The resistance of a metal increases as temperature increases.
Warming a thermistor gives electrons enough energy to escape from atoms, more free electrons available to carry charge so there is a lower resistance.
Resistance is caused by electrons colliding with atoms and loosing energy.
Potential dividers, the voltage is spilt according to how the resistance is split. The higher the resistance, the higher the voltage.
Light dependent resistors have high resistance in the dark and low resistance in the light
A transistor is used as an electronic switch.
Thermistors have hhigh resistances in the cold and low resistances in high temperatures.
Bandwidth is the range of frequencies within a signal.
Plastic stretch is when it is strech past the elastic limit. It results in permanent deformation. Atoms in the material move relative to eachother and don't return to original positions.
Elastic stretch, returns to original shape after forces are removed. Atoms move small distances relative to equilibrium positions without changing position in the material.
Ultimate tensile stress is the maximum stress the material can withstand.
For ductile materials the limit of proportionality is when the graph stops being proportional then there is the elastic limit where the material stops behaving elastically. The yield point is the stress at which a large amount of plastic deformation takes place at a constant or reducing load.
Adding or subtracting fixed values makes a brighter or darker image.
Multiplying or dividing makes it brighter or darker and increases or decreases the contrast.
Mapping a digit of colour onto another one is adding false colour.
To reduce noise and make a smoother image, replace pixels with median of neighbours.
To smooth edges, replace pixels mean of neighbours.
The Laplace rule involves finding edge, multiply by 4 and subtract the above, below and left and right values. Everything that isnt an edge goes black so only the edges are left.
Quality of a digitalised signal depends on 1) The resolution (difference between possible values) and 2) Sampling rate.
Greater number of bits, the greater the resolution.
Sampling rate must be twice maximum frequency, to record all high frequency detail.
The advantages of digital signals over analogue signals is R.I.P.E.
R is for resistance to the effects of noise. I is for information, it can be sound and images. P is for proccessed by computers. E is for easy to send, recieve and reproduce.
Structures of Solids
Metals are crystalline lattice of metal ions making the tough and ductile. Held together by a sea of delocalised electrons which makes them good conductors. The strong electrostatic forces makes metals still.
Ceramics are crystalline or polycrystalline atoms or amorphous (glass). Giant rigid structures makes them brittle. And strong ionic or covalent bonds make them stiff. Examples of ceramics are pottery, brick or glass.
Polymers are strong covalently bonded. Scrunched up or folded branches unfold by rotating around bonds (flexible). Stronger and more cross-linking bonds, the more rigid the material.
Composites combine properties of other materials i.e. reinforced concrete.
Period is the time taken for a whole vibration. Polarised waves only oscillate in one direction. Phase difference is how much one wave lags behind another.
Brittle is breaking suddenly withour deforming plastically.
Ductile can be made into wires without losing their strength.
Malleable is change shape but might lose their strength.
Hard is very resistance to cutting, indentation and abrasion.
Stiffness is high resistance to bending and stretching.
Tough is difficult to break.
Series ciruits. Current is equal everywhere. Voltage adds up. Resistance adds up.
Parrallel circuits the current splits between branches, the higher the resistance the lower the current is). Voltage is equal at all branches. One over total resistance is equal to one over first resistance add one over second resistance and so on.
Negative curvateure is when the radius of the cicrle is moving towards 0. Positive curvature is when the radius of the curve is increasing.
Light can be viewed as beams travelling between points. However, from most light sources, the light radiates
outwards as a series of wavefronts. Light from a light source is bent - wavefronts of light have a property known as
Decreasing curvatures of wavefronts
As light travels further away from its source, its curvature
decreases. Consider a sphere expanding gradually from a point,
which represents a given wavefront of light. As the sphere
expands, the curvature of its surface decreases when we look at
any part of the surface with a constant area. It should be noted at
this point that light from a source infinitely far away has 0
curvature - it is straight. This is useful, as ambient light (light from
a source that is far away) can be assumed to have a curvature of 0,
as the difference between this and its actual curvature is negligible.
