3.2.3 Waves

Progressive Waves

Oscillation - When a stretched spring is vibrated at one end, the oscillations are transferred along the spring in the form of a progressive wave.

Amplitude - the maximum displacement from equilibrium caused by a wave - from the middle of the wave to the peak or trough

Frequency - the number of waves that pass a certain point in one second. The unit is the hertz, Hz. A frequency of 1 Hz is one wave per second.

Wavelength - the distance between two identical consecutive points on a wave. It is measured in metres. We can define it as:

Wave speed - the speed of a mechanical wave depends on the properties of he medium which it travels through. It is the rate at which a wave passes a certain point in a certain time frame.

In phase - when two points (eg. like A and A' on the graph above) are oscillating in time with each other.

Out of phase/anti-phase - when two points (eg. like A and B) are oscillating at different times

Phase difference - The difference in length between 2 points on a wave - eg. there is a phase difference of π between A and A'

Path difference - The difference in length between 2 different waves.


Longitudinal and Transverse Waves

Longitudinal waves have vibrations that are parallel to the direction in which the wave is travelling (created by compressions and rarefactions). An example of longitudinal waves is sound, where air molecules vibrate backwards and forwards parallel to the wave's travel.

Transverse waves have vibrations that are perpendicular to the direction of propagation. An example of this are light waves. Electromagnetic waves are also transverse waves, however they are not vibrations of particles, but are oscillating electric and magnetic fields.

Transverse Waves

Electromagnetic Waves

A transverse wave has oscillations in a plane that is perpendicular to the wave's velocity. However, these oscillations could be in any direction in that plane. The electric field in a light wave, for example can oscillate in any direction in that plane. Such a wave is said to be unpolarised.

We are able to restrict the oscillations of the electric field in an electromagnetic wave to one direction only - we can then say that the wave is polarised.

Radio waves that are used to carry TV signals are transmitted as horizontally or vertically polarised waves. Neighbouring transmitters may emit waves of opposite polarisations. This reduces interference between the two signals.

It is possible to polarise light by passing it through a sheet of polaroid. Two sheets of polaroid that are perpendicular to each other will block off all of the light. Since reflected light is partly polarised, polaroid sunglasses can reduce the glare from reflected light.

Refraction at a Plane Surface

The change of direction when a wave moves from one medium into another is called refraction - eg. when light travels from air through a window, the medium changes from air to glass.

The refractive index of a material is the ratio of the speed of light in a vacuum to its speed in the material.

Light always travels faster in a vacuum compared to any other medium, and so the absolute refractive index is always greater than 1. As it is a ratio, there is no unit.

Materials which a high value of refractive index are said to be optically dense.

Snell's Law of refraction states:

Total internal reflection is when a ray of light, leaving an optically dense material and travelling into a less dense one, is not refracted out of the dense material but is totally reflected back inside.

Total internal reflection occurs when the light ray is moving from one medium to another in which the speed of light is greater, and the angle of incidence is greater than the critical angle.

We can calculate the critical angle by the following:

Total internal reflection occurs in optical fibres which carry TV and telephone communications and provide the skeleton for networks. Optical fibres are used to carry information in the form of digital pulses of infrared light over long distances. It is important that as little light is lost in order to transmit information in it's original form. This is achieved by using glass with an extremely small amount of impurities, as impurities can scatter the light so that it strikes the boundary at less than the critical angle and is refracted out of the fibre.

Scratches can also affect the signal, and one way of preventing this is to use cladding which provides an outer layer to protect the inner fibre. The cladding is made from a glass of lower refractive index so that total internal reflection can still occur.

Optical fibres are also used in endoscopes and medical situations.

Superposition of waves, Stationary waves

When two similar waves meet, the resultant wave depends on the amplitude and the relative phase of the two waves. When two identical waves meet, the amplitude of the resultant wave is double if they are in phase, however if they are out of phase, they will cancel out.

Waves that form a resultant wave with twice the amplitude create constructive interference, whereas waves that cancel out each other create destructive interference.

