Resistance and Resistivity
Introduction


Charge moves more readily through some materials than through others. The resistance of a sample is a measurement of how much the sample impedes the flow of electrons through it. The type of material within a resistor and its physical dimensions are some of the factors which determine its resistance.



Resistivity is a different concept to resistance. Resistivity values are used to compare the conducting properties of different materials. The resistivity value of a specific material is a 'material constant' and, unlike resistance, is independent of the dimensions of the sample.

Conductors and insulators
Good conductors, such as gold, contain many electrons which are not bonded to any particular atomic nucleus. These are called free electrons.

By contrast, in insulating materials all the electrons are used in bonding. Thus, they are not available to contribute to the conducting process.

Click on the button in Fig.1 to close the switch and see what happens when a small electrical potential
energy
A system has energy when it has the capacity to do work. The scientific unit of energy is the joule.
energy
difference (p.d.) is placed across the ends of a metal conductor.

Figure 1.   Free electrons can be made to move.
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Complete the following statements to explain the effect of applying a p.d. across different materials.

  • A small p.d. causes current in materials, such as metals, which are good . In other materials, called , the same p.d. would cause current. Conductors contain many electrons which are , whereas the electrons in insulators are all .
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Decide whether the following statements are true or false.
  • Materials with low resistances are good conductors.
    Resistance is due to the electrons in a material being used in bonding.
    Resistance is caused the atoms of a material impeding the flow of electrons.
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Conduction through metals involves the movement of free electrons. These can be made to move by applying a p.d. Their motion is impeded by the vibration of the atoms in the metal and so even good conductors have some
resistance
The opposition to the flow of current provided by a circuit is called resistance. Resistance is measured in units called Ohms.
resistance
.

By considering situations where these materials are used classify each of the following as conductors or insulators.
  • Glass
    Silver
    Wood
    Copper
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Materials used to make the cases of hairdryers should be …
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If very high voltages are applied across thin samples of good insulators such as glass a small
current
The rate of flow of charge past any specific point in a circuit. The base unit of current is the Ampere.
current
can be produced, so even good insulators can allow some conduction. The terms 'conductor' and 'insulator' describe materials at opposite ends of the resistance spectrum.
Drift velocity
When a lamp in a room is switched on, the light appears almost instantly. From this it is very easy to imagine that electrons travel through the metallic conducting wires very quickly. In actual fact the speed at which the electrons move between the positive and negative sides of a power supply is very small – a few millimetres per second. Fig.2 helps to explain this apparent paradox.
Figure 2.   Fast or slow transfer?
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Complete the following statements.

  • Repeatedly putting in balls at the bottom transfers energy to the top even though the balls themselves travel from the bottom of the tube to the top.
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Applying a p.d. across a conductor is analogous to pushing the ball in at the bottom of the tube in Fig.2. The energy involved is transferred through the system almost instantly yet the electrons themselves travel through the system more slowly.

The actual arrangement within a conductor is more complicated than illustrated in Fig.2. In the absence of a p.d. the free electrons within a metal move quickly in random directions, as illustrated in Fig.3.

Figure 3.   Electrons in random motion in a conductor.
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Adding a p.d. superimposes an overall motion onto the normal random behaviour of the electrons. When the p.d. is applied in Fig.4 the free electrons within the conductor, though still moving quickly between collisions, now also drift slowly towards the positive terminal of the battery.

Figure 4.   A p.d. causing drift.
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The speed at which the electrons drift vd is related to the current I in the wire. A larger current is the result of more free electrons per second passing through the conductor. One way of achieving this is for the free electrons to move faster!

Figure 5.   Examining drifting electrons in a conductor.
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To derive a relationship connecting the current I and drift speeds vd, we must determine the total charge Q passing through a part of a conductor in a time t. We will look at a small section of the conductor with length x and cross-section area A.

Consider the section of conductor in Fig.5. It has a volume of Ax.
There are n free electrons per unit volume.
The total number of electrons in this part of the conductor is then given by nAx.
The charge on one electron is q.
The total charge in this part of the conductor Q may be calculated from Q = nAxq.
The free electrons are moving at a speed vd.
We can say where t is the time taken to travel a distance x
Rearranging gives

From the definition of current we know that

.
We have already determined expressions for Q and t that we can substitute into this equation for the current.