The curvature of a wavefront is given as:
where v is the distance from the wavefront to the in-focus image depicted by the light. Curvature is measured in
Power of lenses
Calculating the power of a lens
The function of a lens is to increase or decrease the curvature of a
wavefront. Lenses have a 'power'. This is the curvature which the
lens adds to the wavefront. Power is measured in dioptres, and is
given by the formula:
where f equals the focal length of the lens. This is the distance
between the lens and the point where an image will be in focus, if
the wavefronts entering the other side of the lens are parallel
Overall, then, the formula relating the curvature of the wavefronts
leaving a lens to the curvature of the wavefronts entering it is:
where v is the distance between the lens and the in-focus image formed, u is the distance between the lens and the
object which the in-focus image is of, and f is the focal length of the lens. The power of the lens can be substituted in
for the reciprocal of f, as they are the same thing.
The Cartesian Convention
If we were to place a diagram of the lens on a grid, labelled with cartesian co-ordinates, we would discover that
measuring the distance of the object distance is negative, in comparison to the image distance. As a result, the value
for u must always be negative. This is known as the Cartesian convention.
This means that, if light enters the lens with a positive curvature, it will leave with a negative curvature unless the
lens is powerful enough to make the light leave with a positive curvature.
Types of Lens
Types of lens
There are two types of lens:
Converging lenses add curvature to the wavefronts, causing them
to converge more. These have a positive power, and have a curved
surface which is wider in the middle than at the rim.
Diverging lenses remove curvature from the wavefronts, causing
them to diverge more. These have a negative power, and have a
curved surface with a dip in the middle
Magnification is a measure of how much an image has been enlarged by a lens. It is given by the formula:
are the heights of the image (or object) before and after being magnified, respectively. If an image is
shrunk by a lens, the magnification is between 0 and 1.
Magnification can also be given as:
where v and u are the image and object distances. Therefore:
An easy way to remember this in the middle of a exam is the formula:
where I is image size, A is actual size of the object M is the magnification factor.
Angles of reflection and incidence
Reflection is when light 'bounces' off a material which is different
to the one in which it is travelling. You may remember from
GCSE (or equivalent) level that we can calculate the direction the
light will take if we consider a line known as the 'normal'. The
normal is perpendicular to the boundary between the two
materials, at the point at which the light is reflected. The angle
between the normal and the ray of light is known as the angle of
reflection (r). The ray of light will be reflected back at the same
angle as it arrived at the normal, on the other side of the normal.
Refraction is when light changes velocity when it travels across the boundary between two materials. This causes it
to change direction. The angle between the normal and the refracted ray of light is known as the angle of incidence
The Refractive Index
The refractive index is a measure of how much light will be refracted on the boundary between a material and a
'reference material'. This reference material is usually either air or a vacuum. It is given by the following formula:
is the speed of light in a vacuum (3 x 10
m/s) and c
is the speed of light in the material.
We can relate the refractive index to the angles of incidence and reflection using the following formula, known as
Total Internal Reflection
Normally, when light passes through a non-opaque material, it is both reflected and refracted. However, sometimes,
rays of light are totally internally reflected; in other words, they are not refracted, so no light goes outside the
material. This is useful in optic fibres, which allow a signal to be transmitted long distances at the speed of light
because the light is totally internally reflected.Refraction 6
The critical angle is the minimum angle of reflection, for a given material, at which rays of light are totally internally
reflected. At the critical angle (C), the angle of incidence must be 90°, as any smaller angle of incidence will result in
Since sin 90° = 1:
In word form, in a material with refractive index n, light will be totally internally reflected at angles greater than the
inverse sine of the reciprocal of the refractive index.
There are two different types of data: analogue and digital. Analogue data can, potentially, take on any value.
