4.4.4 stationary waves

Stationary waves are also known as standing waves. They can be created by longitudinal and transverse waves and form when two progressive waves with the same frequency (and ideally the same amplitude) travelling in opposite directions superpose. At points in antiphase the waves cancel out forming a node. At points in phase the waves cancel out forming an antinode. Where displacement is zero, amplitude and therefore intensity are also zero. The separation between adjacent nodes (or antinodes, for that matter) equates to half the wavelength of the original progressive waves. It is important to realise that there is no net transfer of energy as the two progressive waves are travelling in opposite directions so they sort of cancel each other out.

All the particles between adjacent nodes are oscillating in phase with one another. This is because, although they have different amplitude, they all reach their maximum positive displacements at the same time. On different sides of a node the particles are in antiphase; the particles to the left of a node reach their maximum positive displacement at the same time the particles on the right reach their maximum negative displacement.

This set of graphs nicely demonstrates the motion of a stationary wave:

We need to know how to demonstrate stationary waves using microwaves, stretched strings, and air columns:

Microwaves
We can form a stationary wave by reflecting microwaves off a metal sheet so that two microwaves of the same frequency are travelling in opposite directions. Using a microwave receiver we can detect the changes in intensity between nodes (low/no intensity) and antinodes (maximum intensity). The distance between the transmitter and the metal sheet must be adjusted until the receiver detects a series of notes/antinodes. As we are aware already, the distance between adjacent nodes or antinodes equates to half the wavelength of the microwaves from the transmitter.

Stretched strings
Each string has a fundamental mode of vibration. The frequency of this vibration is the fundamental frequency (f₀). This depends on factors such as the strings mass, tension, and length. When a string is stretched between two points the two points act as nodes. If the string is plucked a progressive wave travels along the string and reflects off its ends which creates two progressive waves travelling in opposite directions and these superpose and a stationary wave is formed. When plucked, the string vibrates in its fundamental mode of frequency, the wavelength of the progressive wave is double the length of the string.

The fundamental frequency (f₀) is the minimum frequency of a stationary wave for a string. However, it is possible to form other stationary waves known as harmonics at higher frequencies. For a given string at a fixed tension the speed of the progressive wave is constant. From v = fλ we can see that as frequency increases λ decreases proportionally. E.d at a frequency of 2f₀ the wavelength is half what it was at f₀. This table (from the kerboodle OCR A Physics textbook) nicely demonstrates this:

Air columns
Most woodwind instruments (sound is longitudinal) produce notes from blowing over the top of a tube creating a standing wave inside. This produces a note at a particular frequency (the length of the tube determines the wavelength of the note it produces). Sound waves reflected off a surface can produce a stationary wave. The original wave and the reflected wave travel in opposite directions and superpose. Stationary sound waves can also be made in tubes by making the air column inside the tube vibrate at frequencies related to the length of the tube. The stationary wave formed depends on whether the ends of the tube are open or closed.

Closed at one end:
In order for a stationary wave to form in a tube closed at one end there must be an antinode at the open end and a node at the closed end. The air at the closed end cannot move so it must form a node whilst at the open end the oscillations of the air are at their greatest amplitude so it must be an antinode. The fundamental mode of vibration has a node at the base and antinode at the top, the wavelength is 4 times the length of the tube.

In a tube closed at one end it is not possible to form harmonics at 2f₀, 4f₀, 6f₀, etc. This is because the open end must be an antinode. The frequencies of the harmonics in tubes closed at one end are always an odd multiple of f₀, 3f₀, 5f₀, 7f₀, etc as demonstrated in this diagram:

Open at both ends:
A tube open at both ends will have an antinode at both ends and a node in the centre (if vibrating at f₀). Harmonics at all integer multiples of the fundamental frequency are possible. This diagram nicely shows this:

4.4.3 Superposition

The principle of superposition of waves
When two waves (of the same type) meet they sort of pass through each other/overlap. This is known as superposition and a single wave is produced whose instantaneous displacement is the sum of the two former waves (the principle of superposition reads: 'when two waves meet at a point the resultant displacement at that point is equal to the sum of the displacements of the individual waves'). It is important to realise that, since displacement is a vector quantity, this resultant displacement can be bigger or smaller than the displacements of the previous waves. This effect is known as interference. If two waves are in phase the maximum positive displacements will line up causing constructive interference. This results in an increase in amplitude since intensity ∝ (amplitude)². If the two waves are in antiphase the maximum positive displacement of one will line up with the maximum negative displacement of another - this results in destructive interference. If the amplitudes of both waves are the same, the resultant will have zero amplitude:

