# Physics

Physics revision

- Created by: Maddie
- Created on: 11-01-12 14:28

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

curvature.

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

dioptres (D).

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

Magnification is a measure of how much an image has been enlarged by a lens. It is given by the formula:

where h

1

and h

2

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.

Refraction

Reflection

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

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

(i).

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:

where c

0

is the speed of light in a vacuum (3 x 10

8

m/s) and c

1

is the speed of light in the material.

Snell's Law

We can relate the refractive index to the angles of incidence and reflection using the following formula, known as

Snell's Law:

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

Critical Angle

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

refraction. Therefore:

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.

Digital Storage

Digital Data

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,

etc.

Pixels

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).

Bits

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

I

Hence:

(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.

Bytes

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

3

as the prefix increases. Instead, we multiply by 1024. So, 1 kilobyte =

1024 bytes, 1 megabyte = 1024

2

bytes, 1 gigabyte = 1024

3

bytes, and 1 terabyte = 1024

4

bytes.

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

8

) 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:

Mean Smoothing

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.

Edge Detection

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

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:

Sampling Rate

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

3

) 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.

Signal Frequencies

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

-1

, 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).

Multiple Frequencies

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

Frequency Spectra

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.

Fundamental Frequency

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

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

formula:

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).

Charge

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

-19

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

particle

, then the total charge Q

total

is given by:

Q

total

= n Q

particle.

Current

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])

i

1

+ i

4

= i

2

+ i

3

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.

Voltage

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

the charge.

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.

Potential Difference

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.

Series Circuits

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).

Parallel Circuits

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

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

-1

. 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

being measured.

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:

and

Ohm's Law

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

if:

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.

Internal Resistance

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

following formula:

PD across battery = EMF of battery - voltage to be accounted for

Let us replace these values with letters to give the simpler formula:

V

external

= E - V

internal

Since V = IR:

V

external

= E - IR

internal

You may also need to use the following formula to work out the

external potential difference, if you are not given it:

V

external

= IΣR

external

You should also remember the effects of using resistors in both series and parallel circuits.

Potential Dividers

Circuit symbols for a

potential divider

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.

Sensors

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

measured.

Temperature Sensor

Use of a potential divider and thermistor to measure

temperature

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

thermistor.

Properties

There are three main properties of sensing systems you need to know about:

Sensitivity

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:

V/°C

Resolution

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.

°CSensors 27

Response Time

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.

Signal Amplification

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:

Amplification

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.

Wheatstone Bridge

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

better sensor.

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

-1

.

Formulae

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.

Semiconductors

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

-1

.

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

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

calculating pressure:

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.

Tensile Strength

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.

Yield Stress

The yield stress is the level of stress at which a material will deform permanently. This is also known as yield

strength.

Strain

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

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 Graphs

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

-3

) 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:

Elastic Region

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

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

Plastic Region

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.

Fracture Point

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

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

about:

Electrical Conductivity

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.

Stiffness

The charge between the negatively charged electrons and the

positively charged ions holds the ions together, making metals

stiff.

Ductility

Since there are no permanent bonds between the ions, they can move about and slide past each other. This makes

metals ductile.

Toughness

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

called plasticity.

Elasticity

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

Brittle

The opposite of tough: a material is likely to crack or shatter upon impact or force. It will snap cleanly due to defects

and cracks.

Transformation

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

structure

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.

Polymers

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.

Types

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

Polycrystalline glass

Amorphous rubber

Properties

Transparency

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.

[1]

Elasticity

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

original shape.

Stiffness

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.

Plasticity

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.

Conductivity

Polymers consist of covalent bonds, so the electrons are not free to move according to potential difference. This

means that polymers are poor conductors.

Boiling Point

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

-1

, 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.

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