Examples include a page of handwritten text, a cassette, or a painting. Digital data can only take on a set range of
values. This enables it to be processed by a computer. Examples include all files stored on computers, CDs, DVDs,
Digital images are made up of pixels. A pixel represents the value of an individual square of the image, and it has a
value assigned to it. The total number of pixels in an image is just like the formula for the area of a rectangle:
number of pixels across multiplied by number of pixels down. When representing text, each pixel is one character
(for example, a letter, a number, a space, or a new line).
Each pixel's value is digital: it takes on a definite value. In a higher quality image, each pixel can take on a greater
variety of values. Each pixel's value is encoded as a number of bits. A bit is a datum with a value of either 0 or 1.
The more values a pixel can take on, the more bits must be used to represent its value. The number of values (N) that
a pixel represented by I bits can take on is given by the formula:
N = 2
(What on earth is this?)
A pixel may be represented by values for red, green and blue, in which case each colour channel will have to be
encoded separately. When dealing with text, the number of values is equal to the number of possible characters.
Overall, for an image:
Amount of information in an image (bits) = number of pixels x bits per pixel.
A byte is equal to 8 bits. The major difference between bytes and SI units is that when prefixes (such as kilo-, mega-,
etc.) are attached, we do not multiply by 10
as the prefix increases. Instead, we multiply by 1024. So, 1 kilobyte =
1024 bytes, 1 megabyte = 1024
bytes, 1 gigabyte = 1024
bytes, and 1 terabyte = 1024
As we have already seen, a digital image consists of pixels, with each pixel having a value which represents its
colour. For the purposes of understanding how digital images are manipulated, we are going to consider an 8-bit
grayscale image, with pixel values ranging from 0 to 255, giving us 256 (2
) levels of grey. 0 represents white, and
255 represents black. The image consists of an edge, and some random noise. There are two methods of smoothing this image (i.e.
removing noise) that you need to know about:
In order to attempt to remove noise, we can take the mean average of all the pixels surrounding each pixel (and the
pixel itself) as the value of the pixel in the smoothed image, as follows:Digital Processing 9
For this image, this gives a perfect result. In more complicated images, however, data will still be lost, although, in
general, less data will be lost by taking the median than by taking the mean.
We can detect the positioning of edges in an image using the 'Laplace rule', or 'Laplace kernel'. For each pixel in the
image, we multiply its value by 4, and then subtract the values of the pixels above and below it, and on either side of
it. If the result is negative, we treat it as 0.
Digitisation of a signal is the process by which an analogue signal is converted to a digital signal.
Digitisation & Reconstruction
Let us consider the voltage output from a microphone. The signal which enters the microphone (sound) is an
analogue signal - it can be any of a potentially infinite range of values, and may look something like this waveform
(from an artificial (MIDI) piano):
When the microphone converts this signal to an electrical signal, it samples the signal a number of times, and
transmits the level of the signal at that point. The following diagram shows sample times (vertical black lines) and
the transmitted signal (the red line):
When we wish to listen to the sound, the digital signal has to be reconstructed. The gaps between the samples are
filled in, but, as you can see, the reconstructed signal is not the same as the original sound:
The sampling rate when digitising an analogue signal is defined as the number of samples per. second, and is
measured in Hertz (Hz), as it is a frequency. You can calculate the sampling rate using the formula:
The higher the sampling rate, the closer the reconstructed signal is to the original signal, but, unfortunately, we are
limited by the bandwidth available. Theoretically, a sampling rate of twice the highest frequency of the original
signal will result in a perfect reconstructed signal. In the example given above, the sampling rate is far too low,
hence the loss of information.
Number of Levels
Another factor which may limit the quality of the reconstructed signal is the number of bits with which the signal is
encoded. For example, if we use 3 bits per. sample, we only have 8 (2
) levels, so, when sampling, we must take the
nearest value represented by one of these levels. This leads to quantization errors - when a sample does not equal the
value of the original signal at a given sample point.
The frequency of a wave describes how many waves go past a certain point in one second. Frequency is measured in
Hertz (usually abbreviated Hz), and can be calculated using the formula:
V = fλ
where V is the velocity of the wave (in ms
, f is the frequency of the wave (in Hz), and λ (the Greek letter lambda)
is the wavelength of the wave (distance from one peak / trough to the next, in m).
