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Electromagnetic Spectrum

• Before the 20th century, physicists made comprehensive models and forms of understanding.
• After the 20th century, physicists opened up with quantum theory and the theory of relativity.

 » Prediction of EM Waves

• Hans Oersted observed a current carrying wire caused a nearby compass needle to deflect. This shows that electricity could produce magnetism.
• Michael Faraday later showed that a changing magnetic field could produce electric current
• James Clerk Maxwell later derived four equations involving electric & magnetic field. He found that the interaction between a changing electric & magnetic field resulting in a wave propagating through space.

                       

• Maxwell called these changing electric & magnetic fields, electromagnetic waves. These waves are predicted to travel around 3 x 10⁸ ms^-1 .
• This speed was similar to the speed of light at the time. Therefore, he
  proposed that visible light is a form of EM wave.
• He hypothesised that there are other forms of EM waves, but this wasn’t
  proven until 1910’s. These waves are also known as electromagnetic
  radiation (EMR).

 » Measuring the Speed of Light

           

• Aristotle & Kepler believed that the speed of light was infinite.
• The first movement of finite speed of light came from Romer in 1675. He noticed that the eclipses of lo, the innermost moon of Jupiter depended on the relative positions of Jupiter & Earth.
• As the Earth is moving away from Jupiter (From L to K), the time at which the eclipse occurred is later & as the Earth is moving towardsJupiter from F to G, the time is earlier.
• Romer reasoned that it was because of light from lo had to travel a further distance to reach the Earth when the Earth is moving away from Jupiter.

                                                                                           

• Using the discrepancies in time for the eclipses of lo, Romer estimated that
  it took about 22 mins for light to travel across the diameter of the Earth’s
  orbit around the Sun. Romer calculated the speed of light to be about
  2.2 x 10⁸ ms^-1
  .
• Hippolyte Fizeau performed the first measurement of the speed of light in 1840’s.

     

• Fizeau shone a bright light through a slot in a toothed cogwheel. This light was then reflected by a mirror placed 8km away.
• As the cogwheel was rotated to a certain speed, the reflected light was eclipsed by the cog. Using the rotation rate (𝑚) & the angle of rotation (𝜽) of the wheel, the time taken for the light to complete its return journey (i.e. 16km) could be determined.

                        

 » Electromagnetic Wave Spectrum

               

• Today there are several forms, due to Hertz’s discovery.
• The EM spectrum is divided into different bands to distinguish their different sources and uses. Radio waves & microwaves can overlap in these terms.
• Earth’s atmosphere filters out specific EMR’s. This is beneficial for safety reasons as high energy EMR is dangerous to living organisms. However, it prevents many Earth-based measurements of objects in space.

 

 » Spectra

• Spectrum: A range of values of a quantity orset of related quantities.

               

• Visible Spectrum:  Range of colours emerging from dispersion of white light.

             

• Electromagnetic Spectrum: A range of frequencies/wavelengths of electromagnetic radiation.

 » Types of Spectrum

- Continuous Spectrum: shows all values on spectrum (analog clock analogy).

• White Light
• Rainbow
• Radiation of a blackbody

- Line Spectrum: shows limited values on spectrum (digital clock analogy).

• Absorption or Emission Spectrum

             

 » Spectrology

• Spectrology: The analysis of spectra produced by a radiant object to obtain
  information such as the chemical composition, temperature & other
  features of that object.
• A spectrometer used in this study. There are two types ofspectrometers:

• • Prism Spectrometer
  • Grating Spectrometer

• How is a visible spectrum produced by a hot object?

• • Input energy causes electrons to be excited. They ‘jump’ to a higher orbital shell.
  • Electrons are unable to stay in the excited state forever.
  • As electrons jump down to their original level, they emit EMR.                   