Simplifying gives .
     Or rearranging gives.




The
velocity
An object's velocity states both the speed and direction of motion relative to a fixed reference point.
velocity
with which free electrons drift through a conductor is given by the formula:



A sample of copper wire has 8 × 1028 free electrons per cubic metre. The size of the charge on each electron is 1.6 × 10−19 C. The current in a sample of wire with a cross-sectional area of 2 × 10−6 m2 is 6 A.

Follow the steps below to see how to calculate the
drift velocity
The speed with which electrons move through a material when a p.d. is applied across the material is called the drift velocity. The drift velocity through a material can be calculated using the formula:

drift velocity
of the free electrons.

    



This example has confirmed that electrons drift through metals at approximately 0.2 mms−1.

If the current I is increased by a factor of 3 and n, A, and q do not change, what will happen to the drift velocity?
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Ohm's Law
Our model of how conductors behave suggests that in these materials resistance occurs because of collisions between the free electrons and the atoms of the material. In such instances these materials are behaving as resistors. Use the slider from the variable power supply in Fig.6 to alter the
voltage
The voltage across a component is the electrical energy transferred by 1 coulomb of charge passing through the component.
voltage
across the resistor.


Click on the figure below to interact with the model.

 Figure 6.  A simple circuit with a variable power supply.



Select the words which best complete the sentence below.

  • As the p.d. across the resistor is increased, the current . If the p.d. decreases, the current .
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The manner in which the current in Fig.6 varies as the p.d. across the component is altered can be summarized in a graph called a current–voltage characteristic.

Alter the slider in Fig.6 and enter pairs of readings as appropriate into the table in Fig.7.

Figure 7.   Current–voltage characteristic for an ohmic resistor.
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Select the words which best describe the current–voltage variation.

  • As the p.d. across the resistor is increased, the current . If the p.d. is doubled, the current .

    The graph of the current the component versus the p.d. the component is a straight line passing through the (0,0) origin.

    Therefore we can say that the current in the resistor is proportional to the applied p.d.
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The straight line passing through the origin allows us to say that:



The constant in this relationship is called the resistance R, so we can say:

 


This law was first stated by Georg Ohm in 1826 and is known today as
Ohm's Law
Ohm's Law states that at constant temperature the current in a conductor is directly proportional to the d.p. across the conductor. The constant of proportionality is called the resistance of the sample.
Ohm's Law
.

Any material which obeys Ohm's Law is called an ohmic conductor.
Ohm's equation calculator
Fig.8 shows a utility which allows you to make calculations using any version of Ohm's equation. Clicking on a particular term in the equation rearranges the equation and calculator to allow that particular quantity to be found.
Figure 8.   Ohm's equation calculator.
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What is the resistance of a lamp when the voltage across the lamp is 9 V and the current flowing is 60 mA?
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Use the Ohm's equation calculator to answer the following questions. Be sure to check that the units selected in the calculator match those of the question.
What is the voltage across a 100 Ω resistor when the current flowing is 50 mA?
  • V    (to 1 d.p.)
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What is the current flowing through a 3.9 kΩ resistor if the voltage across the resistor is 9 V?
  • mA    (to 1 d.p.)
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What is the
power
The power of system is a measurement of the rate at which energy is transferred from one form to another. The scientific unit of power is the watt.
power
dissipated in the 3.9 kΩ resistor if the voltage across the resistor is 9 V?
  • mW    (to 1 d.p.)
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Resistors
Carbon film resistors such as those shown in Fig.9 are ohmic conductors. These resistors are used widely in the construction of electronic circuits.