Let us consider the following signal (time is in ms, and the y-axis represents volts):
This signal is constructed from a number of different sine waves, with different frequencies, added together. These
sine waves are as follows:Signal Frequencies 14
Each of these sine waves has a different frequency. You can see this, as they have different distances between their
peaks and troughs. These frequencies can be plotted against the amplitude of the wave, as in the table, and chart
drawn from it, below:
Wave (y=) Period (ms) Amplitude (V) Frequency (Hz)
3sin x 6.284 3 159
sin(0.5x + 40) 12.566 1 80
2sin(3x - 60) 2.093 2 478
This chart is known as the frequency spectrum of a signal.
The fundamental freqency is the lowest frequency that makes up a signal. In the above example, the fundamental
frequency is 80 Hz. It is always the frequency farthest to the left of a frequency spectrum, ignoring noise. Other
frequencies are known as overtones, or harmonics.
Bandwidth is the frequency of a signal. Although original signals have varying frequencies, when these are
transmitted, for example, as FM radio waves, they are modulated so that they only use frequencies within a certain
range. FM radio modulates the frequency of a wave, so it needs some variation in the frequencies to allow for
transmission of multiple frequencies. Since bandwidth is a frequency, it is the number of bits per. second. The
bandwidth required to transmit a signal accurately can be calculated by using 1 as the number of bits, giving the
where B is bandwidth (in Hz), and t is the time taken to transmit 1 bit of data (in s).
The bandwidth of a signal regulates the bit rate of the signal, as, with a higher frequency, more information can be
transmitted. This give us the formula (similar to the formula for lossless digital sampling):
b = 2B
where b is the bit rate (in bits per. second), and B is the bandwidth (in Hz).
Electrons, like many other particles, have a charge. While some particles have a positive charge, electrons have a
negative charge. The charge on an electron is equal to approximately -1.6 x 10
coulombs. Coulombs (commonly
abbreviated C) are the unit of charge. One coulomb is defined as the electric charge carried by 1 ampere (amp) of
current in 1 second. It is normal to ignore the negative nature of this charge when considering electricity.
If we have n particles with the same charge Q
, then the total charge Q
is given by:
= n Q
Current is the amount of charge (on particles such as electrons) flowing through part of an electric circuit per second.
Current is measured in amperes (usually abbreviated A), where 1 ampere is 1 coulomb of charge per second. The
formula for current is:
([The triangle (Greek letter /delta/) means change in the quantity])
where I is current (in A), Q is charge (in C) and t is the time it took for
the charge to flow (in seconds).
In a series circuit, the current is the same everywhere in the circuit, as
the rate of flow of charged particles is constant throughout the circuit.
In a parallel circuit, however, the current is split between the branches
of the circuit, as the number of charged particles flowing cannot
change. This is Kirchoff's First Law, stating that:
At any point in an electrical circuit where charge density is not changing in time [ie. there is no buildup of charge, as in a capacitor], the sum
of currents flowing towards that point is equal to the sum of currents flowing away from that point.
Charge moves through a circuit, losing potential energy as it goes. This means that the charge travels as an electric
current. Voltage is defined as the difference in potential energy per. unit charge, i.e.
where V is voltage (in V), E is the difference in potential energy (in joules) and Q is charge (in coulombs).
There are two electrical properties which are both measured in volts (commonly abbreviated V), and so both are
known under the somewhat vague title of 'voltage'. Both are so called because they change the potential energy of
Electromotive Force (EMF)
Electrical power sources (such as batteries) 'push' an electric current around a circuit. To do this, they have to exert a
force on the electrons. This force is known as electromotive force, or EMF. The current travels around a circuit (from
the negative pole of the power source to the positive) because of the difference in EMF between either end of the
source. For example, the negative end of a battery may exert 9V of EMF, whereas the positive end exerts no EMF.
As a result, the current flows from negative to positive.
