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

The nucleus contains positive particles called

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 H_{2}O = a
water molecule.

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.

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.

*The size of a current equals the amount charge to pass a given point in a circuit in one second*

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 A

**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:

- A
_{1}reads 0.30 A - A
_{2}reads 0.10 A - A
_{3}reads 0.20 A

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

*The sum of currents entering a junction equals the sum of currents leaving the junction*

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?

- There are 4 amps flowing into the junction.
- There are 7 + 9 + 2 = 18 amps flowing away from the junction.
- The total current into the junction = total current out.
- Thus, A must indicate 18 – 4 = 14 amps flowing into the junction

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

Definition:

*The potential difference between two points in a circuit is one volt if one joule of electrical energy is changed to other forms of energy when one coulomb passes from one point to the other*

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:

- 5C flows through the bulb
- 2A flows for (i) 5 sec, (ii) 2 minutes

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:

- The voltmeter measures the p.d.
*across*R - The ammeter measures the current
*through*R (assuming that none passes through the voltmeter)

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:

This can be stated as:

*The current through a metallic conductor is directly proportional to the p.d. across it, provided that the temperature and other physical conditions are constant*

Thus, Ohm’s law implies that:

*The resistance of a metallic conductor is constant as long as the temperature and other physical conditions are constant*

Note - The equation R=V/I is *not*Ohm’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**

- the switch is closed
- the variable resistor is set to produce the smallest current
- the voltage V and current I are recorded
- the variable resistor is adjusted to produce a range of values of I and V

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.

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:

- the total resistance
- the value of the current I
- the values of V
_{1}and V_{2.}

**Example**

Find:

- the single resistance equivalent to the two resistors
- I
- I
_{1} - I
_{2}

b)

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

Notice that I_{1} + I_{2} = 1 + 0.5 = 1.5 A = the total
current.

**Example**

Calculate:

- the overall resistance
- the value of I
- the p.d. across the 4 ohm resistor
- the p.d. across the other resistors
- I
_{1} - I
_{2}

Thus, the circuit is equivalent to:

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

Notice that I_{1} + I_{2} = 0.41 + 0.14
= 0.55 = I.

**Example**

Find:

- the total resistance
- I
- the p.d. across the 3 ohm
- I
_{1} - I
_{2} - the p.d.s across the 2 ohm and 1 ohm resistors

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 I_{1} + I_{2} = 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.

We assume that all of the current I goes though the resistors
R_{1} and R_{2} 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 R_{1 }and R_{2}, 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.

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 m^{2}/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 ^{0}C^{-1 }or K^{-1}.

Note: Zero kelvins (0K) = -273 ^{0}C, 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**

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 __therm__al
res__istor__. As the name suggests, its resistance depends on its
temperature.

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:

**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

**Example**

A copper wire has a cross-sectional area of
1.0*10^{-7}m^{2}, and carries a
current of 1A. If the number of free electrons is about 10^{29} 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 (= -273^{0}C = 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:

- negative free electrons drifting towards the positive terminal
- positive holes drifting towards the negative terminal

**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 5^{th} electron is not
needed for bonding, and so each Sb atom provides an extra free electron. In pure
germanium, only about 1 atom it 10^{10} provides a conduction electron. Thus, doping of only 1 part in
10^{6} 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 **n**egative electrons.

If trivalent indium is added to germanium, each atom introduces
an extra hole into the lattice structure. The **p**ositive 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.

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 ^{0}C).

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