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.
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.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.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.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.
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.
The speed at which the electrons drift v_{d} 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!
To derive a relationship connecting the current I and drift speeds v_{d}, 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 crosssection 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 v_{d}. 
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 × 10^{28} 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 crosssectional area of 2 × 10^{−6} m^{2} 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}.
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.
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.
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.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.
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.The resistance value for a particular resistor is identified from the coloured bands on the body of the resistor.
Each colour represents a number according to the following scheme:
The first band on a resistor is interpreted as the first digit of the resistor value.
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.
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 goldcoloured tolerance band, indicating that the actual resistance value is within ±5% of the nominal value. Other tolerance colours are:
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.
Alter the slider in Fig.16 and enter pairs of readings as appropriate into the table in Fig.17
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 setup. However, the diode is a nonlinear 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.
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.
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. 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 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.
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.
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 crosssectional area.
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 crosssectional 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.


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 crosssectional
area. Underlying this was the assumption that the temperature of the sample remained constant. Fig.24 represents wires at
different temperatures.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.
Superconductors
In 1911, Heike KamerlinghOnnes, working in his lowtemperature 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, KamerlinghOnnes received the Nobel prize for his discovery of superconductivity.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 (T_{c}) 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.
In the future, superconductors could be used for many purposes including highspeed 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.
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.
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