 » Emission Spectrum        

    

• Emission Spectrum: consists of specific wavelengths observed as bright
  lines against a dark background.
• This occurs when a sample of gas is energised (electricity). The electrons of
  the gaseous atoms absorb some of the energy & make a quantum jump to
  the outer energy shell. They soon release the absorbed energy & fall back to
  their ground state. The energy released is in the form of EM waves. Under
  the visible region, it is observed as colours.

 » Exploring Electromagnetic Spectrum

          

• Absorption Spectrum: consists of dark lines as seen against a rainbow of
  colours as a background.
• This occurs when a white light passes through a cool gas. Some of the
  component wavelengths of white light are absorbed by the electrons of the
  atoms. These electrons then make a quantum jump to the outer energy
  shell. They soon release the absorbed energy & fall back to their ground
  state. Since the energy released by the electrons is in all directions in space,
  the energy obtained in the viewing direction is less than the original energy.
  The spectral lines on these particular wavelengths appear to be darker than
  the other wavelengths.

 » Stellar Spectra

• Continuous spectrum is produced when an object emits all wavelengths of
  the electromagnetic spectrum. Under the visible range, a continuous band
  of colours is observed.

• • An incandescent light bulb produces a continuous spectrum.

                            

- Continuous Spectrum

• A heated solid, liquid or a dense gaseous object such as a star produces a continuous spectrum including a wide range of wavelengths. This is because in a dense object like a solid, liquid or even gas at high pressure the atoms are close together & the electron energy levels overlap, causing the transition of the electrons in all wavelengths.

- Stellar Temperature

• The visible spectrum of the stars shows that hot stars radiate more EMR with higher frequency (lower wavelength) than cooler stars.
• Short wavelengths correspond to the blue of the visible spectrum while longer wavelengths correspond to the red end.

⁛   Surface Temperature

◊    Wein’s Law:

- Translational Velocity

• This is determined by analysing the Doppler Effect on the absorption lines. If a star is approaching the observer, every absorption line in the spectrum of the star is shifted towards the blue end of the spectrum by the same amount. If the star is moving away, all the lines are shifted towards the red end.
• Greater Red Shift = Faster the star is travelling from the Earth

                  

- Rotational Velocity

• The spectral lines of a rotating star are broadened due to the red & blue shift.
• Broader Line = Faster Rotation of Star

                                                                    

- Chemical Composition

• Each chemical element has a unique emission spectrum consisting of lines corresponding to the energy level transitions within the element. The chemical composition of a star can be determined by comparing the absorption spectrum of a star to the emission spectra of the elements on Earth.

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The Wave Model of Light

• There were two competing theories, by Newton & Huygens, about the nature of light in the 17th century.

• Newton’s corpuscular theory prevailed & slowed the acceptance of the wave model of light.
• In the early 19th century, the first convincing experiment proved the wave model of light
• Light is now known to display both properties, particles & waves, depending on the circumstances.

 » Huygens’ Principle

• Huygens’ principle is used to explain how 2D waves propagate
• “Every point on a wavefront can be considered the source of circular secondary wavelets. This new wavefront will be tangential to the wavelets”.

                     

                                      

 » Light Diffraction

• Diffraction is the bending or spreading of waves around the edge of an object or through an opening.
• It is a wave property

                 

 » Path Difference

• It is the extra distance travelled compared to another.

 » Interference

• When waves are diffracted, when it passes through narrow slits, the wavelets will interfere with each other producing interference/diffraction patterns
• There are two types of interference of light waves (Constructive & Deconstructive).
• This can be determined by considering the path difference, a measure of the difference in distance from different sources to the same point.

        

 » Young’s Double Slit Experiment

• Thomas Young first demonstrated the concept of interference of light in 1801. His experiment concluded that light was a wave.
• He allowed light from a monochromatic source to pass through a narrow slit to obtain a narrow beam of light which was then passed through the double narrow slits.
• He predicted that if light was a particle, light would pass through the two slits only & create two bars of light on the background surface.
• However, he observed that light was diffracted from the two slits & interfered to produce a constructive & destructive interference patterns (bright & dark bands) on a distant screen.

