Electricity and Magnetism

D.C. ELECTRICITY (Part 1)


ATOMS, ELEMENTS AND MOLECULES

An atom has a tiny nucleus around which are orbiting electrons.

The nucleus contains positive particles called protons and neutral particles called neutrons. An electron has a negative charge, equal but opposite to that of a proton. The force of attraction between the positive protons and the negative electrons keeps the electrons in orbit.

The lightest element is hydrogen – its atoms have just one proton in the nucleus, with one electron in orbit.

The above diagram represents an atom of the next heaviest element, helium, with 2 protons and 2 neutrons in its nucleus and 2 electrons in orbit.

Increasing the proton number one at a time takes us through all the naturally occurring elements from the lightest, hydrogen, with one proton, to the heaviest, uranium, which has 92 protons in its nucleus, and 92 electrons in orbit.

A molecule is made up of atoms which have joined together. For example, two hydrogen atoms can combine with one oxygen atom to form one molecule of water. In chemical symbols, H = a hydrogen atom , O = an oxygen atom and H2O = a water molecule.


ELECTRIC CURRENT AND CHARGE

The electrons in an atom are at various distances from the nucleus, and the further away they are, the weaker they are attracted to the nucleus. In a piece of metal one or more electrons completely escape from each atom, and become free to move around inside the metal. The random motion of electrons in a metal does not constitute an electric current, since there is no net motion in any particular direction. When a battery is applied across a piece of metal, these so-called ‘free electrons’ start to move towards the positive terminal. This flow of electrons is an electric current. In poor conductors (i.e. insulators), such as polythene, the electrons are tightly bound to the atoms and so cannot form an electric current.

The simple cell

In 1799 Volta discovered how to produce a continuous electric current. His set-up consisted basically of two different metals (‘electrodes’) separated by a liquid that could carry a current (an ‘electrolyte’). The following is equivalent to the set up that Volta used:

The small bulb lights when the electrodes are put in the acid, showing that an electric current is flowing through the bulb.

Electromotive force (‘emf’)

If a high resistance voltmeter, which is one that draws very little current, is put in place of the lamp, it reads about one volt (1V). This is called the electromotive force ('emf') of the cell. A dry cell, such as used in a torch, has an emf of about 1.5volt.

In a school or college laboratory, a lead-acid ‘accumulator’ is often used as a power supply. This has an emf of about 2volts. A car battery is equivalent to six accumulators joined end to end (in ‘series’) – producing an emf of about 12 volts. An accumulator and a car battery can be recharged many times – a car battery charges while the engine is running.

Note: We are concerned here only with direct current ('dc'), which means current that flows in one direction only, such as is produced by a battery (as opposed to alternating current ('ac') which repeatedly changes direction).

Circuit diagrams

All the components that might occur in an electrical circuit are given symbols - the following represents an actual circuit, together with the corresponding circuit diagram:

When wiring up a circuit, one approach is to treat it as a series of loops, and wire up each one in turn. In the above case, the top loop contains the battery, switch, ammeter and bulb. If this is wired first, it can be tested by pressing the switch and checking that the ammeter moves the correct way. The bottom loop is then completed, which means just adding the voltmeter across the bulb.

Current and charge

It was originally decided to treat electric current as being a flow of positive charges, moving from the positive to the negative terminal – this is referred to as the direction of ‘conventional current flow’. However, it was later discovered that electric current in metals is actually the flow of negative electrons moving from the negative to the positive terminal. In circuit diagrams it is still the conventional direction of current flow that is shown.

Definition
The unit of electric charge is the coulomb (C), so the unit of current is coulombs per second.

The equation relating current, charge and time is sometimes remembered as:

This actually represents three equations - put a finger on one symbol, and the position of the other two indicates if they are to be divided or multiplied:

The ammeter reads 0.3A (this is a typical current for a small bulb). How much charge passes through the bulb in (a) 1sec, (b) 3 minutes?


Currents in series

Both A1 and A2 read the same. Thus, the current is the same at every point in a series circuit.

Currents at a junction

In the above, the bulbs are not the same so the currents through them are different. For one particular set-up: Note that 0.10 + 0.20 = 0.30

This is an example of Kirchoff’s first law (or ‘junction law’), which can be stated as:

Example
The diagram represents a set of ammeters indicating currents in amps, with arrows for the direction of the currents. What would the ammeter labelled A read, and what would be the direction of the current?
POTENTIAL DIFFERENCE (‘p.d.’)

The basic thing that happens in an electric circuit, such as that below, is that energy changes from one form to another:

Definition: We can view potential difference as an electrical ‘pressure’ driving a conventional current from a higher to a lower potential. So, in the above circuit diagram, A is taken to be at a higher potential than B.

