Module 7: The Nature of Light PDF

💡 Module 7: The Nature Of Light Electromagnetic Spectrum 📖 Investigate Maxwell’s contribution to the classical theory of electromagnetism, including: – unification of electricity and magnetism ✅ – prediction of electr...

💡 Module 7: The Nature Of Light Electromagnetic Spectrum 📖 Investigate Maxwell’s contribution to the classical theory of electromagnetism, including: – unification of electricity and magnetism ✅ – prediction of electromagnetic waves ✅ – prediction of velocity ✅ Maxwell's Theory Of Electromagnetism James Clerk Maxwell proposed his theory of electromagnetism, one of the most important breakthroughs of all time. His two major breakthroughs were the unification of electricity and magnetism (using Maxwell's Equations), and the prediction of electromagnetic waves. Maxwell's equation existed individually prior to Maxwell but, hes major contribution was to package them together into just four elegant equations. His equations include: Gauss Law for Electricity Gauss Law for Magnetism Faraday's Law Amphere's Law Using his fourth equation, Maxwell predicted the existance of electromagnetic waves - waves that propagate through space with velocity: 1 c= ϵ0 μ0 Module 7: The Nature Of Light 1  📖 Describe the production and propagation of electromagnetic waves and relate these processes qualitatively to the predictions made by Maxwell’s electromagnetic theory ✅ The Electromagnetic Nature Of Light A moving charge generates a magnetic field while a changing mangetic field generated an emf. Maxell combines these ideas to propose if a changing electric field is produced, then it will produce a changing mangetic field. The changing magnetic field would in turn produce a changing electric field and the cycle repeats produing two propagating field that oscillate at equal frequencies. Maxwell's calculations provide a value for the speed at which electromagnetic radiation propagate through a vacuum. Accepted value for speed of light is 299 792 458 m/s. c = fλ where: c = speed of light (m/s) f = frequency of wave (Hz) λ = wavelength of wave (m) 📖 Conduct investigations of historical and contemporary methods used to determine the speed of light and its current relationship to the measurement of time and distance ✅ Galileo's Experiment Module 7: The Nature Of Light 2  Galileo attempted an experiment with two observers 10km apart from one another who had laterns. Observer A would first uncoever his light and then when observer B saw A's light he would then uncoever his own. Observer A would then measure the time between uncovering his own lamp and seeing B's light. The conclusion at the time that light was basically instantaneous. Roemer's Method In 1676, Reomer conducted an experiment concerning the period of revolution if Lo - the innermost moon of Jupiter. The periods of revolution were longer when the Earth was receding fromJupiter and shorter then the Earth was approaching. He realised this was a time dely due to the distance that light had to travel. When movving away the light had to cross the Earth's diameter and hence had a longer period. By calculating the difference in time and diameter if the Earth he was able to estimate the speed as 2.3 x 10^8 m/s. This was important as it determined that light has a finite speed. Fizeau's Method Fizeau used a spinning tooth wheel and a mirror 9km away to measure the speed of light. 1. He first found the speed of rotation where light passed through one gap, travelled to the mirror and back and then passed through the next gap 2. Having found the angular velocity and the angular spearation of the gaps and the distance between the mirrors, he could then determine the time. 3. Then by using the speed formula he could determine the speed of light. He was only 5% over the currently accepted model. 📖 Conduct an investigation to examine a variety of spectra produced by discharge tubes, reflected sunlight or incandescent filaments ✅ Module 7: The Nature Of Light 3  Incandescent Filaments An incandescent light bulb produces light by heating a metal filament to a very high temperature that it produces electromagnetic radiation at a range of wavelengths. Some of the light prodduced is in the infrared part of the spectrum that is invisible to human beings but is detected as heat. Discharged Tubes An example of a discharged tube are Fluorescnet lights which contain low pressure gas through which current is passed casuing light to emit by gas in the ultra violet range. The phosphor (coating inside) is excited and emits light over the entire visible spectrum. Since fluorescne tlight emit less light in infrared range, they are more efficient at converting electrical energy into light energy. 📖 Investigate how spectroscopy can be used to provide information about: – the identification of elements ✅ The Electromagnetism Spectrum Module 7: The Nature Of Light 4  The electromagnetism spectrum is the range of frequencies of electromangetic radiation and their respective wavelengths. Changing the frequencies and wavelengths of the waves changes their properties. The shorter the wavelength the greating the penetrating power. The larger the wavelength, the lower the penetrating power. Spectroscopy Spectrascopy involves investigating the spectra produced when matter interacts with or emits electromagnetic radiation. Each element or molecule has a signiture absoption or emission spectra that helps us to uniquely identify it in spectroscopy experiments. Absorption Spectra Atomic Absorption Spectroscopy (AAS) is commonly used to identify small concentration of metal or ions in samples. Module 7: The Nature Of Light 5  Using this method both the elements identity and its concentration can be accurately determined. The basic principles of AAS are to choose a beam source that emits electromagnetic radiation at a wavelength that is absorbed by the element of interest. The sample interacts with the incident radiation causing electron to absorb energy and promote to a higher level. The transmitted radiation is then detected to determine the absoption spectra. The different wavelength can be separated using a prism. The black lines indicate the wavelengths of light that were absorbed by the sample. By looking at the absoprtion spectra for all posible combination of elements, we can identify the elements present in the sample. To determine concentration, absorption spectra is compared against the calibration curve. This calibration curve is made by measuring the absorption at certain wavelengths for samples where the elements concentration is known. Emission Spectra Module 7: The Nature Of Light 6  Emission spectroscopy is easier to perform but absorption spectroscopy gave more accurate results. In emission spectroscopy, the sample is usually vaporised and contained within a discharged tube. When elements are heated to high temperatures, they produce light. Atoms within the material absorb energy and become excited making them unstable and eventually they return to ground state causing the energy they absorb to be released. The colour of light released depend on amount of energy it has. Since atoms usually have different excited states, they produce many different colours. The combination of these colour is distinct to each element. A large voltage excites the atoms and the resulting electromagnetic radiation emitted from the electrons relaxation is the measured. A similar prism or diffraction grating can be used to separate the emitted wavelenghts of light. Since the interaction for absoroption and emission spectroscopy are the same, their spectra are complements of each other. Module 7: The Nature Of Light 7  📖 Investigate how the spectra of stars can provide information on: – surface temperature ✅ – rotational and translational velocity ✅ – density ✅ – chemical composition ✅ Surface Temperature The light seen from stars is mostly emitted from its photosphere (surface layer). All objects emit blackbody radiation due to thermal energy. The wavelength or frequency depends almost on internal energy and not the characteristics of the material. The peak wavelength of this radiation emitted is inversely proportional to the blackbody's temperature (Wien's Law). This relationship is called Wien's displacement law: b λmax = T where: λmax = peak wavelength (m) b = 2.8977729 x 10−3 m K ← Wien's Displacement Constant Module 7: The Nature Of Light 8  T = temperature (K) Therefore by simply measuring the peak wavelength in a star's spectrum, we can calculate its surface temperature. Translational Velocity In 1920s, Edwin Hubble discovered that all stars are moving away from us by measuring the Doppler shift of steller spectra. When wavelengths are slightly longer they have been red shifted. When wavelengths are slighty shorter they have been blue shifted. For emission and absoprtion spectrum, when a star was moving away from us, the observed spectra lines were moved to a longer wavelength (red shifted) and when a star moved towards use the spectra lines moved to a smaller wavelength (blue shifted). The extent of these shifts reveal how fast the star is moving Rotational Velocity The rotational velocity can also be measured through the Doppler effect. When a star rotates, one side is moving towards us while the other is moving away. Hence due to rotation, light emitted from side moving towards us will be blue shifted and light emitted from side moving away from us will be red shifted. This degree of broadening reveals the rate of rotation. Module 7: The Nature Of Light 9  The faster the star spins the wider the bands Density To determine the density of a star we can utilise the formula density = mass / volume. The volume of the star is calculated using its radius found with the help of the Stefan Boltzmann Law. Then the mass is estimated using brightness and tempeture of stars. Hence, density can then be calculated. Chemical Composition Stars emit light across a range of wavelengths which then can be passed through a cloud of gas causing certain wavelengths to be absorbed leaving dark bands in the spectrum corresponding to the absorbed wavelengths. The absorbed light temporary puts the atoms in the gas in an excited state so when they return to ground state, light is released and form an emission spectrum Module 7: The Nature Of Light 10  Through the analysis of the band the chemical composition can be determined since the lines emitted or absorbed are characteristic of each element. Light: Wave Model 📖 conduct investigations to analyse qualitatively the diffraction of light ✅ Huygen's Principle Huygen's principle states that each point on a wavefront can be considered a source of a secondary smaller waves (wavelets). Each point on the initial wavefront can be treated as if it is a point source producing circular waves. After one period, the circular waves will have advanced a distance equal to one wavelength. When the amplitudes of each Module 7: The Nature Of Light 11  individual circular wave is summed up, the result is another plane wave shown by the new wavefront. This process is repeated at new wavefront causing the wave to propagate. Diffraction Diffraction occurs when a straight plane wave passing through a narrow opening bends. If the wavelength is smaller than size of gap, degree of diffraction is less. If wavelength is equal to or larger than size of gap then degree of diffraction is larger. Diffraction Grating As light passes through gap, some wavelets will diffract at the edge of the gap and some will pass through the centre. Resulting in emerging light waves to interact, casuing interference. Constructive and destructive interferences will occur at different places. When these lights waves are to shone on to a screen, areas of contructive interference apear as bright bands and areas of destructive interference apear as dark bands. This pattern of dark and light bands is called diffraction pattern. Module 7: The Nature Of Light 12  Extent of diffraction is proportional to the ratio wavelength / width. This ratio describes spacing of diffraction pattern. Diffraction pattern is generated by passing light through diffraction grating (piece of metal containing lots of very closely spaced parallel gaps). 📖 conduct investigations to analyse quantitatively the interference of light using double slit apparatus and diffraction gratings 𝑑sin𝜃 = 𝑚𝜆 ✅ Young's Double-Slit Experiment In 1803, Young performed an experiment in which he shone monochromatic light on a screen with 2 tiny slits. On far side of the screen he placed another screen on which he observed the pattern produced by light passing through slits. He observed a series of bright and dark bands. He explained this pattern by treating light as a wave. The monochromatic light was like a plane wave that passed through the narrows slit casuing them to diffract into coherent circular waves which would then interact casuing interference. Interference results in lines of constructive (antinodal) and destructive (nodal) interference that matched the diffraction pattern. Module 7: The Nature Of Light 13  Double-Slit Experiment Analysis Let us understand how the waves produced by the two slits interact with each other when they hit the screen. At a particular point, P, on the screen the wave from slit 1 (S1) will have travelled a different distance compared to the wave from slit 2 (S2). The difference in this path is termed path difference (pd). pd = ∣S1 P − S2 P ∣ At point M, at centre of screen equidistance from each split, each wave will have travelled same distance (pd = 0). The light waves arrive in phase and constructively interfere to produce an antinode where a fringe of bright light is seen (central maximum). Constructive interference occurs when pd = 0 or pd = λ. λ Module 7: The Nature Of Light 14  λ Suppose point N where pd = 2 , waves are completely out of phase and will canel each other out to produce a nodal point, destructive interference occurs and no light is seen. Constructive interference of coherent waves occurs when pd = mλ, where m = 1, 2 3,... Destructive interference of coherent waves occurs when pd = (m − 12 )λ , where m = 1, 2, 3,... Calculating Fringe Separation It is possible to calculate the location of bright fringes using geometry. Considering 2 slits, S1 and S2, seperated by a distance, d. Imagine a perpendicular line from mid point between the slits that reaches to the screen. Another point P is substended from is line at an angle θ. It is assumed that rays will travel parallel to each other since distance to screen is much greater than the slit seperation. Module 7: The Nature Of Light 15  Hence pd = sin θ. Since constructive interference occurs when ever pd = mλ. Then: d sin θ = mλ where: d = distance between slits (m) θ = angle from perpendicular bisector between slits (°) m = any whole number λ = wavelength (m) This is the path difference formula used for constructive interference. To calculate the path difference used for destructive interference the use: 1 d sin θ = (m − )λ 2 d = distance between slits (m) θ = angle from perpendicular bisector between slits (°) m = any whole number λ = wavelength (m) 📖 conduct investigations quantitatively using the relationship of Malus’ Law 𝐼=𝐼max𝑐𝑜𝑠2𝜃 for plane polarisation of light, to evaluate the significance of polarisation in developing a model for light ✅ Polarisation Polarisation occurs when a transverse wave is allowed to vibrate in only one direction. Module 7: The Nature Of Light 16  The above wave is vertically polarised - wave oscillation occur in vertical direction only. Since the vertical wave is passes through a polarising filter in the vertical plane, it is unaffected. The above wave is horizontally polarised - wave oscillation occur in horizontal direction only. Since the horizontal wave passes through a polarising filter in the vertical plane, it is blocked. The above wave is polarised at 45 degree to the horizontal and vertical place producing a vertical polarised plane with a smaller amplitude. An ideal polariser only allows one polarisation of light to pass and absorbs any other polarisation of light. Module 7: The Nature Of Light 17  Malus's Law The intensity of light, depending on the orientation of filter and plane of polarisation, is reduced as the light passes through a polarising filter. This relationship is termed Malus' Law: I = Imax cos2 θ where: I = Intensity of light through filter (cd) Imax = Intensity of light entering the filter (cd) θ = angle between direction of polarisation of the light entering the filter (θ1 ) and the axis of polarisation of the filter (θ2 ). Module 7: The Nature Of Light 18  When the axis of polarisation of light is the same as that of the filter, I = Imax , intensity is not reduced and is maximum. Conversely, if axis of polarisation of light is perpendicular to that of the filter, I = 0, intensity is zero. Light: Quantum Model 📖 analyse the experimental evidence gathered about black body radiation, including Wien’s Law related to Planck's contribution to a changed model of light – 𝜆max=bT ✅ Black Body Radiation All objects with a internal temperature of above 0 K emit electromagnetic radiation. This is because, if the internal temperature is above 0 K, the atoms will have some energy casuing them to vibrate back and forth. Since the charges in the atom are also accelerating, it produces electromagnetic radiation. A blackbody is a hypothetical object that absorbs all incident ER (does not reflect radiation) no matter wavelength and frequency and radiates energy as electromagnetic waves called blackbody radiation. All forms of ER are essentially same, differing only in their frequency and wavelength. ER is emitted by all obects with a temperature above absolute zero (0 K). Module 7: The Nature Of Light 19  The wavelength and frequency of emitted radiation depends entirely on object's internal energy. The higher the temperature, the higher the frequency and shorter the wavelength of emitted radiation Black Body Spectrum A black body spectrum is a contnuous spectrum of the radiation emitted by a black body. Classical wave theory predicted that as the wavelength of radiation emitted becomes shorter, the intensity would increases. Meaning that as the energy decreased in wavelength, the intensity would approach infinity. This would violate the principle of conservation of energy and was called the ultraviolet catastrophe. Module 7: The Nature Of Light 20  However, experimental results showed that the black body curve had a definite peak and as the temperature of a black body increases, this peak shifted towards a shorter wavelength. Max Planck proposed that energy radiated from a blackbody was emitted in discrete packets of energy called quanta to explain and solve the ultraviolet catastrophe. Each packet represented a particular frequency or wavelength of radiation, and size of packets was dependent on the internal temperature of blackbodies. Wien's Law Wien discovered that the peak wavelength at which the object will emit the maximum intensity of radiation is dependant on surface temperature. Wien's Module 7: The Nature Of Light 21  displacement law is used to calculate peak wavelength of an object at a particular surface temperature. b λmax = T where: λmax = peak wavelength of emitted radiation (m) b = 2.898 x 10^-3 m K T = surface temperature The product of surface temperature and wavelength at which peak intensity of the emitted radiation occurs is a constant equal to 2.898 x 10^-3 m K. Planck's Equation German physicist Max Planck discovered that certain features of the EM spectrum could not be explained using a wave model for light and thus proposed that light was emitted as discrete packets of energy called 'quanta'. E = hf where: E = energy of quantum of light (J) h = frequency of ER (Hz) Module 7: The Nature Of Light 22  f = planck's constant (6.626 x 10^-34 J s) This formula can be combined with the wave equation to produce: hc E= λ where: E = energy of quantum of light (J) h = frequency of ER (Hz) c = speed of light (3 x 10^-8 m/s) λ = wavelength (m) This theory was widely disregarded as the wave model of light has become the accepted and correct explaination for light. The Electron Volt An electron volt (eV) is a unit for the amount of energy an electron gains when it moves through a potential difference of 1V. It is used as a replacement for Joules since the amount of energy studying light are usually very small. 1 eV = qe × 1 V = 1.602 × 10−19 J 📖 investigate the evidence from photoelectric effect investigations that demonstrated inconsistency with the wave model for light ✅ Observing Photoelectric Effect Photoelectric effect occurs when a type of ER is incident on a metal plate. The metal metal plate becomes positively charged due to electrons being ejected from the surface of the metal. These electron are called photoelectrons because they were released due to light. Module 7: The Nature Of Light 23  This setup consist of a metal surface (cathode), lit from an external source casuing photoelectrons to be emitted and detected at the anode. This flow of electrons is termed photocurrent and is registered by a sensitive ammeter. A voltage supply is used to make cathode negatively charged and anode positevly charged so when photoelectrons are emitted, they are repelled by the cathode and attracted to the anode. Thus maximum current is measured. Alternatively, voltage may be adjusted to make cathode positively charged and anode negatively charged so when photoelectrons are emitted, they are attracted by the cathode and repelled by the anode. Thus slowing them down. German physicist Phillip Lenard made a number of suprising discoveries about the photoelectric effect. Lenard used a filter to vary frequency of light and observe for particular cathode, there is a certain frequency of light below which no photoelectrons are observed (threshold frequency f0). For f > f0, photoelectrons will be registered. For f < f0, no photoelectrons will be detected. He discovered that for light that as frequency above threshold frequency, the rate at which photoelectrons are emitted is proportionate to intensity of light. Module 7: The Nature Of Light 24  The graph shows: light intensity increases with photocurrent When applied voltage is positive, photoelectrons attract to anode and current reaches a maximum value and remain even if voltage is increased. when applied voltage is negative, photoelectrons repel the anode and current is reduced since fewer photoelectrons have energy to oversome opposing electric potential. The voltage, V0, for which no photoelectrons reach anode is called stopping voltage and for particular frequency of light on a particular metal, V0 is constant. When light sources have same intensity but different frequencies, they produce same max current. However, higher frequency light has higher stopping voltage. As long as incident light has frequency above threshold frequency of cathode, photoelectrons emit without time dely regardless of light intensity. When lit with light above threshold frequency, some photoelectrons are emitted from first layer of atoms at metal surface and have max kinetic energy. Others, come from deeper and lose kinetic energy due to collisions. Explaining Photoelectric Effect Module 7: The Nature Of Light 25  The photoelectric effect could not be explained using wave model of light. According to wave model, frequency of light is irrelevant to ejection of photoelectrons. Since wave is a form of continuous energy trasfer, it would be expected that energy from wave would build up in metal over time, meaning even low-frequency light should transfer enough energy to emit photoelectrons if left on metal for long enough. Similarly, wave model predicts that there should be time delt between light striking metal and emittion of photoelectrons. 📖 analyse the photoelectric effect 𝐾max = ℎ𝑓−𝜙 as it occurs in metallic elements by applying the law of conservation of energy and the photon model of light ✅ Einstein & The Photoelectric Effect Albert Einstein drew on Planck's work by assuming light exist as photon each with energy making properties of photoelectric effect easy to explain. Einstein observed that for a particular metal, amound of energy required to eject a photoelectron is constant value (work function) depending on strength of bonds within metal. Shining light on metal surface is equivalent to bombarding it with protons that when struck transfer all energy to an electron. Each metal has a threshold frequency - the frequency at which photons have energy equal to the work function of the metal. ϕ = hf0 where: ϕ = work function (J or eV) h = planck's constant (6.63 x 10^-34 J s or 4.14 x 10^-15 eV s) f0 = threshold frequency (Hz) The behaviour of the electrons as photons collide depend on whether or not the photon contains enough energy to overcome the work function. If energy of proton is less than work function, then photoelectrons will not release as electrons dont have sufficient energy to break bonds. Module 7: The Nature Of Light 26  If energy of proton is greater than work function, then photoelectrons are released. The remainder of energy in excess is transformed into kinetic energy of the photoelectrons. The greater the frequency of light, the greater the kinetic energy of photoelectrons. Kmax = hf − ϕ where: Kmax = max KE of photoelectrons (J or eV) h = planck's constant (6.63 x 10^-34 J s or 4.14 x 10^-15 eV s) f = frequency of incident photon (Hz) ϕ = work function (J or eV) If energy of protons is equal to work function then electrons are not bonded to metal but have no kinetic energy. Graphing Einstein's equation results in a linear graph that shows the work function and threshold frequency of different metals. Light & Special Relativity Module 7: The Nature Of Light 27  📖 analyse and evaluate the evidence confirming or denying Einstein’s two postulates: – the speed of light in a vacuum is an absolute constant ✅ – all inertial frames of reference are equivalent ✅ Frames Of Reference A frame of reference describes where an observation is being made from. This may be stationary or in motion. A inertial frame of reference is a non- accelerating frame of reference while a non interial frame of reference is accelerating. Michelson - Morley Experiment The aether was though as a medium, a space filling substance, for the propagation of electromagnetic waves. The Michelson - Morley experiment was set out to prove the existence of the aether but instead disproved it and led Einstein to think of his famous thought experiments. Einstein's First Postulates Einstein's first postulates (Principle of Relativity) states: "All inertial frames of reference are equivalent and all laws of physics are the same in every inertial frame of reference" The first postulate is an extension of Newton's to include Maxwell's laws of electromagnetism. This postulate imples there is no absolute frame of reference. The motion of any frame is defined relative to other frames. Hence, there is no experiment that can be performed to distinguish between inertial frames of reference. Einstein's Second Postulates Einstein's second postulate states: "The speed of light, c, has the same value in any interial frame of reference." This postulate imples that the speed of light will remain constant for all observers regardless of their motion or the motion of their source. Consider, person A travelling away from person B at a speed v. If B shines a lazer towards A, then Newtonian physics implies that the speed of lazer should be c - v in A's frame of Module 7: The Nature Of Light 28  reference. However, according to this postulate the speed of the lazer in A's and B's frame of reference is always c. Loss Of Simultaneity The loss of simultaneity is a consequence of one of Einstein's postulates. Consider person A on a train moving at consant velocity v relative to person B on the ground. In A's frame, a light source is placed right in the centre. It is expected that from A's frame of reference, when switched on the light reaches the front and back wall at the same time. However, because of Einstein's second postulate A and B will observe the light travelling at the same speed in both directions. B will measure the light reaching back wall first. This is because the back wall moves towards the light while the front wall moves away from the light, so the light takes longer to catch up to it. Module 7: The Nature Of Light 29  Therefore, loss of simultaneity occurs when two events which are simultaneous in one inertial frame are not simultaneous in another interial frame. Einstein proposed that this occurs because of the amount of time that has elapsed in one frame of reference is not the same the time elasped in another frame of reference. Spacetime In the last example it was shown that time, which has one dimension, depend ont he frame of reference in which it is measured. Since a frame of reference is just a way of defining three dimensional space, time and space are interelated in a four dimensional relationship called spacetime. Spacetime coordinates describe events with spatial reference at a specific time. 📖 investigate the evidence, from Einstein’s thought experiments and subsequent experimental validation, for time dilation 𝑡=𝑡0√(1− 𝑣2𝑐2) and length contraction 𝑙=𝑙0√(1−𝑣2𝑐2), and analyse quantitatively situations in which these are observed, for example: – observations of cosmic-origin muons at the Earth’s surface ✅ – atomic clocks (Hafele–Keating experiment) ✅ ✅ – evidence from particle accelerators – evidence from cosmological studies ✅ The Light Clock Consider person A in spaceship travelling close to speeds of light , v, away from person B. A and B have light clocks that work by bouncing a light pulse between two mirrors located on the floor and on the ceiling. When a light pulse oscillates from one mirror to other and back, a period of time of one unit passes. Let the distance between the mirros be d. As A speeds along, the light will trace a zig zag path. Module 7: The Nature Of Light 30  A's Frame Of Reference A measures a unit of time of ta and sees the light pulse travel at speed of light, c, along a distance of 2d. Hence: 2d ta = c B's Frame Of Reference B measures a unit of time tb and sees thelight pulse travel also at speed of light, c, but along a distance of 2dc. Hence: 2dc tb = c It is clear that dc > d since hypotenuse in right angles triangles is the largest side. Hence tb > ta. This shows that the time B measures is greater than the time that A measures for the same event. Galileo stated that all inertial frames of reference are equivalent. Hence, it follows that according to A's frame of reference that A is stationary and B is receeding from them at speed, v. Meaning if A observed B's clock, it would see that time has slowed down for B aswell. This occurs because the whole point of relativity is that you can measure quantities relative to some particular frame of reference, not in any absolute sense. There is no absolute clock ticking away at the absolute 'right' time but all you can be sure of is that time in your own interial frame of reference is ticking away at a rate of one second per second. Time Dilation Einstein's time dilation equation is: t Module 7: The Nature Of Light 31  t0 t= v2 1− c2 where: t = time observed in stationary frame (s) t0 = time observed in moving frame - proper time (s) v = speed of moving frame of reference (m/s) c = speed of light (3.00 × 108 ms−1 ) Lorentz Factor The lorentz factor is a quantity expressing how much the measurements of time, length and other physical properties change for an object while that object is moving. This factor appears in many equation in special relatvity 1 γ= v2 1− c2 where: γ = lorentz factor v = speed of moving frame of reference (m/s) c = speed of light (3.00 × 108 ms−1 ) Hafele-Keating Experiment Joseph C. Hafele and Robert E. Keating performed an experiment in 1971 to test Einstein's theory of special relativity and time dilation. They used four caesium-beam atomic clocks, placing two in aeroplanes that flew around the world comparing them with the other 2 that remained in the US. The aeroplanes first flew eastwards (same direction as Earth's rotation), and then westwards (opposite direction to Earth's rotation). The theory suggested that a greater time dilation would occur when aeroplanes flew westwards as there is a greater relative motion between Earth and aeroplane. Hafele and Keating's results matched the prediction and comformed time dilation. The Twin Paradox Module 7: The Nature Of Light 32  The twin paradox describes a situation where twin A heads off on a long space journey to a star at close to the speed of light and comes back while twin B stays on Earth. Two contradictions emerge. Twin B percieves twin A travelling away at close to the speed of light and measures time for twin A running slowly, hence when twin A returns twin B will be older. Twin A percieves twin B receding at close to the speed of light and measures time for twin B running slowly, hence when twin A return twin B will be younger. B is always in a interial frame of reference (not accelerating) while A is sometimes in a non-inertial frame of reference like when it accelerated from Earth, decelerates then accelerates back towards Earth. As B watches from an interial frame she views A in the non inertial frame ageing slowly while A measures B's time passing quickly. Resulting in A viewing B ageing more rapidly when it accelerates and more slowly at constant velocity. B sees A age slower and slower during acceleration, then ageing constantly but slowly at constant velocity. A nevers ages rapidly. Therefore, when twin A returns twin B is now older than her. Cosmic-Origin Muons At Earth's Surface Muons are unstable subatomic particles created in the upper atmosphere (15 km) when high energy cosmic rays interact with nuclei of oxygen atoms. Free muons are accelerated to almost the speed of light and have a mean lifetime of about 2.2 microseconds Applying newtonian physics, it would be expected that no muons are detected on the surface of the Earth. However, these muons are actually detected at the surface because of time dilation that occurs due to relativistic speeds. As muons travel really fast, an observer on Earth will measure the lifetime of a muon as far greater than their mean lifetime. Using time dilation equation, an observer on Earth would see the muon's time run much slower meaning that muons will live long enough to reach Earth's surface. Length Contraction Since the speed of light remains constant, time and length are relative to the frame of reference and the direction of the motion. Distances in the Module 7: The Nature Of Light 33  perpendicular direction to the velocity are unaffected since relative motion affects length only in direction of travel. Using the train moving at relativistic speeds, as it approaches you the light reflecting off each end of the train would reach you at different times as the speed of light is finite and constant. The light reflecting from the front of the train will reach you before the light reflecting fromt he back casuing an optical aberration. As train approaches you it appears elongated and squished when it moves away. v2 l = l0 1− 2 c where: l = length observed in movign frame (m) l0 = length observed in stationary frame - proper length (m) v = speed in moving frame of reference (m/s) c = speed of light (3.00 × 108 ms−1 ) Proper Time & Proper Length t0 and l0 are referred to as proper time