Figure 9.   Carbon film resistors.
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The resistance value for a particular resistor is identified from the coloured bands on the body of the resistor.
Figure 10.   Coloured bands on a resistor.
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Each colour represents a number according to the following scheme:
Figure 11.   Resistor colours.
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The first band on a resistor is interpreted as the first digit of the resistor value.
For the resistor shown in Fig.12 , the first band is yellow, so the first digit is …
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Figure 12.   Typical resistor.
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For the resistor shown in Fig.12 , the second band is violet, so the second digit is …
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The third band is called the multiplier and is not interpreted in quite the same way. The multiplier tells you how many zeros you should write after the digits you already have.
For the resistor shown in Fig.12 , the third band is red so the multiplier is …
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The resistor shown in Fig.12, has a resistance of …
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Figure 13.   Typical resistor.
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The fourth band on the resistor is slightly separated from the other three and is called the tolerance band. This indicates the percentage accuracy of the resistor value. Most carbon film resistors have a gold-coloured tolerance band, indicating that the actual resistance value is within ±5% of the nominal value. Other tolerance colours are:
Figure 14.   Tolerance colours.
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The resistor shown in Fig.15 has a resistance of …
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Figure 15.   Resistor
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I–V characteristic for a lamp
The current versus voltage characteristic for a lamp can be determined using the circuit in Fig.16 below.

Click on the figure below to interact with the model.

 Figure 16.  Determining the I–V characteristic for a lamp.



Alter the slider in Fig.16 and enter pairs of readings as appropriate into the table in Fig.17
Figure 17.   The shape of the I–V characteristic for a lamp.
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Select the words which best describe the current–voltage variation of Fig.17.

  • As the p.d. across the lamp is increased, the current . If the p.d. is doubled, the current doubles.
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The graph of the current through the lamp is a curved line passing through the origin (0,0). Since the line is curved we cannot say that the current in the lamp is directly proportional to the applied p.d.

As the p.d. increases it becomes progressively more difficult for current to flow through the lamp. Increasing the p.d. increases the current, but the fractional rise in the current will not be as great as that in the p.d. The resistance in this case is not constant. Instead, as the p.d. and current increase, the resistance increases. This is to be expected. As the metal in the filament of the lamp warms up, its atoms will vibrate more and the free electrons will collide with them more as they try to pass through.

By looking at the graph we can see that the slope of the line decreases. Therefore we can conclude that on this type of I–V characteristic a less steep slope represents a larger resistance.
I–V characteristic for a silicon diode
The current versus voltage characteristic for a silicon
diode
A diode is an electronic component that only allows an electric current to pass in one direction.
diode
can be determined using a similar set-up. However, the diode is a non-linear component and so we want to be able to observe the effects of both positive and negative voltages.


Click on the figure below to interact with the model.

 Figure 18.  Determining the I–V characteristic for a silicon diode.



Alter the slider in Fig.18 so that the p.d. across the diode is as indicated in the table of Fig.19. Enter the corresponding current into the appropriate space in the table.
Figure 19.   The shape of the I–V characteristic for a silicon diode.
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The manner in which the current in Fig.18 varies as the p.d. across the diode is altered can be summarized in the current–voltage characteristic.

When the p.d. across the diode is greater than 0.7 V large quantities of current pass through the circuit. If the p.d. is less than this, or is negative, no current flows. This property means that a diode will only conduct when it is connected into a circuit the correct way around.

Varying resistance
A variable resistor is an electrical component designed to allow the current in a circuit to be altered. The resistance of the variable resistor in Fig.20 depends on the length of the wire.

Figure 20.   A variable resistor.
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Move the slider in the variable resistor of Fig.20. The size of the current is indicated by the thickness of the arrows. Observe how the current changes as you move the slider.

When is the current greatest?
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When the sliding contact is near terminal A, the charge only has to pass through a short length of resistance wire before leaving, so the resistance in the circuit is relatively low. This is why the current is large.

If the sliding contact is at terminal B, the charge has to pass through several turns of resistance wire before leaving. With more resistance in the circuit, the current in the circuit is less.

This type of variable resistor can be used in a circuit as shown in Fig.21. Move the slider to alter the current in the bulb.

Click on the figure below to interact with the model.

 Figure 21.  A variable resistor in a circuit.