As charge travels around a circuit, each coulomb of charge has less potential energy, so the voltage (relative to the
power source) decreases. The difference between the voltage at two points in a circuit is known as potential
difference, and can be measured with a voltmeter.
In a series circuit, the total voltage (EMF) is divided across the components, as each component causes the voltage to
decrease, so each one has a potential difference. The sum of the potential differences across all the components is
equal to the potential difference (but batteries have their own 'internal resistances', which complicates things slightly,
as we will see).
In a parallel circuit, the potential difference across each branch of the circuit is equal to the EMF, as the same 'force'
is pushing along each path of the circuit. The number of charge carriers (current) differs, but the 'force' pushing them
(voltage) does not.
Power is a measure of how much potential energy is dissipated (i.e. converted into heat, light and other forms of
energy) by a component or circuit in one second. This is due to a drop in the potential energy, and so the voltage, of
charge. Power is measured in Watts (commonly abbreviated W), where 1 W is 1 Js
. It can be calculated by finding
the product of the current flowing through a component / circuit and the potential difference across the component /
circuit. This gives us the equation:
where P is the power dissipated (in W), E is the drop in potential energy (in Joules, J), t is the time taken (in s), I is
the current (in A) and V is either potential difference or electromotive force (in V), depending on the component
Since power is the amount of energy changing form per. second, the amount of energy being given out each second
will equal the power of the component giving out energy.
You should be able to substitute in values for I and V from other formulae (V=IR, Q=It) in order to relate power to
resistance, conductance, charge and time.
Resistance and Conductance
Conductance is a measure of how well an artefact (such as an electrical component, not a material, such as iron)
carries an electric current. Resistance is a measure of how well an artefact resists an electric current.
Resistance is measured in Ohms (usually abbreviated using the Greek letter Omega, Ω) and, in formulae, is
represented by the letter R. Conductance is measured in Siemens (usually abbreviated S) and, in formulae, is
represented by the letter G.
Resistance and conductance are each other's reciprocals, so:
Ohm's Law states that the potential difference across an artefact constructed from Ohmic conductors (i.e. conductors
that obey Ohm's Law) is equal to the product of the current running through the component and the resistance of the
component. As a formula:
V = IR
where V is potential difference (in V), I is current (in A) and R is resistance (in Ω).
In terms of Resistance
This formula can be rearranged to give a formula which can be used to calculate the resistance of an artefact:
In terms of Conductance
Since conductance is the reciprocal of resistance, we can deduce a formula for conductance (G):
The Relationship between Potential Difference and Current
From Ohm's Law, we can see that potential difference is directly proportional to current, provided resistance is
constant. This is because two variables (let us call them x and y) are considered directly proportional to one another
where k is any positive constant. Since we are assuming that resistance is constant, R can equal k, so V=RI states
that potential difference is directly proportional to current. As a result, if potential difference is plotted against
current on a graph, it will give a straight line with a positive gradient which passes through the origin. The gradient
will equal the resistance.Resistance and Conductance 22
In Series Circuits
In a series circuit (for example, a row of resistors connected to each other), the resistances of the resistors add up to
give the total resistance. Since conductance is the reciprocal of resistance, the reciprocals of the conductances add up
to give the reciprocal of the total conductance. So:
In Parallel Circuits
In a parallel circuit, the conductances of the components on each branch add up to give the total conductance.
Similar to series circuits, the reciprocals of the total resistances of each branch add up to give the reciprocal of the
total resistance of the circuit. So:
When considering circuits which are a combination of series and parallel circuits, consider each branch as a separate
component, and work out its total resistance or conductance before finishing the process as normal.
Batteries, just like other components in an electric circuit, have a resistance. This resistance is known as internal
resistance. This means that applying Ohm's law (V = IR) to circuits is more complex than simply feeding the correct
values for V, I or R into the formula.
The existence of internal resistance is indicated by measuring the potential difference across a battery. This is always
less than the EMF of the battery. This is because of the internal resistance of the battery. This idea gives us the
PD across battery = EMF of battery - voltage to be accounted for
Let us replace these values with letters to give the simpler formula:
= E - V
Since V = IR:
= E - IR
You may also need to use the following formula to work out the
external potential difference, if you are not given it:
You should also remember the effects of using resistors in both series and parallel circuits.
Circuit symbols for a
A potential divider, or potentiometer, consists of a number of resistors, and a voltmeter.
The voltage read by the voltmeter is determined by the ratio of the resistances on either
side of the point at which one end of the voltmeter is connected.
To understand how a potential divider works, let us consider resistors in series. The
resistances add up, so, in a circuit with two resistors:
If we apply Ohm's law, remembering that the current is constant throughout a series circuit:
Multiply by current (I):
So, just as the resistances in series add up to the total resistance, the potential differences add up to the total potential
difference. The ratios between the resistances are equal to the ratios between the potential differences. In other
words, we can calculate the potential difference across a resistor using the formula:Potential Dividers 25
In many cases, you will be told to assume that the internal resistance of the power source is negligible, meaning that
you can take the total potential difference as the EMF of the power source.
A potential divider may work by combining a variable resistor such as an LDR or thermistor with a constant resistor,
as in the diagram below. As the resistance of the variable resistor changes, the ratio between the resistances changes,
so the potential difference across any given resistor changes.
Alternatively, a potential divider may be made of many resistors. A 'wiper' may move across them, varying the
number of resistors on either side of the wiper as it moves.
A sensor is a device which converts a physical property into an electrical property (such as resistance). A sensing
system is a system (usually a circuit) which allows this electrical property, and so the physical property, to be
Use of a potential divider and thermistor to measure
A common example of a sensing system is a temperature sensor in
a thermostat, which uses a thermistor. In the most common type of
thermistor (an NTC), the resistance decreases as the temperature
increases. This effect is achieved by making the thermistor out of a
semiconductor. The thermistor is then used in a potential divider,
as in the diagram on the right. In this diagram, the potential
difference is divided between the resistor and the thermistor. As
the temperature rises, the resistance of the thermistor decreases, so
the potential difference across it decreases. This means that
potential difference across the resistor increases as temperature
increases. This is why the voltmeter is across the resistor, not the
There are three main properties of sensing systems you need to know about:
This is the amount of change in voltage output per. unit change in input (the physical property). For example, in the
above sensing system, if the voltage on the voltmeter increased by 10V as the temperature increased by 6.3°C:
This is the smallest change in the physical property detectable by the sensing system. Sometimes, the limiting factor
is the number of decimal places the voltmeter can display. So if, for example, the voltmeter can display the voltage
to 2 decimal places, the smallest visible change in voltage is 0.01V. We can then use the sensitivity of the sensor to
calculate the resolution.
This is the time the sensing system takes to display a change in the physical property it is measuring. It is often
difficult to measure.
Sometimes, a sensing system gives a difference in output voltage, but the sensitivity is far too low to be of any use.
There are two solutions to this problem, which can be used together:
An amplifier can be placed in the system, increasing the signal. The main problem with this is that the signal cannot
exceed the maximum voltage of the system, so values will be chopped off of the top and bottom of the signal
because it is so high.
A wheatstone bridge, using a thermistor
This solution is far better, especially when used prior to
amplification. Instead of using just one pair of resistors, a second
pair is used, and the potential difference between the two pairs
(which are connected in parallel) is measured. This means that, if,
at the sensing resistor (e.g. thermistor / LDR) the resistance is at
its maximum, a signal of 0V is produced. This means that the
extremes of the signal are not chopped off, making for a much
Resistivity and Conductivity
Resistivity and conductivity are material properties: they apply to all examples of a certain material anywhere. They
are not the same as resistance and conductance, which are properties of individual artefacts. This means that they
only apply to a given object. They describe how well a material resists or conducts an electric current.
Symbols and Units
Resistivity is usually represented by the Greek letter rho (ρ), and is measured in Ω m. Conductivity is usually
represented by the Greek letter sigma (σ), and is measured in S m
The formula relating resistivity (ρ) to resistance (R), cross-sectional area (A) and length (L) is:
Conductivity is the reciprocal of resistivity, just as conductance (G) is the reciprocal of resistance. Hence:
You should be able to rearrange these two formulae to be able to work out resistance, conductance, cross-sectional
area and length. For example, it all makes a lot more sense if we write the first formula in terms of ρ, A and L:
From this, we can see that the resistance of a lump of material is higher if it has a higher resistivity, or if it is longer.
Also, if it has a larger cross-sectional area, its resistance is smaller.
Silicon, doped with phosphorous
A semiconductor has a conductivity
between that of a conductor and an
insulator. They are less conductive than
metals, but differ from metals in that, as a
semiconductor heats up, its conductivity
rises. In metals, the opposite effect occurs.
The reason for this is that, in a
semiconductor, very few atoms are ionised,
and so very few electrons can move,
creating an electric current. However, as the
semiconductor heats up, the covalent bonds
(atoms sharing electrons, causing the
electrons to be relatively immobile) break
down, freeing the electrons. As a result, a
semiconductor's conductivity rises at an
increasing rate as temperature rises.
Examples of semiconductors include silicon
and germanium. A full list of semiconductor materials is available at Wikipedia. At room temperature, silicon has a
conductivity of about 435 μS m
Semiconductors are usually 'doped'. This means that ions are added in small quantities, giving the semiconductor a
greater or lesser number of free electrons as required. This is controlled by the charge on the ions.
Stress, Strain & the Young Modulus
Stress is a measure of how strong a material is. This is defined as how much pressure the material can stand without
undergoing some sort of physical change. Hence, the formula for calculating stress is the same as the formula for
where σ is stress (in Newtons per square metre but usually Pascals, commonly abbreviated Pa), F is force (in
Newtons, commonly abbreviated N) and A is the cross sectional area of the sample.
The tensile strength is the level of stress at which a material will fracture. Tensile strength is also known as fracture
stress. If a material fractures by 'crack propagation' (i.e., it shatters), the material is brittle.
The yield stress is the level of stress at which a material will deform permanently. This is also known as yield
Stress causes strain. Putting pressure on an object causes it to stretch. Strain is a measure of how much an object is
being stretched. The formula for strain is:
where is the original length of some bar being stretched, and l is its length after it has been stretched. Δl is the
extension of the bar, the difference between these two lengths.
Young's Modulus is a measure of the stiffness of a material. It states how much a material will stretch (i.e., how
much strain it will undergo) as a result of a given amount of stress. The formula for calculating it is:
The values for stress and strain must be taken at as low a stress level as possible, provided a difference in the length
of the sample can be measured. Strain is unitless so Young's Modulus has the same units as stress, i.e. N/m² or Pa.Stress, Strain & the Young Modulus 31
Stress–strain curve for low-carbon steel.
Stress (σ) can be graphed against strain (ε).
The toughness of a material (i.e., how much
it resists stress, in J m
) is equal to the area
under the curve, between the y-axis and the
fracture point. Graphs such as the one on the
right show how stress affects a material.
This image shows the stress-strain graph for
low-carbon steel. It has three main features:
In this region (between the origin and point
2), the ratio between stress and strain
(Young's modulus) is constant, meaning that
the material is obeying Hooke's law, which
states that a material is elastic (it will return
to its original shape) if force is directly
proportional to extension.
Hooke's law of elasticity is an approximation that states that the Force (load) is in direct proportion with the
extension of a material as long as this load does not exceed the elastic limit. Materials for which Hooke's law is a
useful approximation are known as linear-elastic or "Hookean" materials.
The relation is often denoted
The work done to stretch a wire or the Elastic Potential Energy is equal to the area of the triangle on a
Tension/Extension graph, but can also be expressed as
In this region (between points 2 and 3), the rate at which extension is increasing is going up, and the material has
passed the elastic limit. It will no longer return to its original shape. After point 1, the amount of stress decreases due
to 'necking', so the cross-sectional area is going down. The material will 'give' and extend more under less force.