and proper length and are quantities that are measured by observers who are in the same frame reference as the event or object being measured. Proper Time: Proper time is the time between two events that occur at the same point in space. Proper Length: The proper length is the distance between two points whose position are measured by an observer at rest with respect to the two points. Module 7: The Nature Of Light 34  📖 describe the consequences and applications of relativistic momentum with reference to: – 𝑝𝑣=𝑚0𝑣√(1−𝑣2𝑐2) ✅ – the limitation on the maximum velocity of a particle imposed by special relativity ✅ Approaching The Speed Of Light At low speeds, γ is so close to 1 that the effects of special relativity can be ignored, but as v approaches speed of light, c, γ rapidly increases and approaches a value of infinity. According to Einstein's equation at speed of light, the length shrinks to 0 and time appears to stop altogether. Relativistic Momentum Like length and time, momentum is a relativistic quantity that needs to be modified when objects move with relativistic speeds. m0 v pv = γm0 v = v2 1− c2 where: pv = relativistic momentum in moving fram measure by oberver (kg m/s) γ = lorentz factor m0 = stationary mass (kg) v = speed of moving frame of reference (m/s) c = speed of light (3.0 x 108 m/s) Module 7: The Nature Of Light 35  The momentum increases very rapidly as velocity approaches the speed of light. The graphs shows the relativistic momentum increases at a rate far greater than for classical momentum. This can be intrepreted by thinking of the mass as quanitity that also increases at velocity increases close to the speed of light. m0 mv = γm0 = v2 1− c2 where: mv = relativistic mass (kg) γ = lorentz factor m0 = stationary mass (kg) v = speed of moving frame of reference (m/s) c = speed of light (3.0 x 108 m/s) As the lorentz factor increases with the increase in the velocity than the relativstic mass also increases. Now consider a rocket ship travelling at 0.99c. Why can this rocket ship simply turn in thrusters and accelerate up to c? This is because at its speed approaches c, there is an increase in relativistic mass making it harder for force of engines to cause change in velocity. Infact, As v approaches c, the relativistic mass approaches infinity. Module 7: The Nature Of Light 36  📖 Use Einstein’s mass–energy equivalence relationship 𝐸=𝑚𝑐2 to calculate the energy released by processes in which mass is converted to energy, for example: – production of energy by the sun ✅ – particle–antiparticle interactions, eg positron–electron annihilation ✅ – combustion of conventional fuel ✅ Mass Energy Equation As momentum increases so does kinetic energy. Einstein developed an expression for the kinetic energy at relativistic speeds. K = (γ − 1)mc2 where: K = kinetic energy (J) γ = lorentz factor m = stationary mass (kg) c = speed of light (3.0 x 10^8 m/s) This expression can be manipulated to give: Etotal = γmc2 where: Etotal = total energy (J) γ m = relativistic mass (kg) c = speed of light (3.0 x 10^8 m/s) Nuclear Fusion Nuclear fusion occurs when two light nuclei are combined to form a larger nucleus. The mass of reactants is greater than the mass of products. This is because the binding energy of the nucleus appears as a loss of mass. Module 7: The Nature Of Light 37  The energy created by this missing mass (mass defect) can be determined from: ΔE = Δmc2 where: ΔE = change in energy (J) Δm = mass defect (kg) c = speed of light (3.0 x 10^8 m/s) Nuclear fusion is a very difficult process because nuclei are positively charged, and thus repel one another. Fusion will only occur if nuclei have enough kinetic energy to over come the repulsive force. Typically, temperature of the order hundreds of millions of degrees are required which are exactly the conditions inside our Sun, where many different fussion reactions are taking place. The main reaction is a fusion of hydrogen nuclei to form helium. Each second , 657 million tonnes of hydrogen and hydrogen isotopes fuse to form 653 million tonnes of helium. Each second, a mass defect of 4 million tonnes results in enormous amounts of energy to be released where a tiny proportion reaches Earth and sustains life. Module 7: The Nature Of Light 38  Electron-Positron Annihilation A positron is an anitparticle of the electron (equal mass, opposite charge) produced when proton-rich radioactive nuclei decay as the result of a proton decaying to a neutron. If positron collides with an electron at low energies, annihilation occurs and gamma rays are produced. 0 0 −1 e + +1 e = γ + γ In this interaction, charge and momentum are conserved. Initially mass appears not to be conserved but since Einstein's equation show us mass of an object is a measure of its energy constant, the mass of electron and positron are coverted into energy. Fuel Combustion The combustion of fuel such as the burning of coal is another example of converting mass into energy. However, the mass difference is to smal that the change in mass goes unnoticed. Module 7: The Nature Of Light 39