                                

 

     

  

 » Bandwidth

• Bandwith: The distance between successive bright (or dark) fringes.
• From the diagram & the equation of interference, we have:

        

                   

 » Diffraction Grating

• This consists of large numbers of equidistant parallel lines engraved on a glass or metal surface.
• Due to a large number of slits on a grating, it produces a sharper image on a screen.
• Diffraction experiments usually use only monochromatic light (light of only one colour/wavelength). When white light, which contains different colours/wavelengths, is used in a diffraction grating, each different colour is diffracted by a different amount and forms its own set of coloured fringes.

                            

                       

 » Polarisation

• This evidence for the wave nature of light.
• Polarisation: When the oscillations of electromagnetic waves are restricted to one dimension.
• A polariser is a device that allows oscillation in one plane.

• Light is a transverse wave which consists of electric & magnetic fields that oscillate perpendicularly to each other & to the direction of propagation.
• Light can be polarised by restricted its vibrations to one particular plane.

              

• Only transverse waves can be polarised.
• A polarised beam of transverse wave is one whose vibrations occur in one direction (perpendicular to its propagation).
• Longitudinal waves can’t be polarised, since its vibrations occur in the same direction as propagation.

                                   

                                

 » Polarisers

• A polaroid sheet is a common polariser which is made by embedding in a sheet of plastic, crystals of certain substances & then stretching the plastic so that the crystals or molecules all align in one direction.

                                  

• When unpolarised light falls on a polariser such as a polaroid sheet, one component is transmitted. Thus, the intensity of light is reduced by 50%.

 

• If a second polaroid sheet (called the analyser) is place after the first polaroid sheet, with the polarising axes of both vertical, the plane-polarised light emerging from the first polaroid sheet will go through the analyser without any change in the nature of its polarisation & intensity.

                       

• However, if the second polaroid sheet is arranged that its polarising axis is at a right angle to that of the first polaroid sheet, no light does through the second polaroid.                                  

 » Malus’ Law

• The intensity of light transmitted through the second polaroid sheet depends on the angles of the polarising axes between the first and second polaroid sheet.

 

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The Quantum Model of Light

 » Blackbody Radiation

• An object’s colour is determined by light (EMR) being reflected, light being radiated or a combination of both of these methods.
• Blackbody: A perfect absorber & emitter of radiation or energy. The radiation emitted from a blackbody is entirely due to its temperature.

                                                  

                                     

• The intensities of the emitted radiation obtained at different temperatures was graphed against wavelengths.
• At a given temperature, the curve will have a peak, representing the wavelength with the highest intensity.
• As the temperature increases, the electromagnetic radiation emitted not only increases in total intensity but is strongest at shorter wavelengths (higher frequencies).

                      

 » Classical Predications of Blackbody

• Classical physics predicted that as the wavelength of the emitted radiation ↓, the intensity of the radiation would ↑, without limit.
• This was known as the ultraviolet catastrophe, as it violated the Law of Conservation of Energy & didn’t match experimental observation.

                        

 » The Quantum Theory

• In 1900, Max Planck, offered an alternative hypothesis that explained the experimental observations.
• His theory made a new radical assumption that the energy from that blackbody can’t possess just any value, but rather has energy which is a multiple of a minimum value related to the frequency of oscillation.
• Planck suggested that energy absorbed or released by an object exists in small bundles.

 » The Quantisation of Energy

• The energy of each packet (also known as quantum of energy) is given by:

           

•  The energy released or absorbed by an object could only be a whole number multiple of ‘hf’

        

 » Analogy for Classical Physics

• According to 20th century physics, the energy liberated or absorbed by an object could be any arbitrary value.

 

• However, calculations using this idea can’t reproduce the graphs obtained experimentally from the radiation of a blackbody.