In general, if W joules of electrical energy changes to other forms of energy when Q coulombs passes between two points, then the p.d. (potential difference) V between (or across) the two points is given by:

From this, the unit of p.d. is joules per coulomb (J/C or J C-1 ), which we call the volt (V): i.e. 1 V = 1 J C-1.

The above equation is sometimes remember as:



This gives the amount of energy, W joules, changed from electrical energy to other forms of energy when a voltage, V volts, produces a current, I amps, which flows for a time, t seconds.

Example

The voltmeter reads 12V. How many joules of electrical energy are changed to light and heat when:
  1. 5C flows through the bulb
  2. 2A flows for (i) 5 sec, (ii) 2 minutes


RESISTANCE

Resistance means opposition to something. In the case of electricity it is the opposition a material has to the flow of electric current.

A metal consists of millions of atoms, which are in fixed positions, and millions of  'free electrons'. An electric current through a metal is a flow of the free electrons. The electrons undergo many collisions with the atoms as they move though the metal and, as electrons are very much lighter than atoms, they are slowed down a lot. This is the reason that a metal has resistance to the flow of the electrons, i.e. to the flow of current.

The electrons moving through the filament in a bulb only move at an average speed of perhaps a few mm per second, but over the short distance they travel between collisions they travel very fast, typically about 100000 ms-1.

Consider:


This is a sensible definition since it means that if we apply the same voltage to two different conductors, the one that has the smaller current has the bigger resistance, R.

From the definition, resistance is in ‘volts per amp’. We call this the ohm:


Example

Using a small bulb for R in the above circuit, when V = 1.95 V, the ammeter read 0.35A. What was the resistance of the bulb?

R = V/I = 1.95/0.35 = 5.6 ohms

The definition of R is sometimes remembered as:


OHM'S LAW

This can be stated as:

This means that, for a conductor obeying this law, a graph of voltage against current is a straight line passing through the origin:

Thus, Ohm’s law implies that:

Since R is equal to V/I, it equals the slope (or gradient) of the above graph. Thus, resistance R = a/b.

Note - The equation R=V/I is notOhm’s law. It is simply the definition of resistance. For any conductor, if we measure a value of V and I we can work out a value for R, regardless of whether the conductor obeys Ohm’s law or not. Ohm’s law is as stated earlier (though the exact wording may vary).

Experiment to verify Ohm’s law

A variable resistor (see below) used in the laboratory can be quite big (typically about 40cm long). If only two of its three contacts are used, as below, a current enters at A, flows through the coil, and leaves at C, via the sliding contact and the top bar. The sliding contact allows the amount of coil that the current flows though to be varied, and so the current itself can be varied. When used to control a current in this way, the variable resistor is often referred to as a 'rheostat'.
 

R is kept at room temperature - if it warms when the current is increased, it should be allowed to cool again before the current and voltage are recorded.

Procedure

Typical results
Average R = 1.89 ohms.

We can plot I upwards or V upwards:

The points are found to lie (almost) on a straight line – showing that the voltage is proportional to the current, which verifies Ohm’s law for a metallic conductor. Also, we see in the table that R is (nearly) constant, which verifies the second version of Ohm’s Law.

In measuring voltage and current, there is always some inaccuracy, and the fact that the points may not lie exactly on a straight line and that R is not exactly constant is considered to be due to ‘experimental errors’.

For the right hand graph, if the ‘best fit’ straight line is drawn through the points, then the slope of the graph ( = a/b) equals the resistance R. If the right hand graph is actually plotted, it is found that R is (close to) 1.90 ohms.


RESISTOR NETWORKS

When resistors are arranged in combinations, it is possible to find a single resistor which would have the same effect – i.e. which would produce the same overall current for the same applied voltage.

Resistors in series

The applied voltage V produces a current I.


Using fact (1), we need no subscript on I since the same current passes through all the resistors.

Using fact (2):

Resistors in parallel

The applied voltage V produces the current I.


Using fact (1), we need no subscript on V since the same p.d. is across all the resistors.

Using fact (2):

Example

Find:
  1. the total resistance
  2. the value of the current I
  3. the values of V1 and V2.

Example

Find:
  1. the single resistance equivalent to the two resistors
  2. I
  3. I1
  4. I2

b)

c) When resistors are in parallel, the p.d. is the same across both of them. In the above case 2V.


Notice that  I1 + I2 = 1 + 0.5 = 1.5 A = the total current.

Example

Calculate:
  1. the overall resistance
  2. the value of I
  3. the p.d. across the 4 ohm resistor
  4. the p.d. across the other resistors
  5. I1
  6. I2
a) Firstly, we can work out the single resistance equivalent to the two in parallel. This is then in series with the 4 ohm, and can simply be added to it to get the overall resistance.