The principle of superposition says that the resultant displacement caused by two waves arriving at a point is the vector sum of the displacements cased by each wave at that instant.

When two continuous similar waves are travelling in opposite directions, they can superpose to form a stationary wave. A stationary wave is a fixed pattern of vibration. Unlike a progressive wave, no energy is transferred along the wave.

There are several points on a standing wave which do not vibrate at all called nodes. At a node, the waves travelling in opposite directions add together to give zero displacement. The distance between two nodes is always half a wavelength. In between each node is a point of minima or maxima, known as antinodes.

All the points between any two nodes vibrate in phase, whereas in a progressive wave the phase changes with position along the wave.

The simplest way that a string can vibrate is with one antinode in the middle of the string. This wave pattern is known as the fundamental mode. The frequency of this fundamental mode is given as:

The fundamental mode

Interference

Waves that always have the same phase difference have coherent sources.

For a maximum: path difference = nλ, where n is a positive integer

For a minimum: path difference = 2n+1 lλ/2 where n is a positive integer


Two Slit Interference
We aren't able to arrange two separate coherent light sources, as the light from each source is emitted in random bursts.

Instead a single laser can be shone through two slits, as a laser light is coherent and monochromatic, with only one wavelength present.

The slits have to be narrower than the wavelength of the laser light so that it is diffracted then the light from the slits is equivalent to two coherent point sources.

A pattern of light and dark fringes are formed, depending on whether constructive or destructive interference occurs.

The light bands occur whenever the path difference between light waves from the two slits is a whole number of wavelengths. The distance between two successive maxima depends on:


-The distance between the two slits. Increasing the distance makes the interference fringes closer.

-The wavelength of light. The fringes are closer together if the wavelength is shorter.

-The distance between the slits and the screen. If we observe the interference pattern at a greater distance, the fringes spread out more.

They can be linked by the formula:

where w is the distance between 2 maxima (m), lambda is wavelength (m), D is distance between the slits and the screen (m), and s is the distance between the two slits (m)

Laser light is monochromatic - all the light is emitted at a single wavelength

Laster light is coherent - the light waves emitted by the laser are all in phase, whereas light from an ordinary light bulb is emitted with random phase differences.

Laser light beams are highly directional - the beam emitted by a laser diverges very little, whereas the light from a lightbulb is emitted in all directions.


Diffraction

When waves pass through a gap, or travel past an obstacle, the waves spread out. This is known as diffraction. Diffraction is a phenomenon common to all types of waves.

The diffraction pattern from a single slit has a pattern of bright and dark fringes. There is a broad, bright central maximum with narrower, less bright secondary maxima on either side. More diffraction occurs if the wavelength increases in length and the gap size decreases.

A diffraction grating is a series of narrow parallel slits usually formed by ruling lines on glass. When light shines on the diffraction grating, a set of bright sharp lines are seen. We can derive an expression to calculate the directions in which these maxima will occur.

If a bright fringe to occurs, the follow equation is valid:

were d is the slit separation (m), theta is the angle between the two lines (degrees), n is the path difference

where n is an integer

The first maximum will occur when n =0, i.e. when there is zero path difference. This is the straight through position and is referred to as the zero order maximum.

Diffraction gratings are useful for analysing light from far away planets and stars and enables astronomers to determine elements present, temperatures and speeds of objects.

3.2.2 Materials

Bulk Properties of Solids

Density is the mass of a given volume, and can be defined as:

When opposing forces are exerted on an object, it becomes stretched or squashed. Tensile forces stretch an object, whereas compressive forces squash an object.

Hooke's Law states that the extension produced by a force in a wire or spring is directly proportional to the force applied and can be defined as:

If we are to test whether a spring obeys Hooke's Law (which it does up to a certain extent), we would set up the experiment as seen below and gradually add more weight to the end of the spring, recording the extension of the spring against the known mass. Should the spring obey Hooke's Law, this would create a straight line on the graph.

Breaking Stress - The maximum amount of stress that can be applied to a material without it breaking.