For good conductors and moderate currents, the resistance of a piece of wire R is directly proportional to its length L, provided its temperature remains constant. This relationship can be written as:



This relationship allows us to compare the resistances of different lengths of wires. The p.d. across both wires in Fig.22 is the same, but the shorter wire allows more charge to pass in a given time, so the current in it is larger than the current in the longer wire.

Figure 22.   Current in short and long wires.
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The current in the shorter wire of Fig.22 is higher. Since the p.d. across both wires is the same, the shorter wire's resistance is lower than the resistance of the longer sample. For comparisons to be valid the samples of wire must be made from the same material and have the same cross-sectional area.

Complete the following statements.

  • For wires of equal length of the same material, the resistance of a thinner sample is than the resistance of a thicker wire. If the p.d. across both wires is the same but the wire allows more free to pass through every second. The current in it is therefore higher and its resistance is than the resistance of the thinner sample.
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We can use our model of current as a flow of charge to understand that less charge each second will be able to flow through narrower openings. So the resistance of thin wires is greater than the resistance of thick wires.

For samples of a specific material having the same length the resistance R of the sample is inversely proportional to the cross-sectional area A.

 
For different wires of the same material we can say, and

We can combine these relationships to state that,

We can further state that,

     The constant in this equation is called the Resistivity (symbol ). The units of resistivity are Ωm−1.



The
resistivity
Resistivity is a material property and is numerically the same as the resistance between opposite faces of cube of the material of side 1m.
resistivity
of a sample of material can be stated as:

 


From this equation we can state that the resistivity of a material is numerically the same as the resistance between the sides of a cube with sides 1 m long.


Material Resistivity / Ωm
Copper 1.7 × 10−8
Gold 2.4 × 10−8
Pure silicon 2.3 × 103
Glass of the order of 1012

Figure .  Resistivity values for common materials.

What is the resistance of a copper wire that is 6 km long and has a cross-sectional area of 2 × 10−5 m2?
  •  Ω
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As you can see from the table, resistivity values vary greatly. Good conductors have very small values while good insulators have very high resistivities. Between these two extremes are a group of semiconductors such as pure silicon. The properties of these materials will be discussed later.

Resistance and temperature
When introducing resistivity we looked at how the resistance of samples of a material depended on the length and cross-sectional area. Underlying this was the assumption that the temperature of the sample remained constant. Fig.24 represents wires at different temperatures.

Figure 24.   Resistance and temperature.
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Complete the following statements.

  • The p.d. across both wires in Fig.24 is the same, but the cooler wire allows free electrons to pass through every second. The current in it is therefore and its resistance is consequently than that of the hotter wire. The atoms in the hotter wire vibrate more than in the cooler wire. As a consequence, the free electrons collide with more atoms and are impeded more as they move through the conductor. We can therefore conclude that the resistance of conductors increases as the temperature .
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Sometimes conductors will change their resistance when we don't want them to. Passing a large current through a conductor causes it to heat up, changing its resistance. Designers of integrated circuits use a variety of techniques to control the operating temperatures of microprocessors and other components.

 
Figure 25.   Keeping cool!
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Superconductors
In 1911, Heike Kamerlingh-Onnes, working in his low-temperature laboratory, discovered that at temperatures a few degrees above
absolute zero
Absolute zero is the lowest possible temperature: −273 °C. At this temperature all particles stop moving, and gases exert no pressure at all.
absolute zero
(−273 °C) an electrical current could flow in mercury without any discernable resistance! He named this remarkable new phenomenon superconductivity. In 1913, Kamerlingh-Onnes received the Nobel prize for his discovery of superconductivity.

Figure 26.   Heike Kamerlingh-Onnes
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At high temperatures superconductors exhibit resistance just like conventional materials. However, when the temperature of the
superconductor
A material through which current can flow without experiencing any resistance is called a superconductor. Certain materials cooled below their transition temperature behave as superconductors.
superconductor
is decreased below the transition temperature (Tc) its resistance 'disappears'. A full understanding of why this happens had to wait until 1957, and was achieved using quantum mechanics.

