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Resistivity and ConductivityConsider the sample of material shown below:

Its resistance R (in ) is given by the formula:

R = l/A REMEMBER!

..... where l = its length (in m), A = its cross-sectional area (in m2), and (in m) is a constant for the material known as its resistivity.

The conductivity (in -1m-1) of the material is given by the equation:

     = 1/

It therefore follows that:

     R = l/A

Log ScalesThe logarithm (or log for short) of a number is the power to which 10 must be raised to give that number. 10 is called the base of the logarithm. (Strictly speaking we should say "log to the base 10" or log10 - but the 10 can be left out.)

For example:

1000000 = 106 log 1000000 = 61000 = 103 log 1000 = 3

500 = 102.699 log 500 = 2.6991 = 100 log 1 = 00.1 = 10-1 log 0.1 = -1

etc etc

Simply use the "log" button on your calculator to find the log of a number.

{Later in the course we will use "log to the base e" (shortened to loge or ln) - where e = 2.718 - a mathematical constant (like ). You will probably find a "ln" button on your calculator as well as a "log" button.}

One important use of logarithms is to make a very large range of numbers more manageable - so that they can all be plotted on one scale. For example, resistivities range from 1.7 x 10-8 m (copper) to 1.0 x 1016 m (perspex). It would not be possible to plot both figures on a conventional scale - but on a log scale the numbers range from about -8 to 16 - which is perfectly manageable.

Potential DividersConsider the circuit below, known as a potential divider:

The current I in the circuit = V/R = 12/(300 + 150) = 0.0267 A

The p.d. across the 150 resistor = I x 150 = 0.0267 x 150 = 4 V

The p.d. across the 300 resistor = I x 300 = 0.0267 x 300 = 8 V

The ratio of the p.d.s (8:4) is the same as the ratios of the resistances (300:150). (Both are 2:1 in this example.)

i.e:

     V1:V2 = R1:R2

Also:

     V1 + V2 = V   (the total voltage)

The above 2 principles apply to any potential divider circuit.

Sometimes there may be just one conductor (wire) with the possibility of making a connection at any point along it, as shown below:

As stated previously:

     V1:V2 = R1:R2   (the resistances of the 2 sections of wire)

But: R is proportional to l.

It therefore follows that:

     V1:V2 = l1:l2

A graph of potential against distance along the wire would be as shown below:

Half-way along the wire: l1 = l2

Therefore: V1 = V2 = 6 V in the above example

i.e. The potential would = 6 V half-way along the wire.

Suppose the conductor (or wire) is non-uniform. For example, it may vary in thickness from one to the other, as in the example below:

In this case a graph of potential against distance along the conductor would be as follows:

If we consider a point half-way along the conductor, the resistance to the left of that point is greater than the resistance to the right - because the left-hand half of the conductor is thinner.

Therefore, the p.d. across the left-hand half is greater than that across the right-hand half (V1 > V2)

Hence the shape of the graph above.

DiffractionDiffraction is the bending of waves that occurs when they pass through apertures or around obstacles.

If the wavelength is considerably less than the size of the aperture, the following occurs:

If the wavelength is only slightly less than the size of the aperture, the following occurs:

The following points should be noted:

There is no change of wavelength after diffraction The smaller the aperture (relative to ), the greater the angle of spread of the waves. If is very small relative to the size of the aperture, diffraction effects are not noticeable (e.g. light waves passing through a window). It follows from the above, that for appreciable diffraction, must be similar to (and a bit less than) the size of the aperture or obstacle. (X-rays ( ~ 10-11 m) passing through the gaps between atoms in crystals (~ 2 x 10-10 m wide) is a good example of a situation where and the size of the aperture are similar. X-ray diffraction can give information about the structure of the crystal. The diffracted X-rays form a pattern on a photographic film.)

Finally, the diagram below shows waves diffracting around an obstacle:

ThermoluminescenceWhen a small sample of ancient pottery, roof-tile, brick etc is ground up and heated rapidly to about 500oC a small extra amount of light is emitted (in addition to the normal glow associated with a hot object). This phenomenon is known as thermoluminescence - and can be used to date the pottery etc. (Only electrical insulators exhibit thermoluminescence. Furthermore, if the material is reheated, the effect is no longer observed.)

Band theory is used to explain thermoluminescence. Atoms in solids have energy bands rather than energy levels (which are a feature of atoms in a gas). The close proximity of the other atoms in a solid has the effect of smearing the levels out into bands. The atoms of a gas are much further apart.

