Interactive textbook: Current and resistance

Everything is backwards

Benjamin Franklin made an unfortunate choice when he decided on which charge to call positive and which negative. It's strange to think now that the electron could have been positive... but Franklin didn't discover that it was negative. He decided it was negative. We are so used to seeing electrons with minus signs attached to them that if we were ever able to see one we would expect it to be a small ball with a minus sign in the middle. We should remember that this is a model.

What the electron has is some properties that make it interact with other particles in a certain way. These properties are associated with a small volume of space rather like a ball occupies a small volume - that's why we think of them as particles.

It is very convenient that there are two types of charge that add and cancel exactly like our system of positive and negative numbers. If there had been three types of charge it would not have been so easy to model. When you get to the section on particle physics you will learn about quarks. They have a property that comes in three varieties so positive and negative numbers are not enough; we have to use colour to model this property!

So, why is everything backwards? Well, we like to have models that help us to visualise how matter behaves. The potential wells and hills introduced in the last section are a good example of this.

We know that the charge in the image above will accelerate to the right by thinking how it would roll down the potential hill. If we released a bunch of positive charges in the field represented below we can imagine that they would either gather at the bottom of the well or sit on top of the hills in the same way as balls would if released on an actual hill.

This is fine as long as we consider the movement of positive charges. The problem is that negative charges do the opposite. This is because potential is defined in terms of positive charge. Therefore, negatives roll up hills. This is not very helpful when it comes to visualising the behaviour of particles. 

We could imagine the hill is inverted but it's bit messy!

This wouldn't matter if it wasn't for the fact that it is electrons that move about in circuits not protons, so whenever there is a force acting on electrons (or current) they flow 'uphill'.

It would be nice if we could visualise the flow of charge like the flow of water in a river - but water doesn't flow uphill! So, we pretend it doesn't. We ignore the flow of electrons and consider instead conventional current which flows in the correct direction, downhill. This makes no difference to our calculations and is much easier to visualise.

Conduction in some liquids is due to the movement of positive ions.

In this case conventional current and the flow of charge carriers are:

Positive charges flow "down hill".

 

Conduction

A conductor is a material that allows charge to flow through it. Metals contains electrons able to move around freely (free electrons). These are not 'extra' electrons; they are just the highest energy electrons of the metal atoms.

The blue balls in this animation are the electrons and the green ones the lattice of metal atoms. 

There are several interesting things to note about the motion of the electrons:

  • Notice how the electrons are closer to each other on the right; the electrons are accelerated by the electric field towards the left. Why? The field is from left to right and so the left side is at a higher potential than the right. As electrons are negative they have a higher potential energy when placed in a position of low potential. As they travel from right to left they lose potential energy.
  • The lattice atoms get in the way of the electrons so the electrons only accelerate for a short distance. Overall, the electrons move with constant velocity. This is rather like what happens when we fall down the stairs; as we hit each step we lose kinetic energy and so fall from step to step in the same amount of time (better than falling out the window!).
  • When the electrons collide with the lattice they transfer energy to the lattice causing the lattice atoms to vibrate. This results in an increase in temperature. When the metal gets hot the lattice atoms get in the way of the electrons more so there is more resistance to their motion.

 

If a metal is cooled down to almost 0 K the metal starts to superconduct, the reistance becomes zero.

This is due to:

 

No vibration does not imply no collisions.

 

When a semiconductor is heated its resistance decreases.

This is due to:

A semiconductor does not have many free electrons, increasing temp. allows more of them to be liberated from the lattice.

Current

Current (\(I\)) is defined as the amount of charge flowing (\(Q\), measured in Coulombs, \(\text{C}\)) per unit time (\(t\)): 

\(I = {Q\over t}\)

The unit of current is \(\text{C s}^{-1}\), also known as the Ampere, \(\text{A}\). It seems a bit strange given that this unit is derived, but the Ampere is the fundamental unit of electricity.

Drift velocity

On a small scale electrons are continually accelerating and colliding. On a large scale they appear to have a constant velocity. This is called the drift velocity.

Consider a section (length \(L\)) of conductor with current \(I\) flowing:

The time taken for the rightmost electron to move to the other end is \(t\). This means that in time \(t\) all the electrons in this section will pass through the left hand end.

In time \(t\) the electrons travel a distance \(L\) so the drift velocity, \(v\), is:

\(v={L\over t}\)

The volume of this section is \(AL\), where \(A\) is the cross-sectional area. There are \(n\) electrons per unit volume so the charge (\(Q\)) of all the free electrons in this cylinder is equal to the charge per electron (\(e\)) multiplied by the total number of electrons:

\(Q=nAL\times e\)

\(I={Q\over t}={nALe\over t}=nA{L\over t}e\)

\(\Rightarrow I=nAve\)

 

Potential difference

It is worth recalling what we mean by potential difference. Potential difference is the work done per unit charge in taking a small test charge from A to B. So when a charge flows from high potential to low potential it will experience a change in energy (and a loss of electrical potential energy). The bigger the potential energy, the more electrical energy the charge will lose.

The relative atomic mass of calcium is 40 and its density 1500 kg m-3.

Calculate the volume of 1 mole of calcium

1 mole = 40 g = 0.04 kg

V = mass/density = 0.04/1500

 

If each calcium atom has 2 free electrons, how many free electrons are there per m3?

2 x 6 x 1023/2.7 x 10-5

If the calcium is made into a 1m long rod what will its cross sectional area be?

V = LA

A = V/L = 2.7 x 10-5 m2 = 2.7 x 10-5 x 104 cm2

If 1A flows through the rod what will the drift velocity of the electrons be?

I = nAve

v = I/nAe

v = 1/4.4 x 1028 x 2.7 x 10-5 x 1.6 x 10-19 = 5.3 x 10-6 m s-1

Resistance

The atomic lattice gets in the way of the electrons, opposing the flow of current. Different conductors oppose the flow by different amounts. If an aluminium wire and a sliver wire, of the same dimensions, were connected between the same two points the silver wire would have more current flowing through it.

The resistance (\(R\)) of the wire is defined as the ratio of potential difference (\(V\)) to current:

\(R={V\over I}\)

The unit of resistance is Ohm (\(Ω\)).

In this (exaggerated!) example the resistance of silver is \(1\text{ Ω}\) and the resistance of aluminium is \(2\text{ Ω}\).

Investigating the factors affecting resistance

The line made by a graphite pencil conducts electricity. By measuring the resistance of pencil lines with a multimeter we can  investigate the relationship between resistance and length or cross sectional area.

Using a pencil line of uniform width we can show that \(R\propto L\). The longer the line the more collisions there are with lattice atoms.

If we increase the width of the line by adding parallel lines to the side, we increase the cross-sectional area (the area of the end of the cuboid prism). Cross-sectional area is the product of the number of lines, the width of one line and the thickness of graphite on the paper. Here we can see that the resistance is inversely proportional the number of lines: \(R\propto{1\over A}\)

From these results we can calculate the thickness of the pencil line. It's \(1.9 \times 10^{-8}\text{ m}\) thick, which is about \(100\) atoms.

Resistivity

As seen in the investigation graphs, resistance of a conductor is dependent on its length and cross sectional area. Combining the relationships so far:

\(R\propto{L\over A}\)

The constant of proportionality is the resistivity (\(ρ\)):

\(R=\rho{L\over A}\)

In its own right, resistivity is defined as the resistance of a \(1\text{ m}\) cube of a material, and it has different value for each material.

Silver has resistivity \(1.6 \times 10^{-8}\text{ Ω m}\). This value is constant for any shape or size of silver, and is equal to the resistance of a \(1\text{ m}^3\) block of silver (not easy to come by!).

The resistivity of calcium is:

Same order of magnitude as silver (it's a metal) but smaller.

Resistors

Resistors are electrical components used to set the current in a circuit. Lab resistors are small cylinders with coloured bands. The bands are a code used by non-colour blind electricians to determine the resistance. There is also a silver or gold band which gives the uncertainty (tolerance) of this value. The circuit symbol for a resistor is a rectangle:

Insulators

All the electrons in an insulator are located within molecules and are not free to move around.

Glass is an example of an insulator (until you heat it up to around \(1000\text{ K}\) at which point it starts to conduct - because atomic electrons receive enough energy from the vibrating atoms to become free electrons).

Semiconductors

A semiconductor is something that exhibits behaviours between a conductor and an insulator e.g. silicon.

Semiconductors contain some free electrons and can be impregnated with impurities to either add free electrons or "holes" that accept electrons. Holes enable conduction since they allow otherwise fixed electrons to move into them. When junctions between two different semiconductors are made you have the basis of computer chips.

Superconductors

As the name suggests, superconductors have zero resistance.

They are made by cooling metals down to a few Kelvin. At these temperatures, electrons pair up. When the pairs of electrons collide with the lattice, one of the pair loses momentum and the other gains it. The overall result is no change in momentum and so no resistance. Superconductors can carry large currents without getting hot, which makes them ideal for constructing very strong electromagnets. Perhaps one day superconductors will be designed for day-to-day temperatures, reducing the energy loss in the transmission of electrical power.

Ohm's law

Ohm's law states that the current flowing through an Ohmic conductor is directly proportional to the potential difference across it, provided that temperature remains constant.

\(V\propto I\)

\(V=IR\)

Experimental verification

To verify Ohm's law you need to measure the current flowing through a conductor for varying potential difference. The potential difference can be supplied with a power supply, the current measured with an ammeter (connected in series) and the potential difference measured with a voltmeter (connected in parallel).

If an Ohmic conductor is used this will result in a linear relationship between current and potential difference:

The resistance, \({V\over I} = 0.5\text{ kΩ}\).

In this example it looks as though resistance is the same as \(1\over\text{gradient}\) but don't be fooled! Resistance is a ratio (\(V\over I\)) and not a rate of change of \(V\) with \(I\) (\(ΔV\over ΔI\)). This difference becomes important when using a non Ohmic conductor like a light bulb.

Here we can see that the resistance increases from \(1\text{ Ω}\) at \(1\text{ V}\) to \(2\text{ Ω}\) at \(6\text{ V}\). If we took the gradient at these points we'd get a different result. The reason for this increase is because the light bulb gets hot.

A current of 3 A flows through a resistor when a potential difference of 12 V is applied across it.

What is its resistance?

R = V/I = 12/3

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