At point 3, the material finally breaks/fractures and the curve ends.
Other Typical Graphs
In a brittle material, such as glass or ceramics, the stress-strain graph will have an extremely short elastic region, and
then will fracture. There is no plastic region on the stress-strain graph of a brittle material.
Metals are constructed from positive ions in a sea of
electrons. This explains many of their properties.
There are several physical properties of metals you need to know
Metals consist of positive metal ions in a 'soup' or 'sea' of free
(delocalized) electrons. This means that the electrons are free to
move through the metal, conducting an electric current.
The charge between the negatively charged electrons and the
positively charged ions holds the ions together, making metals
Since there are no permanent bonds between the ions, they can move about and slide past each other. This makes
Metals are tough for the same reason as they are ductile: the positive ions can slide past each other while still
remaining together. So, instead of breaking apart, they change shape, resulting in increased toughness. This effect is
When a metal is stretched, it can return to its original shape because the sea of electrons which bonds the ions
together can be stretched as well.Metals 33
The opposite of tough: a material is likely to crack or shatter upon impact or force. It will snap cleanly due to defects
Diffusive transformation: occur when the planes of atoms in the material move past each other due to the stresses on
the object. This transformation is permanent and cannot be recovered from due to energy being absorbed by the
Diffusionless transformation: occurs where the bonds between the atoms stretch, allowing the material to deform
elastically. An example would be rubber or a shape memory metal/alloy (often referred to as SMA) such as a
nickel-titanium alloy. In the shape memory alloy the transformation occurs via the change of phase of the internal
structure from martensitic to deformed martensitic, which allows the SMA to have a high percentage strain (up to
8% for some SMA's in comparison to approximately 0.5% for steel). If the material is then heated above a certain
temperature the deformed martensite will form austenite, which returns to twinned martensite after cooling.
A simple polymer consists of a long chain of monomers (components of molecules) joined by covalent bonds. A
polymer usually consists of many of these bonds, tangled up. This is known as a bulk polymer.
A bulk polymer may contain two types of regions. In crystalline regions, the chains run parallel to each other,
whereas in amorphous regions, they do not. Intermolecular bonds are stronger in crystalline regions. A
polycrystalline polymer consists of multiple regions, in which the chains point in a different direction in each region.Polymers 34
Polymers which are crystalline are usually opaque or translucent.
As a polymer becomes less polycrystalline, it becomes more
transparent, whilst an amorphous polymer is usually transparent.
In some polymers, such as polythene, the chains are folded up.
When they are stretched, the chains unravel, stretching without
breaking. When the stress ceases, they will return to their original
shape. If, however, the bonds between the molecules are broken,
the material reaches its elastic limit and will not return to its
Polymer chains may be linked together, causing the polymer to
become stiffer. An example is rubber, which, when heated with
sulfur, undergoes a process known as vulcanization. The chains in
the rubber become joined by sulfur atoms, making the rubber
suitable for use in car tyres. A stiffer polymer, however, will
usually be more brittle.
When a polymer is stretched, the chains become parallel, and
amorphous areas may become crystalline. This causes an apparent
change in colour, and a process known as 'necking'. This is when the chains recede out of an area of the substance,
making it thinner, with fatter areas on either side.
Polymers consist of covalent bonds, so the electrons are not free to move according to potential difference. This
means that polymers are poor conductors.
Polymers do not have boiling points. This is because, before they reach a theoretical boiling point, polymers
decompose. Polymers do not have melting points for the same reason.
Velocity, frequency and wavelength
You should remember the equation v = fλ from earlier in this course, or from GCSE. v is the velocity at which the
wave travels through the medium, in ms
, f (or nu, ν) is the frequency of the wave, in Hz (no. of wavelengths per.
second), and λ is the wavelength, in m.
This equation applies to electromagnetic waves, but you should remember that there are different wavelengths of
electromagnetic radiation, and that different colours of visible light have different wavelengths.