Electricity Tutorial 4  Resistivity
Contents 
The
resistance of a wire depends on three factors:
the
length; double the length, the resistance doubles.
the
area; double the area, the resistance halves.
the
material that the wire is made of.
Resistivity is a property of the material. It is defined as the resistance of a wire of the material of unit area and unit length.
The formula for resistivity is:
In physics code we write this as:
We can rewrite this to give:
There are three bear traps

Constantan has a resistivity of 47 ´ 10^{8} W m. How much of this wire is needed to make a 10 W resistor, if the diameter is 0.50 mm? Give your answer to an appropriate number of significant figures. 
Measuring Resistivity
This is a required practical. You will have a sample of a resistance wire (usually constantan or nichrome). You will need to measure the diameter of the wire with a micrometer screw gauge. This should be done in three separate places to reduce the uncertainty. You work out the area of the wire using:
Then you will measure the resistance of lengths of the resistance wire, using a circuit like this:
You measure lengths of 0.10 m, 0.20 m, etc. You measure the voltage and current. You should do repeat readings, and take averages. Then you need to process the data to get the resistance. (You know how to do this, don't you?) Then the data are plotted on a graph like this:
The gradient is measured. To work out the resistivity, you need to multiply the gradient by the area of the wire. This can be compared with the data book value of the resistivity.
The reciprocal or inverse of resistivity is conductivity. It has the physics code s, (“sigma”, a Greek letter ‘s’), and units Siemens per metre (S m^{1}).
Conductivity is given by the relationship:
Superconduction
A superconductor is a material that has zero resistance. A current flows when there is no potential difference. The piece of metal floating above the magnet shows that there must be a current flowing.
Picture from Wikimedia Commons.
Authors: Julien Bobroff and Frederic Bouquet
For all metals the resistivity (hence resistance) decreases as they get colder.
For some metals like copper and silver, there is still a tiny bit of resistance left at very low temperatures.
Very low temperatures have to be maintained, which is expensive. Room temperature superconductivity has not been seen.
Superconductivity is seen in:
Aluminium
Tin;
Some alloys;
Some heavilydoped semiconductors.
All superconductors have a critical temperature above which the phenomenon stops. The graph below shows the idea:
Above the boiling point of liquid nitrogen, 77 K (196 ^{o}C), superconductivity can be observed in a few materials. These are called high temperature superconductors.
Very large magnets such as those found in the large hadron collider have coils made of superconducting materials. It is believed that the superconductivity will last 100 000 years, as long as the coils don’t go above their critical temperature.
The mechanism for superconduction is complex, and cannot be explained in terms of electrons colliding with ions. The Meisner Effect and flux trapping are not required on the syllabus.
The mechanism for superconduction is complex, and cannot be explained in terms of electrons colliding with ions.
Explain what happens when a superconducting metal reaches its critical temperature. 
Superconduction is used in very powerful magnets used is MRI Scanners and machines used in high energy particle physics (e.g. cyclotrons, and accelerators).
Metallic Conduction (Extension)
Electricity moves due to the movement of charge carriers. If we think about an ionic solution, the positive ions are attracted to the negative terminal (the cathode), while the negative ion are attracted to the positive terminal (the anode).
In a metallic conductor (wire), the simplest model of conduction is to consider the metal as a lattice of metal ions in a sea of free electrons. The electrons move about randomly.
When a voltage is applied across the ends of the wire, the electrons continue to move randomly, but there is an overall drift to the positive end of the wire. So you will (rightly) think that electrons go from negative to positive. The protons don't move. So this idea is opposite to what you have been told. The explanation is that the earliest physicists got it wrong. They didn't know about electrons in the Eighteenth Century. So instead of rewriting all the rules of electricity, people talked about conventional current going from positive to negative. All currents are regarded as conventional.
Resistivity is the resistance of a wire of 1 metre length and of 1 m^{2} cross sectional area. According to this model, resistance arises due to the vibration of the metal ions, and the probability of the ions colliding with the free electrons. Resistance is a representation of the probability of collisions. Such collisions involve a change of energy from kinetic energy to internal energy. The increase in internal energy makes the temperature in the ions rise, hence increases the probability of a collision. Therefore the resistance increases as the wire gets hotter.
How fast do electrons move?
When a voltage is applied across the ends of the wire, the electrons continue to move randomly, but there is an overall drift to the positive end of the wire. So you will (rightly) think that electrons go from negative to positive. The protons don't move. So this idea is opposite to what you have been told. The explanation is that the earliest physicists got it wrong. They didn't know about electrons in the Eighteenth Century. So instead of rewriting all the rules of electricity, people talked about conventional current going from positive to negative.
We can write an equation for the conduction of a current in a wire. The current depends on:
The speed of the charge carriers (v (m/s));
The area of the wire (A (m^{2}));
The charge on the electrons (e = 1.6 × 10^{19} C);
The number of charge carriers per unit volume (n (m^{3})).
The symbols in italics are the physics codes for the various quantities, and the units are given as well.
The formula is:
I = nAve
The number of charge carriers per unit volume is probably the hardest quantity to get your head round. It means the number of electrons in a cubic metre of the material. For example, copper has 8 × 10^{28}electrons in each cubic metre of material. You can look up the number of electrons per cubic metre in a data book.
Derivation
Consider a piece of metal that is l m long and has an area of A m^{2}. It is made of a metal that has N free electrons m^{3}. Let's suppose that a current of I A is flowing. Each electron has a charge of e C.
We know that current is the flow of charge every second.
The total charge is the number of electrons multiplied by the volume of the metal and the charge on each electron:
So we can substitute for Q:
The term l/t is a distance over time which is speed, v m s^{1}. So we write:
I = nAve
Worked Example A wire is carrying a current of 200 A. Its cross sectional area is 7.85 × 10^{5} m^{2}. If copper has 8.0 × 10^{28} electrons per cubic metre, what is the speed of the electron drift? Give your answer to an appropriate number of significant figures. 
Answer Rearrange the equation:
v = 200 A ÷ (8.0 × 10^{28} m^{3} × 7.85 × 10^{5} m^{2} × 1.6 × 10^{19} C) = 2.0 × 10^{4} m s^{1} The answer is given to 2 significant figures as the data are given to two significant figures. 
The result from this calculation shows that the speed of electron
drift is slow, 1 mm every 5 seconds. The propagation of the message "that
the current is flowing" is very fast, not far off the speed of light. But the
drift speed of electrons is not at all fast. We can show this by putting a
small crystal of potassium permanganate (a dark purple ionic compound) onto a
filter paper which has been soaked in sodium chloride solution. When a current
flows, the purple permanganate ions move towards the positive terminal, but it
takes some time.
In questions on this, you may be given a diameter of a wire. To work out the area, you must use the formula:
You must convert millimetres to metres when using the equation. 
Now try this example:
A copper wire of diameter 1.4 mm connects to the tungsten filament of a light bulb of which the diameter is 0.020 mm. The current in both materials is 0.52 A. Find the speed of an electron in each of the two materials. Copper has 8 × 10^{28} electrons per cubic metre. Tungsten has 3.4 × 10^{28} m^{3}. 
The tungsten filament has a higher resistance than the copper wire. For the same current to pass through, the electrons have to go faster, because:
The number of charge carriers per cubic metre is smaller;
The area is smaller.
This is opposite to the generally perceived idea that electrons are slowed down by high resistances. There is a greater chance of a collision, so the wire will hotter.
Models of semiconduction are to be found in Tutorial 6.