Interference patterns can be seen when, for example, raindrops fall on a pond. As the waves ripple outward they overlap with waves caused by other drops. At this point they superpose and can interfere constructively (if in phase) or destructively (if out of phase). If is important to realise that this does not produce a stable interference pattern but rather one that changes all the time, for a stable pattern the waves must be coherent. Coherence refers to waves emitted from two sources that have a constant phase difference. For example, two filament lamps cannot produce stable interference patterns as they emit light of a range of different frequencies and changing phase difference between different waves (in other words, they do not produce coherent light).

Interference patterns contain a series of fringes known as maxima (louder/brighter etc, where constructive interference occurs) and minima (quieter/dimmer etc, where destructive interference occurs). Maxima and minima are the result of two waves that have travelled different distances from their sources. This difference is known as the path difference. If the path difference to a point is a whole number (or 0) the two waves will arrive in phase (producing constructive interference) producing a point that has maximum amplitude. If the path difference to a point is an odd number of half wavelengths (n+0.5 where n is an integer) the two waves will arrive in antiphase (producing destructive interference) producing a point that has minimum amplitude.

At the central maxima the path difference is zero so the phase difference (the difference between displacements of particles along a wave) is zero. At the first order maxima the path difference is one wavelength, so the phase difference is 360° (the peaks from the first wave line up with the peaks from the second wave so constructive interference occurs). At the first order minima the path difference is half a wavelength so the phase difference is 180° (the peaks from the first wave line up with the troughs from the second waves which results in destructive interference).

Experiments:
Okay so there are a series of experiments we need to know regarding superposition/two-source interference. We need to know techniques and procedures to investigate superposition experiments using sound, light and microwaves.

Light
One way to observe the interference patterns of light is to observe the pattern of coloured light on thin oil films on water. Basically, light reflecting off the bottom surface of the oil interferes with the light reflected off the top surface. If the thickness of the oil results in a path difference that is a non-integer half number of wavelengths of light the two sets of light wavs are out of phase and destructive interference occurs and the waves cancel out. The colours result from the different wavelengths in white light and the differences in the thickness of the oil layer. The distance the light travels through the oil before reflecting off the surface differs. Different wavelengths of light are cancelled out by different thicknesses of oil. The wavelengths are not cancelled out from the colours we observe.

Credit: Kerboodle OCR Physics A textbook

Sound

• Connect two loudspeakers to the same signal generator (this means they will emit coherent sound waves)
• the sound waves will travel out from each loudspeaker and overlap forming an interference pattern
• this interference pattern comprises a series of maxima (louder areas) and minima (quieter areas)
• the positions of maxima and minima can be detected with a microphone (or your ears, but a microphone is more accurate)

Microwaves

• Introduce a pair of slits in front of a single microwave source
• the microwaves will diffract and overlap forming an interference pattern
• the interference pattern can be detected using a microwave receiver connected to a voltmeter or an oscilloscope
• if the receiver were to me moved in an arc around the double slit the maxima and minima created as part of the interference pattern can be detected
• the positions of each maxima and minima can be marked on a piece of paper situated below the apparatus

The Young double-slit experiment

As we know from above we need two coherent waves to form an interference pattern. Young used a monochromatic light source (by using a light filter). This means only light of a specific frequency can pass through a narrow slit that follows to diffract the light. Light diffracting from this slit then arrives at a double slit in phase. It diffracts again at the double slit. Each slit acts as a source of coherent waves which spread from each slit overlapping and forming an interference pattern that can be seen on a screen as fringes (alternating bright and dark regions). This experiment demonstrates the wave nature of light - it can also determine the wavelengths of various different colours of visible light.

