] a b The current is not directly proportional to the p.d., therefore X does not obey Ohm's law. [1] V = 2.0 V and I = 60 mA [1] R= c V [1]; I R= 2.0 = 33 [1] 0.060

The resistance of Y decreases as the current increases. [1] V I 12 = 150 [1] 0.080

3

a R = [1]; R =

b

The resistance of the thermistor is higher at a lower temperature. [1] 1 Since R = , the graph is a straight line of smaller gradient. [1] gradient of line R=

4 5

a

l A

and

A = r 2 =

d 2 l 4l [1]; R = = 2 [1] 4 A d

l = 1.0 m, r = 1.5 × 10­4 m, R = 5.33 [1]

=

b

RA 5.33 × × (1.5 × 10­4)2 = [1]; l 1.0

= 3.8 × 10­7 [1]; Unit: m [1]

There is no change. [1] The resistivity depends on the material and not on its dimensions. [1]

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Information

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