Thus, the circuit is equivalent to:

So, the total resistance = 4 + 1.5 = 5.5 ohm



Notice that I1 + I2 = 0.41 + 0.14 = 0.55 = I.

Example

Find:
  1. the total resistance
  2. I
  3. the p.d. across the 3 ohm
  4. I1
  5. I2
  6. the p.d.s across the 2 ohm and 1 ohm resistors
a) For the two resistors in series, total resistance = 2 + 1 = 3 ohm

The circuit is therefore equivalent to:

We find the combined resistance R using:


  c) The p.d. across the 3 ohm is the same as that of the battery, i.e. 2 V.


Notice that I1 + I2 = 0.67 + 0.67 = 1.34 = the total current I (the small difference is due to rounding).

f) The current of 0.67A flows through both the 2 ohm and the 1 ohm.
    p.d. across the 2 ohm = R I = 2 * 0.67 = 1.34 V.
    p.d. across the 1 ohm = R I = 1 * 0.67 = 0.67 V.
    Notice that these add up to 2.0 volts, which equals the battery voltage.


POTENTIAL DIVIDER

Theory

We assume that all of the current I goes though the resistors R1 and R2 and none goes through the voltmeter, and so:


A variable resistor can be used as a potential divider:

The sliding contact at C means that R1 and R2, the parts of the coils to either side of C, can be continuously varied, and the output V can therefore be any desired fraction of the emf E.


RESISTIVITY

By experiment we find that the resistance, R, of a material is:


Thus, for example, if we double the length, R is doubled, if we double the area, R is halved.

An expression such as the above can be turned into an equation by introducing a ‘constant of proportionality’. In this case, the constant is called the 'resistivity' of the material. Thus:


From this, the unit of resistivity  = ohm m2/m = ohm metres

Note: Resistivity is a property of a material, rather than of a particular conductor. Given the value of resistivity of a material, we can determine the resistance of a particular wire sample from its dimensions using the equation above for R.

Example


Germanium is an example of a 'semiconductor', its resistivity being between that of a very good conductor and a very good insulator.


TEMPERATURE COEFFICIENT OF RESISTANCE


The unit of temperature coefficient can be 0C-1 or K-1.

Note: Zero kelvins (0K) = -273 0C, and is called 'absolute zero', and is believed to be the lowest possible temperature.

The definition produces the average value of the temperature coefficient over the temperature range used (it may not the same over all temperature ranges).

Rearranging the definition gives:


Example


NON-OHMIC BEHAVIOUR

The circuit described earlier for verifying Ohm's law can be used to investigate the V-I characteristics of other conductors. The conductor labelled R in that circuit now represents one of the following:

1. A filament lamp

Results are obtained as previously described. The I-V and V-I graphs have the shapes:

The graphs indicate that the resistance increases as the current increases.

A mentioned before, when an electric current flows through a metal, the electrons undergo repeated collisions with metal atoms which are at fixed sites, which impedes the progress of the electrons. For the filament, as the current increases, the filament gets hotter. As the filament gets hotter, the atoms vibrate more strongly. Thus, they effectively present a bigger cross-sectional area to electrons, resulting in more collisions, thus impeding electrons even more, resulting in a greater resistance.

2. A thermistor

The term thermistor comes from thermal resistor. As the name suggests, its resistance depends on its temperature.

A thermistor is a semiconductor. One commonly used type is an NTC type, for which resistance falls significantly as the temperature rises. When such a thermistor is heated, more electrons are liberated from their bonds and are free to then act as current carriers, hence the drop in resistance.

In the previous Ohm's law set up, R would be replaced by a thermistor. This would be coated in varnish to make it water resistant, and then immersed in water which would be gradually heated. The resistance could be determined over a range of temperatures - for example, the thermistor could be initially be placed in ice-water which is then gradually heated to boiling.

Thermistors can be used to detect very small temperature changes and, for example, be used in a thermostat.

3. A semiconductor diode

This device allows a current to flow in one direction through itself, but not in the opposite direction. The circuit symbol indicates the direction that (conventional) current flows easily:

The previous experiment would be done with the diode in place of R. It could be done with the diode 'forward biased' and then 'reverse biased'. A graph of current against voltage is typically like:

Light emitting diodes (‘LEDs’) are used as indicators in many electronic devices, and often glow red or green when a current passes through them - circuit symbol:

A light dependent resistor ('LDR') is also a non-ohmic device. Its resistance depends upon the amount of light falling on it - circuit symbol:

METALS, SEMICONDUCTORS AND INSULATORS

Resistivites compared

1. Conduction in metals

Metals have low resistivities, i.e. they are very good conductors of electricity.