 » Analogy for Quantum Physics

• Energy is not absorbed or emitted continuously but rather, in small packets known as quantum (plural is ‘quanta’). Energy is quantised.

    

 » Consequences of Blackbody Radiation

• Even though Planck’s quantum theory successfully explained the blackbody radiation curve, it was based on a radical assumption that had no experimental evidence to support it.
• In 1905, Einstein proposed a particle theory of light based on Planck’s Quantum Theory. The particles of light (& other EM waves) would have an energy determined by Planck’s Equation (E = hf).
• Einstein’s theory successfully explained the photoelectric effect & provided a strong experimental support for Planck’s Quantum Theory.

 » Einstein’s Particle Theory of Light

• The energy of each photon particle in an EM wave is: E = hf.

         

 » The Electron-Volt

• Electron-Volt (eV): An alternative unit for measuring small amounts of energy. 1eV is defined as the amount of energy gained by an electron when it moves through a potential difference of 1 volt.

        

 » Hertz’s Experiments

• In 1873, Maxwell proposed the existence of EM waves
• In 1887, Heinrich Hertz experimentally proved that EM waves existed.
• Hertz set up this experiment:

                                  

• Using a high voltage induction coil, Hertz was able to produce an oscillating spark in the gap between the electrodes of his transmitter.
• His receiver was a small loop of wire, with a small gap, which was placed at some distance from the transmitter.
• Hertz observed that when sparks were jumping across the gap in the transmitter, sparks would also jump across the gap in the receiver, even though the receiver was not connected to a power supply. AC electricity creates EMR.
• Since the loops were not connected, Hertz hypothesised that the oscillating sparks in the transmitter produced an EM wave, which then induced the sparks across the gap in the receiver.
• He also demonstrated that the invisible radiation in his experiment (which were radio waves) had the same properties as light. Hertz proved Maxwell’s theory
• Hertz also noticed that the intensity of the sparks in the detecting loop increases when it was illuminated with ultraviolet light.
• Light seemed to facilitate the escape of charges from the surfaces. Hertz failed to investigate this further.
• This is now known as the photoelectric effect.

                          

 » The Photoelectric Effect

• In 1905, Einstein proposed a new theory of light.
• Einstein proposed that light consists of tiny particles called photons. Each photon has a discrete amount of energy which is proportional to the frequency of light (E=hf).
• When a photon collides with an electron, at or just within the surface of a metal, it transfers energy to the electron and follows the ‘all or nothing’ principle.
• By building on the ideas proposed by Planck, Einstein’s theory was able to explain the photoelectric effect.
• Photoelectric Effect: the liberation of electrons from the surface of a conductor when light strikes the surface.
• The electrons absorb energy from the incident radiation and can overcome the potential energy barrier that normally confines them to the material.

                                       

 » Prediction Based on the Wave Properties of Light

• ↑ intensity of the incident light, the more energy is transferred to the surface, hence more electrons are liberated with ↑ energy.
• Below a certain intensity, no electron is liberated no matter what the frequency of the incident light is.
• There is a time lag between when the light strikes the metallic surface to the liberation of electrons.

 » Actual Observations

• No photoelectrons are emitted unless the frequency of light is greater than the critical value. The corresponding minimum frequency is called the threshold frequency of the surface.
• Above the threshold frequency, electrons are liberated no matter how small the intensity of the incident light is.
• The liberation of electron is instantaneous.
• The KE of photoelectrons is proportional to the frequency of the incident light

 » The Photoelectric Effect (Continuation)

• The minimum energy required to cause the emission of an electron is called the work function. (which is dependent on the material from which the electron is being ejected).
• If the energy of the photon exceeds the energy required to overcome the electrostatic forces holding the electrons in place, the excess energy will appear as KE of the now emitted electron.
• The energy of an emitted electron is given by:

        

• There is one photoelectron produced per photon absorbed. Unless the photon energy is high enough it will make no difference how intense the beam becomes; electrons will not be emitted.
• Thus, emission is independent of intensity.

                     

         

 

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Light & Special Relativity

• Frame of reference: a coordinate system used to measure velocity & observe.
• Inertial frame of reference: one which is stationary or moving at a constant velocity. (non-accelerating)
• Non-inertial frame of reference: one which is accelerating.

 » Galilean Experiment

• A cannonball is dropped from the top of the mast of a stationary ship.
• An observer on the ship observed the cannonball landed next to the base of the mast.
• The experiment was then repeated on a ship moving at a constantspeed.
• The same result was obtained by observer on the ship.

 » Galilean Relativity

• All inertial frames of references are equivalent.
• The laws of motion are the same in all inertial frame ofreference.
• The results of Galileo’s experiment implied that it is impossible to detect the state of motion of an inertial frame of reference by carrying out any mechanical experiment within that inertial frame of reference.

 » Relative Velocity

- Relative Velocity is the velocity of an object from one frame of reference.

• A car travelling at 60kmhr^-1 , is travelling at 60kmhr-¹ relative to the ground.

                            

 » Light in Inertial Frames

• According to Galilean principle of relativity, all velocities are relative, all inertial frames of reference are equivalent and there is no absolute frame of reference.
• Maxwell’s equations however indicated that EM waves travelled at a constant speed in a particular medium, irrespective frame of reference of the observer or the source of light.
• To resolve this issue the concept of Aether/Luminferous Aether was proposed.
• Initially physicists thought that EM waves (such as light) travelled through a medium called Aether. Aether was thought to be fixed in space and acted as a stationary absolute frame of reference. All objects in space are moving relative to this absolute frame of reference.
• The speed of light is thought to be measured relative to the Aether.

 » Aether

• A hypothetical medium that is responsible for propagation of light waves. It acts as a stationary absolute frame of reference.

•  Properties:

• Permeate all of space and yet was completely permeable to all objects.
• Stationary, low density, and perfectly transparent.
• Had great elasticity to support the propagation of light waves.
• If c was 3 x 10⁸ ms^-1 in this aether, it should be faster or slower in a frame of reference moving through aether such as the Earth.
• In 1887, Michelson & Morley performed an experiment to test this idea.

 » Michelson & Morley Experiment

• This experiment is designed to measure the relative motion of Earth through Aether using a device called interferometer.
• A single beam of light was allowed to split into two identical components, moving perpendicularly to each other.
• When the two identical beams recombined, the interference pattern was observed.
• The whole apparatus was then rotated through 90^o .

                                

                                               

• Because the beams are perpendicular, they cannot both be parallel to Earth’s motion through the aether. Therefore, if aether never existed, the speed of light relative to the Aether, would be different along the two perpendicular paths.
• As we rotated the interferometer, the relative speed of light along these two paths would keep changing, resulting in the changing interference patterns.
• The interference pattern was expected to appear the same again after 90o rotation since the two paths of light now resumed the original position only switching roles.
• Despite conducting the experiments at various times and locations, and the use of extremely sensitive interferometer, there was no change in the interference pattern whatsoever. This was known as the ‘null’ result.
• Implication: The ‘null’ result of M-M experiment questioned the existence of aether and provided the experimental evidence to support Einstein’s special theory of relativity.

 » Einstein’s Thought Experiment

• Einstein was able to solve the discrepancy between Galileo’s relativity & Maxwell’s EM equations.
• Einstein used thought experiments to reach conclusions, this helped him visualise complex problems that were not experimentally possible to test during his time.
• Thought experiments are hypothesis, theories, principles that are proposed to consider consequences. They generally involve a situation, multiple scenarios and a conclusion.
• The thought experiment involves a person sitting in a moving train travelling at the speed of light. When the person holds a mirror in front of them, do they observe their reflection?

                

• There are two cases:
• • Case 1: If they did not see their reflection then the Galilean principle of relativity was violated.
  • Case 2: If they saw their reflection then according to Galilean /Newtonian relativity, the stationary observer on the ground would see the speed of light to be twice its usual speed.

• Einstein believed:
• • The passenger must observe their reflection,so as not to violate general relativity. Einstein believed that Galilean’s relativity applied to not only mechanical systems by also to EMR.
  • Both the stationary observer and the observer in the moving train measured the speed of light to be c. (to satisfy Maxwell’s equations and not to contradict the first statement).

• Given this:
• • The two observers had to disagree on something else.
  • c = d/t

• Einstein coined this as the Special theory of relativity, and it has two postulates
• • The Laws of Physics apply equally in all inertial frames of reference.
  • The speed of light is constant as measured in any inertial frame of reference.

 » Consequences of Special Relativity

• Relativity of Time

• The time taken for an event to occur within its own rest frame is called the proper time (t_o). Measurements of this time made from any other inertial frame of reference (t_v) in relative motion to the first, are always greater.
• Rest Frame: a frame of reference within which an event is occurring. An observer outside a moving FOR will observe anything within that FOR to be occurring slower.

•  Twin Paradox

• There are two twins. One is placed on a fast-moving spaceship while the other remains on Earth. After a long period of time, the spaceship returns and the ages of the two twins is compared.
• From the perspective of the Earth twin, t_v > t_o. The age of the Space twin is 21 yrs. The age of the Earth twin is 50 yrs.
• We must recognise who is an inertial frame for the while time. The Earthbound twin remains in a nearly inertial frame at all times. The spaceship is not an inertial frame at all times. The prediction in the relative ages of the twins made by the space twin is not justified.

 » Relativity of Length

  

    

                         

 » Relativistic Momentum

       

 » Particle Accelerators

• Measurements inside particle accelerators show that velocities of particles with mass don’t exceed the speed of light, no matter how much energy is provided.

            

 » Mass – Energy Equivalence

• As an object approaches the speed of light, the work/energy supplied will no longer act to accelerate the object, but instead is converted to inertial mass. (E = mc² )
• Einstein was able to show the energy associated with an object’s mass.
• Thus, the total energy of an object consists of two parts: the kinetic energy & the rest energy which is related to its mass.

          

• Analogy: suppose we supply 100kJ of energy to accelerate a particle, 80kJ is converted into its kinetic energy & 20kJ is converted into mass then:

                             

• We can see that when the particle is stationary (E_k = 0), its energy is mc² . This implies that a particle at rest has energy & related to its mass.

 » Relativistic Mass

• From relativistic momentum,

                                         

                           

• The mass of an object measured within its own rest frame is called the rest mass (m_o). Measurements of this mass made from any other inertial frame of reference (m_v) in relative motion to the first, are always greater (mass dilation).

 » Antiparticle Annihilation

• It has been discovered that for every subatomic particle, there exists a corresponding anti-particle.
• For example, there exists an anti-electron (positron) which has the exact same mass but opposite charge.
• There are also anti-protons, anti-neutrons & etc.
• When any particle & anti-particle meet, they mutually annihilate each other. All the mass is converted into energy via, E = mc² .

                           

 » Nuclear Reactions

• Nuclear Fusion: the process in which two or more small nuclei combine to form a larger nucleus with the release of a large amount of energy (more energy released compared to nuclear fission).
• Nuclear Fission: the process in which a heavy unstable nucleus splits to form more stable, lighter nuclei. It also emits neutrons & energy.
• Both reactions, fusion & fission release vast amounts of energy which can be calculated using Einstein’s equation & mass-defect.

                  

• Mass Defect: the difference between the mass of the constituent nucleons & the mass of the nucleus
• Einstein’s equations, E = mc² , does not apply to nuclear reactions, but also to other mass-energy reactions such as the combustion of fuel. In many such situations, the mass defect/difference is so small that it is usually unnoticed.

Definitions

 

Formulas