In 1986, the discovery of superconductivity at substantially higher temperatures launched a major drive to find materials which would act as superconductors at temperatures closer to room temperature. By 1995 researchers had developed superconductors with transition temperatures as high as −135 °C.

Figure 27.   MagLev train.
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In the future, superconductors could be used for many purposes including high-speed computers, transporting electricity over large distances, storing energy in magnets, more efficient and lighter generators and motors. On 24 December 1997 the Japanese Maglev train achieved a speed of 550 kilometres per hour! Superconducting materials were used in the magnets that provided the magnetic levitation for the train.

Summary


The speed at which free electrons drift through materials is very small, whereas electrical energy transfer is very fast.

This speed at which electrons drift through a material is given by the formula:



The resistance of a sample of material is the ratio of the p.d. across the sample and the current in the sample.

The resistance of a sample of material depends on the type of material and the physical dimensions of the sample.

Resistivity values allow the conducting properties of different materials to be compared directly. The resistivity of a sample can be calculated by the equation:



In some materials, the p.d. across a sample V is directly proportional to the current I in the sample. These materials are called ohmic conductors because they obey Ohm's Law.



Passing a current through some materials warms them up and changes the resistance.

At very low temperatures some materials exhibit superconductivity.

Exercises
1. In the circuit in Fig.28 resistors with values 2.2 kΩ and 4.7 kΩ are connected in parallel with a 5.0 V power supply. What is the potential difference across each of the resistors?
  • V   (to 1 d.p.)
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Figure 28.  
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2. Calculate the current in the 2.2 kΩ resistor in Fig.28.
  • mA   (to 2 d.p.)
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3. Calculate the current in the 4.7 kΩ resistor in Fig.28.
  • mA    (to 2 d.p.)
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4. Calculate the total current from the supply in Fig.28.
  •  mA   (to 2.d.p.)
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5. Calculate the single value of resistor which will take a current of 3.33 mA from this 5 V supply.
  • kΩ    (to 2 d.p.)
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6. A cube of a certain material has sides of length a. What is the electrical resistance between opposite faces of this cube directly proportional to?
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7. A 240 volt, 60 watt light bulb filament is made from metal wire which has a resistivity 75 × 10–8 Ωm. If the diameter of the wire is 0.1 mm, what is the length of the wire needed to make each filament?
  • m   (to nearest whole number)
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8. A piece of wire of length 50 cm and diameter 1 mm, has a resistance of 10 Ω. Calculate the resistivity of the material.
  •  × 10−5 Ωm   (to 2.d.p.)
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9. A heater rated at 3 kW is used to heat water. How much electrical energy does this heater supply in 5 minutes?
  • kJ   (to nearest whole number)
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10. A small bulb on a control panel is rated 3.5 V and 300 mA. It is connected with a series resistor of resistance R to a 5 V supply of negligible
internal resistance
All batteries or power supplies have internal resistance. This resistance has the effect of reducing the output p.d. as the current supplied increases.
internal resistance
. Which one of the following options shows the correct resistance for R and the power delivered to R when the bulb is working normally?
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11. A sample of copper wire has 8 × 1028 free electrons per cubic metre. The current in a sample of wire with a cross-sectional area of 1.2 × 10–6 m2 is 3.5 A. (The size of the charge on each electron is 1.6 × 10−19C.)

Calculate the drift velocity of the free electrons.
  •  × 10−4 ms−1   (2 d.p.)
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12. Electrons drift through a certain sample of wire with a speed of 0.45 mms−1. The current in the wire is 0.3 A and its diameter is 2.6 mm. (The size of the charge on each electron is 1.6 × 10−19C.)

Calculate the number of free electrons per cubic metre in the sample.
  •  × 1026 m−3   (2 d.p.)
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13. The resistor shown in Fig.29 has a resistance of ...
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Figure 29.   Resistor
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14. An electrical transmission cable is made out of 1 strand of steel wire (resistivity of steel = 1.5 × 10−7 Ωm.) surrounded by 6 strands of aluminium wire (resistivity of aluminium = 3.2 × 10−8 Ωm.). The diameter of each wire is 1 cm. The resistance of 1 km of this wire would be:

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