Okay so now for some maths. The separation between the double slits is denoted as 'a'. The interference pattern is observed on a screen at distance 'D'from the slits (D>>a). A bright fringe (maxima) is seen on the screen at position 'Y' and the next an position 'X', the distance between 'Y' and 'X' is x. The path difference S1P is one wavelength and the angles θ1 and θ2 are almost the same (they are very VERY small). We can use trig to show that:

sinθ1 ≈ sinθ2 ≈ tanθ2

where sinθ1 = wavelength/a and tanθ2 = x/D

This means that wavelength/a ≈ x/D. We can roughly express this as:

λ = ax/D

Provided D>>a.

Isaac Newton had a theory in which light was made up of tiny particles. Christiaan Huygens (a Dutch physicist) believed that light was made up of waves vibrating up and down perpendicular to the direction of the light travels. From this he formulated a way of visualising wave propagation (known as 'Huygens' Principle').  Huygens theory was the successful theory of light showing wave motion in three dimensions. He suggested that in a vacuum, or other uniform mediums, the light waves are spherical, and these wave surfaces advance or spread out as they travel at the speed of light. This theory explains why light shining through a pin hole or slit will spread out rather than going in a straight line (diffraction). Huygens theory better describes early experiments. Huygens' principle lets you predict where a given wavefront will be in the future, if you have the knowledge of where the given wavefront is in the present.

4.4.1 Wave motion

A progressive wave is an oscillation that transfers energy, but not matter, from one place to another. The particles of matter do not move in the direction of the wave. Instead they move from their equilibrium position to a new position and back. The particles exert forces on each other - a displaced particle experiences a restoring force meaning it is pulled back to its equilibrium position. 

There are two types of progressive wave, transverse waves, and longitudinal waves.

Transverse waves
In transverse waves, oscillations/vibrations are perpendicular to the direction of energy transfer. They can be in any orientation - up and down, side to side, etc - provided that they occur at right angles to the direction of energy transfer. The peak/trough is where the oscillating particles have maximum displacement from their equilibrium positions. Examples of transverse waves include water waves/electromagnetic waves/waves on a stretched string (e.g a guitar string)/S- waves (produced in earthquakes).

Longitudinal waves
In longitudinal waves, oscillations are parallel to the direction of energy transfer. When they travel through a medium they crease a series of compressions and rarefactions. Examples include sound waves and P-waves (produced in earthquakes). Since the displacement of particles occurs in the same plane as the direction of energy transfer you may be wondering how the restoring forces work. Well lets take sound for an example. Air particles are displaced and bounce off their neighbors - this provides the restoring force. As the wave moves region of higher pressure (compressions) and regions of lower pressure (rarefactions). again, the particles are still oscillating around their equilibrium positions.

Key terms
Okay so there are quite a few key terms we have to commit to memory for this topic - but we will use them loads so i'm sure you'll remember them soon enough:

• Displacement - the distance from the equilibrium position in a particular direction
• Amplitude - the maximum displacement from the equilibrium position
• Wavelength - minimum distance between two points in phase on adjacent waves
• Period (of oscillation) - the time taken for one oscillation/the time taken for a wave to move one whole wavelength past a given point
• Frequency - the number of wavelengths passing a given point per unit time
• Wave speed - the distance travelled by the wave per unit time

NOTE: wavespeed has the unit v, but if we're talking about electromagnetic waves then it has the unit c (for the speed of light, 3 x 10^8 ms^-1).

The wave equations

We can see from the definition above that the frequency of a wave and its period of oscillation are reciprocals of eachother. From this we can form an equation that relates the frequency of a wave to its period:

f = 1 / T

We also know that if a wave has a frequency of say 10Hz, then there are 10 complete oscillations each second. Say we have a wavelength of 1m, this means that the wave has travelled 10m in each second meaning its speed must be 10ms^-1. this means that for a certain frequency the wave has trvelled a distance of f x λ (frequency x wavelength) per second which is equal to the wavelength. From this information we can form another important equation...

V = f λ

Graphical representations
So, like in forces and motion, we can show the displacement of the particles of a wave against the distance along the wave on a displacement-distance graph (this can be called a wave profile). The wave profile can be used to determine the wavelength and amplitude of both longitudinal and transverse waves. The wave profile of a transverse and longitudinal wave will look the same (well, the same shape anyway (sinusoidal), not necessarily the same numbers) because it is a measure of the displacement and distance of the wave/particles NOT how the wave looks.