Brittleness - A brittle material cannot be permanently stretched. When a tensile force is applied the material breaks soon after the elastic limit is reached. Brittle materials may be strong in compression. Widespread use of brittle materials, such as cast iron, concrete and house bricks is only possible if the design keeps them in compression.

Ductility - A ductile material can be easily and permanently stretched. Copper is a good example, and can easily be drawn out into thin wire.

Elasticity - An elastic material returns to its original size and shape when the force which is stretching or compressing it is removed.

Elastic limit - This is the maximum stress that can be applied to a material without causing a permanent deformation.

Hooke's Law - The extension produced by a force in a wire or spring is directly proportional to the force applied. This only applies up to the limit of proportionality.

Plasticity - A plastic material is the opposite of an elastic material. A plastic material does not return to its original size and shape when the force which is stretching or compressing it is removed. There is permanent deformation.

Strength - Some materials can withstand large stresses before they fracture. These are strong or high-strength materials.

Stiffness - This is a measure of how difficult it is to change the size or shape of a material

Tensile Strain - This is the ratio of extension to original length. It is the fractional change in length. It has no unit.

Tensile Stress - This is the force per unit area of cross section when a material is stretched. It is measured in Pascals (Pa)

Yield Point - Beyond the elastic limit, a point is reached at which there is a noticeably larger permanent change in length. This results in plastic behaviour.

A - This region shows where the object obeys Hooke's Law: extension is proportional to force

B - Proportionality limit - Beyond this point, the extension is no longer proportional to force

C & D- Elastic limit - material begins to behave plastically. This is the point beyond which, when the stress is removed, the material does not return to its original form.

E - Yield point - the material shows large increase in strain for small increase in stress.

F - The material is put under so much stress that it breaks completely

When a wire is stretched by a force, provided the elastic limit is not exceeded, then the work done is stored as elastic strain energy. The area below the graph (in green) is the total work done or the elastic strain energy stored, and can be defined as:

Where E represents the elastic strain energy, F represents the force (N), and delta L is the change in length (m)

We can determine a material's property by looking at their stress/strain graphs and comparing them to other materials eg. brittle against ductile

The Young Modulus

The Young modulus (E) is a measure of stiffness of a material and is defined as:

where F is the force/weight exerted on the wire (N), L is the length of the wire (m) and A is the cross sectional area (m^2)


The Young Modulus can be used to compare the stiffness of different materials, even if the samples have different measurements, however it only applies up to the limit of proportionality.

From a stress-strain graph, we can identify Young's Modulus by:

- calculating the gradient
- multiplying the gradient by length/Area

We can create an experiment in order to determine the Young modulus of a material by setting up as the diagram above. (We can make it more accurate by using two wires instead of one, and using a spirit level in-between them to compare two lengths)

- initially load out the wire to straighten them
- use a micrometer to measure the diameter of the wire
- use a metre rule to measure the initial length of the wire
- slowly add another weight, noting down the total weight on the wire
- use the metre rule to measure out the new length
- calculate the extension
- repeat again until you obtain a varied set of data
- take care and wear eye protection, as the wire may easily snap

3.2.1 Mechanics

Scalars and Vectors

A vector quantity has magnitude and direction, whereas a scalar quantity has magnitude only.

Scalar quantities include: distance, speed, energy, power and mass
Vector quantities include: displacement, velocity, force, acceleration and momentum

When two vectors are added, we need to take account of their direction as well as their magnitude. Two vectors can be added by drawing a scale diagram by representing each vector as a line, and drawing each vector tip to tail.

The sum of a number of vectors is known as the resultant, The resultant is the single vector that has the same effect as the combination of the other vectors. It is important to take into account which direction the vector is, as the result could change completely.

If we are given two vectors and an angle between them, we are able to easily find the resultant force using trigonometry.

An object is said to be in equilibrium if it is stationary, or moving at constant velocity.

Weight is the force that acts on a mass due to the gravity. It is defined as:

where m represents mass (kg), W is weight (newtons), and g is gravity (N/kg)


An alternative method of investigating these problems is to resolve all the forces into two perpendicular directions, for example horizontal and vertical. If the object is to be in equilibrium, two conditions must be satisfied:

the sum of the horizontal components must equal zero

the sum of the vertical components must equal zero

Moments

The moment of a force about a point is equal to the magnitude of the force, F, multiplied by the perpendicular distance of the force from the pivot, s.

Where f represents force (newtons), and d is distance (m).

The principle of moment states that if an object is in equilibrium the sum of the moments about any point must be zero. Sum of clockwise moments = sum of anticlockwise moments.

Two parallel forces which act in opposite directions will tend to make an object rotate. If these forces are equal in magnitude, they are known as a couple.

The turning effect, or torque, of a couple is Fs, where F is the magnitude of one of the forces and s is the perpendicular distance between the forces.

All the mass of a body can be thought of as acting at a single point, known as the centre of mass of a body. If the resultant force on an object passes through the centre of mass it will accelerate without rotating. If the resultant force does not pass through the centre of mass the object with spin.

Motion Along a Straight Line

Displacement, s, is the distance travelled in a given direction.

Speed is the distance covered in unit time, and can be defined as:

The speed at any given instant in the journey may be above or below the average speed. The speed at a certain time is known as the instantaneous speed. If we measure the distance covered in a very small time interval, the value for speed approaches the instantaneous value.

Velocity is a vector quantity. It has a magnitude and a direction. Velocity is the speed in a given direction and is defined as:

where v represents velocity (m/s), d is displacement (m), and t is time (s)

Acceleration is the rate at which velocity changes, and can be defined as:

The gradient of a displacement - time graph is the instantaneous velocity.

A velocity time graph for a journey can be used to calculate the acceleration and the displacement. The gradient of the graph is v/t which is the instantaneous acceleration. A straight line represents constant acceleration. A line with a negative gradient represents a negative acceleration. This could be slowing down or it could mean that the object is speeding up in the opposite direction.

Where s represents displacement (m), u is initial velocity (m/s),v is final velocity (m/s), a is acceleration (m/s^2), t is time (s)

Projectile Motion

An object thrown through the air follows a parabolic path. Even though this is not a straight line, we can still use the equations of motion (SUVAT). This is because horizontal motion does not affect vertical motion.

This means that a two dimensional problem can be solved by treating two one dimensional problems - keeping the horizontal and vertical motions separate.


Newtons Laws of Motion

Newton's First Law of motion states that every object will continue to move with uniform velocity unless it is acted upon by a resultant external force.

Newton's Second Law of motion states that the rate of change of an object's linear momentum is directly proportional to the resultant external force. The change in momentum takes place in the direction of the force.

Where p stands for momentum (kg m/s), m is mass (kg), and v is velocity (m/s)

Where F is force (newtons), m is mass (kg), and a is acceleration (m/s^2)

Where F is force (newtons), m is mass (kg), v is final velocity (m/s^2), u is initial velocity (m/s^2) and t is time (seconds)


Newton's Third Law of motion states that if an object, A, exerts a force on a second object, B, then B exerts an equal and opposite force back on object A.

Work, Energy and Power

The work done is equal to the force multiplied by the distance through which the force moves, in the direction of the force.

Energy is the ability to do work.

Power is the rate at which energy is transferred. A power of 1 watt means that 1 joule of energy is transferred every second.

Power, P, is the rate at which work is done, and can be defined as:

where P is power (watts), W is work done (joules), and t is time (seconds)

For a moving machine, such as a motor or a car, it is often useful to relate the power output to the velocity at which the machine is moving.

We can then deduce that:

Where P is power (joules), F is force (newtons) and v is velocity (m/s)

Conservation of Energy

The principle of conservation of energy states that the total energy of a closed system is constant

The kinetic energy of a moving mass depends on the mass and the velocity squared. We can define kinetic energy as:

Gravitational potential energy is the energy that an object has because of its position in a gravitational field. The work done in lifting a mass, m, through height h is

W = force x distance = m x g h

This is the same as calculating the potential energy by the mass:

When a mass falls from a height, its potential energy is transferred into kinetic energy. If we can ignore energy losses due to air resistance, then the potential energy will end up as kinetic energy.

During an energy transfer, the total energy stays constant. However, the energy may not all be transferred as useful energy. For example, the engine of a car transfers chemical energy into kinetic energy, however a lot of energy is transferred to thermal energy. The percentage of input energy that is transferred into useful energy is known as it's efficiency.

We can define efficiency as:

3.1.3 Current Electricity

Charge, Current and Potential Difference

Electric current is the rate of flow of charge

In metallic conductors the charge carriers are electrons which move from the negative terminal to the positive terminal. This should not be confused with current, which moves from the positive terminal towards the negative terminal, also known as conventional current.

We can also define current as:

Where I is current (amps), Q is charge (coulombs) and t is time (seconds)

This can be arranged to:

Potential Difference is work done per unit charge
A charge gains energy when it passes through a cell. It releases the gained energy as it passes through components in a circuit, and a potential difference exists across the component.

Where V is voltage (volts), q is charge (coulombs) and W is energy (joules)

Charge faces opposition when it flows around a circuit. This is called resistance and it is measured in ohms (Ω). The potential difference needed to make a current flow in a circuit depends on the resistance in the circuit. The bigger the resistance, the more potential difference is required to make a certain current flow.

We can define this as:

Where R is resistance (ohms), V is voltage (volts), and I is current (amps)


Current Voltage Characteristics

We can measure current using an ammeter and voltage with a voltmeter in order to obtain information about the following characteristics.

The circuit symbol for an ammeter

The circuit symbol for a voltmeter

Ohmic conductors are components which follow Ohm's law, and show that voltage is directly proportional to current. These display a straight line on V-I graphs.
Examples include: all metals

The shape of a semiconductor diode depends on the direction in which the current is flowing.
When the diode is facing forward, there is a large resistance between 0 and 0.7V, however the resistance decreases quickly between 0.7 and 1V, and a large current flows, making the graph increase sharply.

On the other hand, when the diode is reversed and is facing backwards, there is high resistance, and so an extremely small amount of current flows. At the breakdown voltage, generally between 50 and 500V, a large current flows, however most diodes break down due to a high heating effect at such a high current.

Filament lamps do not follow Ohm's law. They show that current is proportional to voltage at small readings, however they start to curve as the current increases in both directions. This is due to the heating effect when a high current flows through - a high temperature increases the resistance of the filament in the lamp, and so there is a decrease in current.

Ohm's law states that the current in a conductor is directly proportional to the potential difference across it, provided that the temperature and other physical conditions remain the same.


Resistivity

Two factors which affect the resistance of a conductor are its length and it's cross sectional area
Resistance is directly proportional to length, and indirectly proportional to the cross sectional area - if you double the length of the wire, the resistance doubles. If you double the area, the resistance will half.

The resistivity of a wire is defined as

where rho is resistivity (ohm metres), R is resistance (ohms), A is the cross sectional area (metres squared), and L is the length of the wire (metres).

Metal wires and resistors have delocalised electrons that move through the metal when a potential difference is applied, causing a current to flow. The metal also has vibrating positive ions. Electrons collide with these ions, causing the ion to have resistance to the current.

As the temperature of the wire increases, the positive ions and electrons will both absorb the heat energy, causing the both to vibrate more. This means that there is a higher number of collisions between electrons and ions, and thus resistance increases.

However, in the case of a thermistor, the resistance decreases as the temperature increases.

Small increases in temperature produce large changes in resistance of the thermistor. The thermistor is made from semiconductor material and therefore has few free electrons to produce a current. As the temperature of the thermistor increases, the thermal energy is enough to release further electrons from the ions to make the material conductive, and thus resistance decreases.

Uses for a thermistor include: temperature sensors eg. fire alarms

The circuit symbol for a thermistor

If the temperature of a conductor is decreased until it reaches absolute zero (0K or -273 degrees celsius) the electrical resistance becomes non negligible. The conductor is then rendered as a superconductor - since it's resistivity has become zero, an electric current can pass through without transferring any energy to the conductor. The temperature at which the material becomes a superconductor is know as the critical temperature.

Circuits

In order for a current to exist, a potential difference must exist. Potential difference is the amount of electrical energy that must be transferred to the charge and is measured in joules per coulomb or volts.

The charge releases the gained energy as it passes through components in a circuit. All the potential energy lost by the charge is generally changed into heat.

Energy is measured in Watts

From formulas, we can deduce that:

where W is work done/energy (watts), V is voltage (volts), I is current (amps) and t is time (seconds)

Power is the rate of change of energy, and is measured in joules per second or Watts. It can be defined as:

We can then substitute V= IR into the equation above and arrive at alternative formulas.

In all circuits, electric charge is conserved - all the charge which arrives at a point must leave it. Current is the flow of charge.

At any point in a circuit where conductors join, the total current towards the point must equal the total current flowing away from the point.

This means that in the circuit above, the current in question is 2A. 

We can deduce this from 3 - 0.67 - 0.33 = 2A

In circuits, energy differences are expressed as potential differences and measured in volts. Energy is always conserved in all circuits.


The algebraic sum of potential differences around a closed circuit is always zero.

For a series circuit, this is pretty simple - add up all the values

For components in parallel, the voltage is exactly the same over each "arm" of the circuit

The total resistance in a series circuit is just the sum of resistance.

The total resistance in a parallel circuit can be calculated by the following equation:

Potential Dividers

A potential divider splits up the potential difference from a source. This can be done using two or more resistors in series. In this case, the potential difference across the output terminals varies according to the ratio of the resistances in series.

We can then deduce that:

Uses for a potential divider include: an audio volume control


We can also incorporate light and temperature sensors in order to vary the output voltage depending upon the conditions.

A light sensor (LDR) has a low resistance in bright light and a high resistance in darkness. Using a potential divider, the output voltage can be made to change with light intensity.

As the light gets dimmer the resistance of the LDR gets higher, the ratio of resistances changes, and so does the voltage ratio.

A temperature sensor (thermistor) has a low resistance when hot and a high resistance when cold. Using a potential divider can vary the output voltage with temperature changes.

As the temperature decreases, the resistance of the thermistor increases and the voltage ratios changes such that the output voltage decreases

Electromotive Force and Internal Resistance

We can define the electromotive force (EMF) as the potential difference across the source when no current flows, and is the energy per coulomb produced by the source. We can also define it by the equation:

Where V is EMF (volts), W is energy (watts), and Q is charge (coulombs)
(EMF is normally given an epsilon as a symbol rather than voltage)

The materials inside a cell offer resistance to the flow of current. This is known as the internal resistance of the cell, and is measured in ohms.

When no current flows in the circuit, then the EMF = the potential difference across the cell

When current flows in the circuit there is a potential difference across R, and a potential difference across r.

Both energy and charge are conserved in the circuit.

The current should be the same in any part of the circuit.

We can then deduce that:

Car batteries require low internal resistance in order to have as much current in order to start up the car - internal resistance limits the amount of current.

Alternating Currents

Root mean square: since the alternating current or potential difference is continually changing in value it is impossible to assign a fixed value over a number of cycles. The root mean square current produces the same heating effect in a resistor as the equivalent direct current, ie. the Irms produces the same heating effect as Id.c.

The peak value of an a.c. current or potential difference is the maximum displacement from the zero line in either direction and is labelled either I0 or V0 respectively on the graph.

In order to convert peak values to rms values, the following equation can be used:

The peak to peak value of an ac current or pd is the maximum displacement across both directions and is labelled either Ipp or Vpp respectively on a graph.

The time period of an a.c. current or pd is the time taken for one complete cycle. The unit is seconds or ms.

The frequency of an ac current or pd is the number of complete cycles per second. The unit is Hz.

Oscilloscope

An oscilloscope is used to display waveforms.

It can measure a.c. and d.c. current, small time intervals and frequencies of alternating currents and voltages.