Each atom in a metal has one or more loosely bound electrons which can escape from the atom and wander throughout the metal. It is these ‘free electrons’ which take part in conduction when a potential difference is applied across a metal. In the absence of a p.d., the random motion of the free electrons does not constitute an electric current since there is no net movement in any particular direction.

Since about one electron escapes from each atom, there are many millions of electrons available to act as ‘current carriers’ or ‘conduction electrons’.

Insulators do not have free electrons (at room temperature) and so are poor conductors.

Drift velocity

When a current flows through a metal, the electrons repeatedly collide with atoms, so they do not have a constant velocity. However, they do have an average velocity, called their drift velocity.

Consider a conductor of length l and cross-sectional area A having n free electrons per cubic metre, each with a charge e.


We visualise all the electron moving with the same drift velocity v. In the time t in which charge Q passes through A, the extreme left electron in the diagram has moved a distance l. So this electron moves a distance l in time t. Its drift velocity, v = l/t. This is the same for all the electrons.


Example

A copper wire has a cross-sectional area of 1.0*10-7m2, and carries a current of 1A. If the number of free electrons is about 1029 m-3, calculate the drift velocity of the electrons.
(electron charge, e = 1.6*10-19 C)

Charge transfer rate

If a charge Q flows through R in time t, then:

2. Conduction in semiconductors

The most commonly used semiconductors are germanium and silicon. These are both tetravalent, i.e. each atom has four electrons in its outermost shell that can take part in bonding with neighbouring atoms.

In a pure semiconductor at absolute zero there are no free electrons - they are all in bonds. So a semiconductor at absolute zero is an insulator.

Note: Absolute zero (= -2730C = zero kelvins, OK) is believed to be the lowest possible temperature.

Intrinsic semiconduction

Above absolute zero some electrons are liberated by thermal vibrations, and are free to act as current carriers. Thus, the conductivity of a semiconductor increases with temperature - or equivalently, its resistivity falls.

When an electron escapes from a bond, it leaves behind a vacancy called a ‘hole’, which is a region of net positive charge. When a p.d. is applied across a semiconductor, the free electrons drift towards the positive terminal. But also a bound electron can jump into a nearby hole, and the hole is transferred to the atom just vacated. This happens repeatedly.

We can represent this as:

For convenience we can think of the holes as being real positive particles (charge equal but opposite to that of electrons) moving to the left:
Thus, in a pure semiconductor we visualise conduction being due to two types of current carriers:
  1. negative free electrons drifting towards the positive terminal
  2. positive holes drifting towards the negative terminal
The mechanism is called intrinsic semiconduction. Notice that an intrinsic semiconductor is pure and has equal numbers of electrons and holes (each liberated electron produces one hole).

Extrinsic semiconduction

Small amounts of certain elements can be added to, say, germanium without distorting its crystal structure. This is called ‘doping’ and the added atoms are called ‘impurities’.

The added atoms are either pentavalent (i.e. have 5 valence electrons, e.g. antimony, Sb) or trivalent (i.e. have 3 valence electrons, e.g. indium, In).

If antimony is added, the extra 5th electron is not needed for bonding, and so each Sb atom provides an extra free electron. In pure germanium, only about 1 atom it 1010 provides a conduction electron. Thus, doping of only 1 part in 106 produces a large increase in conduction electrons. The extra electrons now outnumber the original holes, and are referred to as the ‘majority carriers’, the holes being the ‘minority carriers’. The result is called an n-type’ semiconductor, since the majority carriers are negative electrons.

If trivalent indium is added to germanium, each atom introduces an extra hole into the lattice structure. The positive holes are the majority carriers, and the semi-conductor is called a p-type.

Semiconductor diodes and transistors are based on n-type and p-type semiconductors.


SUPERCONDUCTIVITY

At extremely low temperatures, the resistance of some materials drops to almost zero. The phenomenon is called superconductivity and such a material is called a superconductor.

In 1911 Onnes showed that a superconductor carried a current for several hours after the supply had been turned off. Thus, a superconductor carrying a current generates negligible heat loss.

A superconductor carrying a current is surrounded by a magnetic field. Such a conductor could float over a magnet for as long as the current persisted. This has suggested the possibility of superconductors as a basis for frictionless bearings. Many machines have rotating parts supported by bearings, and these are a source of wear and heat due to friction.

The temperature at which a material becomes superconducting is called its transition temperature. For some materials this temperature is not far above absolute zero (-273 0C).

Materials are now known which have transition temperatures much higher, and it is hoped that ‘room temperature’ superconductors will be found.

 

A Level Physics - Copyright © A C Haynes 1999 & 2004