Phase difference describes the difference in displacements of particles along a wave (or on different waves). One complete cycle is 360° (2π radians). If particles reach their maximum positive (or negative)displacements at the same time they are in phase and their phase difference is zero. Similarly, if one particle reaches its maximum positive displacement at the same time another reaches its maximum negative displacement the particles are in antiphase and their phase difference is 180° (π radians). There is an equation that we can use to determine phase difference:

ϕ = (x/λ) x 360°

We can also use displacement-time graphs to show how the displacement of a given particle varies with time (duh). They look the same for transverse and longitudinal waves. These types of wave can be used to determine the period (and therefore frequency) of a wave.

Oscilloscope experiment

So we need to know techniques and procedures used to use an oscilloscope to determine frequency. Basically, using the set up below we can see that using a microphone produces a trace on the oscilloscope screen. Each horizontal square on the screen represents a certain time interval known as the timebase. This is set to a certain ms cm^-1 (e.g 10 ms cm^-1) - this means that each square represents a time interval of 10 mc cm^-1. The up/down squares represent the y sensitivity which is measured in V cm^-1. E.g a setting of 10 V cm^-1 means that each square will represent a pd. of 10V. Using the timebase we can do f = 1/T to determine the frequency.

Reflection, refraction, polarisation, and diffraction.

Reflection: this occurs when a wave changes direction at a boundary between two different media but remains in he original medium. The law of reflection states that whenever waves are reflected the angle of incidence is equal to the angle of reflection. When waves are reflected their frequency and wavelength do not change.

Refraction: this occurs when a wave changes direction as ti changes speed when it passes from one medium to another. There is always some refection off the surface (partial reflection).If a wave slows down as it enters the medium it will refract toward the normal, if it speeds up it will refract away from the normal. Sound waves speed up when they enter a denser medium whereas electromagnetic waves usually slow down. Since the speed of the waves changes and frequency is constant, this means that wavelength also changes as V = f λ. water waves can also be refracted - when a water wave enters a shallower bit of water is slows down and it's wavelength gets shorter.

Diffraction: this is the spreading out of a wave as it passes through a gap/travels around an obstacle. ALL waves can be diffracted and the speed, wavelength, and frequency are all constant (they do not change). The effects of diffraction are most significant when the gap the wave travels through is the same as the waves wavelength.

Polarisation: this means that the particles oscillate in one plane only. We cannot polarise longitudinal waves as their oscillations already act in one plane only (the direction of energy transfer). If a wave is plane polarised its oscillations occur in one plane only (e.g some sunglasses contain polarising filters so you can only see in one plan only). Partial polarisation can also occur (this happens when  transverse waves reflect off a surface). This means that more waves oscillate in one particular plane compared to others/another but they wave is not completely plane polarised.

Most naturally occurring electromagnetic waves are unpolarised. we can polarised them using polarising filters (each filter only allows waves with a particular orientation of oscillations through). We need to know how to observe polarising effects with microwaves and light:

• Unpolarised microwaves can be polarised by placing a metal grille in front of the transmitter (in between the transmitter and the receiver).
• If you take two pieces of polaroid filter and place them together (at right angled orientations to each other) you can nicely see the effect of polarisation. Unpolarised light travels through the first filter and becomes plane polarised. It cannot pass through the second filter as the second filter is not in the same plane as the first (it is 90° sideways). This means that the intensity of the light transmitted drops - no light is in fact transmitted through the second filter and the intensity falls to zero.

NOTE: think we need to know what wave fronts are, they are just lines joining all the points on a wave that are in phase

Intensity

This nicely leads me on to intensity. The intensity of a progressive wave is the radiant power passing through a surface per unit area. It has the units W m^-2 and is calculates with the following equation:

I = P/A

where A is the cross sectional area of the surface, P is the radiant power passing through the surface, and I is the intensity of the wave at the surface.

for a point source the radiant power will spread out uniformly in all directions (e.g over the surface of a sphere). This makes the equation I = P / (4πr²). From this we can see that intensity has an inverse square relationship with the distance from the source. 

Intensity drops as the energy becomes more spread out and the wave height (amplitude) decreases. Decreased amplitude means a reduced average speed  of the oscillating particles. For example, if you were to half th amplitude you would have particles that oscillate with half the speed which means a quarter of the kinetic energy and energy is proportional to intensity so intensity is proportional to amplitude squared...

intensity ∝ (amplitude)²

A ripple tank can be set up with a camera above it to take photos of the wavefronts. The frequency is changed and the images allow the wavelength to be measured. This allows the wave equation to be investigated and also shows that the wave speed does not depend on frequency.

4.5.3 Wave–particle duality

The wave-particle duality is a model used to describe how all matter has both wave and particle properties. De Broglie realised that all particles travel through space as waves and anything with mass that is moving has wave-like properties (these waves are known as matter waves/de Broglie waves).

Usually we would describe electrons as particles (as they have mass and charge) and as a result they can be accelerated and deflected by electric and magnetic fields. But under certain conditions we can also make electrons diffract - they spread out like waves as they pass through a tiny gap and even form diffraction patterns. If an electron gun fires electrons at a thin piece of polycrystalline graphite the electrons pass through the gaps between individual carbon atoms. The gaps are so small the electrons diffract and form a diffraction pattern. The electrons are behaving as particles when they are accelerated by the p.d. and they behave as waves when they diffract. They then behave as particles again when they hit the screen.

De Broglie realised that the wavelength of a particle was inversely proportional to its momentum. Further investigation lead to the de Broglie equation…

λ = h/p

The de Broglie equation can be applied to all particles. For example, protons and neutrons have also been shown to have wave properties and form diffraction patterns. However, as particles become larger their wave properties become harder to observe. The mass of a proton is much greater than the mass of an electron so at the same speed their momentum is significantly greater so their wavelength is much smaller and therefore harder to observe.

4.5.2 The photoelectric effect

The photoelectric effect: when Heinrich Hertz shone UV radiation onto zinc in 1887 electrons were emitted from the surface of the metal. The emitted photons are known as photoelectrons.

We need to know a simple demonstration of the photoelectric effect - this can be done with a gold-leaf electroscope. If we briefly touch the top place with the negative electrode from a high voltage power supply we will charge the electroscope. Excess electrons are deposited onto the plate and stem of the electroscope. Any charge developed on the plate and stem of the electroscope spreads to the stem and the gold leaf - since the stem and gold leaf now both have the same charge they repel each other and the leaf lifts away from the stem. If a clean piece of zinc is placed on top of a negatively charged gold-leaf electroscope and UV radiation shines onto the zinc surface the gold leaf slowly falls back toward the stem because the electroscope gradually loses its negative charge. This is because the UV (incident radiation) has caused the free electrons to be emitted from the zinc. These are photoelectrons. Three key observations resulted:

1. Photoelectrons were only emitted if the incident radiation was above a certain frequency (the threshold frequency) - no matter how intense the radiation was.
2. If the incident radiation was above the threshold frequency emission of photoelectrons was instantaneous.
3. If the incident radiation was above threshold frequency, increasing the intensity of the radiation increased the number of electrons emitted (not the kinetic energy at which they were emitted). To increase maximum kinetic energy you increase the frequency of the incident radiation.

These observations cannot be explained with the wave model of light so Einstein published the photon model in 1905. He suggested that each electron in the metals surface requires a certain amount of energy in order to escape from the metal and that each photon could transfer its exact energy to one surface electron only in a one-to-one reaction. Remember, if threshold frequency is not met (this depends on the energy of the photon, E=hf) the photoelectron will not be released regardless of the intensity (number of photos per second).

Depending on their positions within the metal electrons would require different amounts of energy to free them. Einstein defined a constant for each metal - the work function. This is the minimum energy required to free an electron from the surface of the metal.

Provided threshold frequency is met, increasing the intensity of incident radiation means more photons hit the metal surface per second so more photoelectrons are emitted per second. The rate of emission of photoelectrons is directly proportional to the incident radiation intensity.

Using the principle of conservation of energy there must be some leftover energy after the electron was freed from the metal - this is the maximum value of kinetic energy that any emitted photoelectron can have. The only way to increase maximum kinetic energy is to increase the frequency of the incident radiation - each photon has more kinetic energy so each electron has more kinetic energy after it has been freed from the metal. From this, Einstein derived his photoelectric effect equation. The energy of each photon must be conserved - it frees a single electron (in a one-to-one reaction) then any remainder is transferred into the kinetic energy of the photoelectron. According to the principle of conservation of energy he produced this equation…

hf = ϕ + KEmax

NOTE: Since all terms are energies, they should all be in joules, or all be in electronvolts. 

It is important to realise that some electrons in the surface of the metal are much closer to the positive metal ions than others. Their relative positions affect how much energy is required to free them. The work function is the minimum energy required to free an electron from the metal - most electrons need a little more energy than the work function to free them. This means that only a few of the emitted photoelectrons have the maximum kinetic energy, most travel a little slower. Also, if a photon strikes the surface of the metal at the threshold frequency for the metal then no energy will be left over from the incident photon to be transferred into kinetic energy. The equation becomes hf₀ = ϕ.

Lastly….. so we already know that the only way to increase the maximum kinetic energy of photoelectrons is to increase the frequency of the incident radiation. A graph of maximum kinetic energy against frequency of radiation on the surface gives a gradient equal to Plancks constant and a y-intercept equal to the negative work function. It is important to realise that since energy metal has a different work function the threshold frequency for each metal is different.

4.5.1 Photons

In 1900 Planck discovered that electromagnetic energy could only exist in certain values - it appeared to come in quanta (little packets). This proposed that electromagnetic radiation had a particulate nature (tiny packets of energy) rather than a continuous wave. Einstein called these ‘packets’ photons.

Nowadays we have more of an understanding that we can use different models to describe electromagnetic radiation. E.g we can use the photon model to explain how electromagnetic radiation interacts with matter and the wave model to explain it’s propagation through space.

So, now we know a photon is like a little packet of energy. We also need to know that the energy of each photon is directly proportional to its frequency. We can use the following equation to show this:

E = hf

NOTE: E (energy of the photon) is in joules, f (frequency of electromagnetic radiation) is in Hz, and h is the Planck constant.

If we were to combine E = hf with the wave equation (c= fλ) we are able to express the energy of a photon in terms of its wavelength and the speed of light through a vacuum:

E = (hc)/λ

NOTE: It is important to note that this equation has both wave elements (λ) and particulate elements (the energy, E, of a photon).

From the equation E = (hc)/λ we can see that the energy of a photon (E) is directly proportional to its wavelength (λ) so the smaller the wavelength the larger the energy of the photon.

Okay so if we think about it, one joule is pretty big at the subatomic scale of the quantum level. To combat this, we use electronvolts (eV) when measuring energies at the quantum scale. The energy of 1eV is defined as the energy transferred to or from an electron when it moves through a potential difference of 1V. But what actually is the quantity of 1eV? Well, we know the work done on an electron is VQ (W=VQ=Ve (e standing for the elementary charge)). So…

W = 1V x 1.60 × 10^-19 C = 1.60 × 10^-19 J.

This means that 1eV is equal to 1.60 × 10^-19 J

Using LEDs

We need to know a little experiment with LEDs to determine a value for the Planck constant. We can do this by considering the energies of the photons they emit. LEDs convert electrical energy into light energy by emitting visible light photons (of a specific wavelength) when the p.d. across them is above a critical value. At this p.d., work is being done (this is given by W=VQ) - this energy is about the same energy as the emitted photon. 

Connect a voltmeter across an LED. Add a safety resistor next to the LED (outside the voltmeter connections) and connect the whole system to a variable resistor/potentiometer connected to a power supply to vary the p.d out. We can us the voltmeter to measure the minimum p.d. that is required to turn of the LED. Place a black tube over the LED to help show exactly when the LED lights up. Provided we know the wavelength of the photons emitted by the LED then we can determine the Planck constant because…

At the threshold p.d. the energy transferred by an electron in the LED (work done) is approximately the energy of the single photon…..

W = VQ = Ve = E = hf = (hc)/λ…………. Ve=(hc)/λ

To obtain a more accurate result we can use a variety of LEDs that emit a known wavelength of photons then we can plot a graph of V against 1/λ. The gradient will be (hc)/e.

4.4.2 Electromagnetic waves

The electromagnetic spectrum
Electromagnetic waves are able to travel in a vacuum - they do not need a medium. It is a transverse wave that can be thought of as alternating electric and magnetic fields oscillating at right angles to eachother. the electromagnetic spectrum is a spectrtum of all the types of electromagnetic wave - they are classified according to the wavelength of each wave. We need to learn the frequencies and wavelengths of each wave class. However, we know that v = f λ and v for amm electromagnetic waves is 3x10⁸ m^s. This means that if we learn just the frequencies then we can work out the wavelengths (and vice versa), meaning we only hae to learn one (frequencies or wavelegths). That being said, here are the wavelengths (commit these to memory even if s the last thing you do - it is a must!! tbh I'm just willing myself on here as I havetn learnt them yet):

• Radiowaves; 10⁶ - 10^-1
• Microwaves: 10^-1 - 10^-3
• Infrared radiation: 10^-3 - 7^-7
• Visible light: 7^-7 - 4^-7
• Red: 7^-7
• Blue: 4^-7
• Ultraviolet: 4^-7 - 10^-8
• X-rays: 10^-8 - 10^-13
• Gamma rays: 10^-10 - 10^-16

NOTE: The  wavelength range of X-rays and gamma rays overlap so these waves are classified by their origin (X-rays are emitted by fast moving electrons and gamma rays are emitted from unstable atomic nuclei).

All electromagnetic waves can be reflected, refracted, diffracted, and polarised. I have covered polarisation of electromagnetic waves (microwaves) in post 4.4.1.


The refractive index and the refraction of light
The angle at which light is bent depends on the relative speeds of light through the two materials. Each material has a refractive index which can be calculated using the following equation:

n = c / v

The law of refraction: The product of the refractive index and sinθ (sin of angle between the normal and the incident ray) is constant. In other words:

n₁sinθ₁ = n₂sinθ₂

Investigating total internal reflection
Total internal reflection occurs at the boundary between two different media provided the angle at which the light strikes the boundary is above the critical angle (depends on the refractive index) and the light is travelling through a medium with a higher refractive index as it strikes a boundary with a lower refractive index. The relationship between C (the critical angle) and n (the refractive index) is as follows:

sin C = 1/n

You can determine the refractive index and the critical angle and also investigate total internal reflection all at the same time by measuring the critical angle of a semi-circular block. By directing a ray of light towards the center of the semi-circular block and moving it until TIR occurs is a good way to accurately measure the critical angle.

Optical fibers totally internally reflect very well and such have many uses including the fast transmission of data and keyhole surgery! A simple optic fibre has a fine glass core surrounded by a glass cladding which has a lower refractive index. This ensures that light travelling in the fine glass core reflects at the cladding boundary and stays in the core.

4.3.3 Potential dividers

Potential divider circuits can vary the p.d. across an output (e.g a lamp) when connected to a fixed point. These are useful in circumstances such as having a 10V battery but only needing 6V for the task you are about to perform.

Basically, there are two resistors in a series circuit. All you have to do is connect a circuit across one of these resistors. The p.d. into this circuit (which is Vout of the potential divider circuit) can be varied from zero to maximum depending on the resistances of R1 and R2.

This  means that the p.d. across each resistor in a potential divider depends on the resistances of the individual resistors...

V1/V2 = R1/R2

We can determine Vout by the potential divider equation:

Vout = (R2/(R1+R2)) x Vin

You can also 'load' a potential divider circuit. This refers to adding an additional resistor/component to the Vout part in parallel which overall decreases the total resistance of this part which lowers Vout. A small additional resistance in parallel significantly reduces Vout.

The potentiometer
To vary Vout we can replace one of the fixed resistors with a variable resistor. A potentiometer is a type of variable resistor that has 3 terminals and a sliding contact. If you adjust the contact the p.d. between two of the terminals will vary. Potentiometers are very compact which is useful so they can be used for portable electronic devices etc.

Temperature sensing circuits
To vary Vout we can replace one of the fixed resistors with a variable resistor.

Replacing the variable resistor with a fixed resistor allows Vout to vary depending on the temperature of the surroundings (as temperature increases resistance of the thermistor decreases as thermistors have a negative temperature coefficient).

Light sensing circuits
A similar principle to temperature sensing circuits only replace the variable resistor with an LDR. As light intensity increases, resistance falls.