A typical energy band diagram for an insulating crystalline solid is shown below (most solids are crystalline):

The insulator is an insulator because it has very few electrons in the conduction band - i.e. very few electrons with enough energy to move freely throughout the solid. This, in turn, is because the forbidden gap for an insulator is very wide. (It is not possible for electrons to have energies in the forbidden gap, so any electron in the conduction band would have to have been promoted from the valence band. This would take a lot of energy (thermal energy) because the gap is so wide.

(In conductors the valence and conduction bands overlap - so there is no shortage of electrons that can move freely.)

The above diagram is for a perfect crystal. Real crystals have defects in the crystal structure (e.g. foreign atoms, missing atoms, etc). The effect of these imperfections is to create extra levels (called defect levels) within the forbidden gap, as shown below:

When an archaeological pot etc is buried in the ground it is constantly subject to background radiation (see below). The effect of this background radiation is (1) to create defects, and hence defect levels, and (2) to give electrons enough energy to promote them from the valence band to the defect levels. Thus, the longer a pot has been buried in the ground the more electrons it has in the defect levels. When the pot is heated this provides the necessary disturbance to make the electrons "fall" back from the defect levels to the valence band. When they do this, photons of light

are emitted whose frequency is given by the formula E = hf, where E is the energy drop. This is thermoluminescence. It follows from the above that the longer a pot has been buried the ground the more intense is the thermoluminescence. i.e. The light intensity can be used to date the pot.

All of the above, of course, assumes that background radiation has remained more or less constant since the pot was first buried.

Ionizing RadiationThere are 3 main types of ionizing radiation - , and . Their properties are summarised in the table below:

name nature charge penetration ionizing ability

alpha ()He nucleus2 protons + 2 neutrons

+ve a few cm of airblocked by paper high

beta () fast-moving electron -vea m or so of airblocked by ~ 3 mm of Al

medium

gamma ()

electromagneticradiation none

~ unlimited in airblocked by a few cm of Pb

low

Background radiation is the radiation which is present all around us - even in the absence of specific radioactive sources. Some of this background radiation comes from space - and is known as cosmic radiation. The remainder originates on Earth from radioactive isotopes in rocks, soil and the atmosphere. (Granite is a famous example of a radioactive rock.) Medical equipment and nuclear power stations also make a small contribution. Before doing any experiment involving radioactive count rates you always have to measure the background count and subtract it from all your subsequent readings.

The Photoelectric EffectWhen light of a high enough frequency hits the surface of a metal which is sufficiently reactive (e.g. Zn, Mg, K etc), electrons are emitted from the surface. This

phenomenon is called the photoelectric effect:

The photoelectric effect is explained by considering the interaction between an incident photon and an electron in the surface of the metal:

(the maximum kinetic energy that the emitted electron can have) = (the energy of the incident photon) - (the minimum energy needed to cause an electron to escape from the surface)

In symbols:

  1/2mvmax2 = hf -

..... where m = mass of electron, vmax = maximum velocity of an emitted electron, h = Planck's constant, f = frequency of the incident light and = the minimum energy needed to cause an electron to escape from the surface

is a constant for a given metal - called its work function

Re-arranging the above equation:

hf = + 1/2mvmax2      [1]

From the above equation it can be seen that, if the frequency of the light is such that hf = , then there is only just enough energy to cause an electron to be emitted (with zero k.e.)

This frequency is a constant for a given metal, and is called its threshold frequency fo. It is the minimum frequency that will cause the photoelectric effect for that metal.

fo is lower for more reactive metals, because their work functions () are lower.

It is possible to do an experiment to measure 1/2mvmax2 for the electrons:

The top plate is negative - i.e. it repels electrons. Its potential is increased until the electrons no longer have enough energy to reach it. The potential at which this occurs is called the stopping potential V. V is measured for a range of different frequencies of light (f).

     energy = charge x voltage (see SPC)

Therefore:

     energy needed to reach top plate = charge on electron x V = eV

Therefore, when the electrons are just stopped:

     1/2mvmax2 = eV

Therefore equation [1] above becomes:

     hf = + eV

or:

     V = (h/e)f - (/e)

Compare the above with the standard equation for a straight line:

y = mx + c

It follows that a graph of V against f will be a straight line and that its gradient (m) will equal h/e and the intercept (c) on the V-axis will equal -/e, as shown below: