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Lecture 19: Interference When two electromagnetic waves meet, the electric fields add up with sign. Namely they can reinforce (constructive interference), they can cancel each other (destructive interference), or anything in between. Suppose the electric fields have the same strength: Constructive, maxima coincide: Destructive, maximum coincides with minimum: Random: Sometimes constructive, sometime destructive, in average the intensity adds up, namely it is double. This is what happens in most everyday situations. The reason is that light is made out of a series of wave packets. Wave packets interfering: Length of packet=coherence length Examples: White light: ~ 1mm Single spectral line: ~ 1m Lasers: 10 cm to 10 Km. The only practical way to observe interference is to use light from the same source by splitting and rejoining a beam. Interferometer: LIGO Interferometer to detect gravitational waves. Washington state site: Main rule of interference: Here DL is the difference in lengths of the two paths the light takes. In the interferometer it is DL=2 (L2-L1). We also need DL > l where l is the coherence length. A very important point is that the wave-length is determined by the colors. Different colors have different wave-length. Example: lyellow= 570nm, lgreen=510nm, lblue= 475nm This means that the same setup can give a maximum for one color and a minimum for another, colors separate. In everyday life this is seen for example in soap bubbles, oil films on water, peacocks. Thin film interference: We could now use the main rule of interference: But there are two important subtleties. 1) The wave length depends on the medium. Frequency is the same. 2) There can be a phase shift when light from medium 1 is reflected by an interface with medium 2 (same as extra half wave-length). As a result we find the rule for thin film interference: Newton’s rings. Consider an analogous of the Newton rings by putting to pieces of glass at an angle using a thin wire to separate them at the end. Newton rings are done by putting a lens over a flat glass. Maxima Clicker question In the previous set up L=15cm, h=0.24mm, l=600nm. How many fringes (bright bands) do you see A. B. C. D. 400 600 800 1000 Two slit experiment Bright bands (fringes) for those angles such that Lecture 20: Fresnel equations Reflection and refraction. Assume E parallel to interface. Continuity of parallel Electric field Continuity of Magnetic field Standard relation for electromagnetic waves We derive the following equations for the electric field: But: So: Remember: Then , Snell’s law!! But we have one more equation Given incident we can find transmitted and reflected. Consider normal incidence For general angle Again: In any case, how much is reflected and how much refracted is completely determined by these equations!!. Lecture 21: Diffraction Gratings We can reinforce the effects of two slits interference by including many slits. This is called a grating and has much sharper maxima. Maxima whenever Application: To separate the color of an incident light. The same effect can be obtained with a prism, however a prism is a thick (triangular section) slab of glass and absorbs too much light for certain applications (for example star light). The spectrum of distant stars can be obtained by using a grating. It turns out that different elements have a special signature in the spectrum (certain wave-lengths) and therefore the composition of far away stars can be determined. In fact Helium was discovered first in the Sun!!. Diffraction A particular case of interference is when light goes thought a hole and it has its own name: diffraction. In this case it is useful to look at the minima. Whenever a ray in the top interferes destructively with a ray in the middle, the same will be true for every ray in the upper half, namely there will be another ray which cancels it. First minimum Other minima are at qn=n l/a, where n is an integer (order). If we put a screen a distance D away the width of the bright band in the middle is: The width of the other bright fringes is half the width of the central one. Important effect. Parallel rays going through a hole are not parallel any more afterwards. They have a range of angles from zero to For a round aperture hole the formula is In the case of a convergent lens, parallel rays produce an image of size In fact the image is a solid dot surrounded by diffraction rings. Observing those rings is the indication that this particular aberration is the effect of diffraction. It limits the resolution of optical instruments including our eye!. For ground based telescopes, the resolution is actually given by the atmospheric fluctuations since they are large enough. For the eye, assuming a 4mm pupil: in radians. Did Nature gave us the maximum possible resolution for our eyes size?. Clicker question Assume that the resolution of the eye is indeed the maximum allowed by diffraction. If the two headlights of a car are 1m apart, at what maximum distance would you be able to resolve them? A. 1 Km B. 10 Km C. 100 Km D. 1000 Km Lecture 22: Light-Matter interaction Photoelectric effect Light is emitted and absorbed by matter. At the end of the 1900 this is was seen as one of the few phenomena left to understand in physics. It turned out to cause a revolution in physics that we still live today. Photoelectric effect: Electrons are free to move inside a metal but normally do not leave it since the metal would be positively charged an attract them back. Light has a powerful localized electric field that can kick the electrons far enough so they do not come back to the metal. Once electrons are away one can direct them with electric fields. Certain night vision systems for example work that way. In physics we are concern with two main quantities, how many electrons are kicked out and with which energy. The number of electrons, not surprisingly, is proportional to the intensity of light. But their energy is proportional to the frequency! The surprising explanation was proposed by Einstein. He proposed that light is made out of quanta, or particles, called photons whose energy is given by (h is called Planck’s constant) Now the explanation is very simple. A photon is absorbed by an electron which acquires an extra energy Eph. Part of the energy is used to leave the metal (called W, work function), the rest becomes kinetic energy. We defined Now, it follows that This provides a simple explanation of the experimental results. To be truthful to Einstein genius, however, we should point out that the formula was derived by Einstein nine years before the first experiment that actually verified it!. (It was known that the energy of the electrons increased with frequency but not this simple formula). Photons The photoelectric effect shows that light interacts with matter as if made of tiny quanta called photons. They have energy and momentum given by With h c = 1.2 x 10-6 eV m (c is the speed of light) An electron Volt, eV is the energy gained by an electron when going through a potential difference of 1 V. 1eV = 1.6 x 10-19 J Clicker question Light of wave-length l=100nm shines on Copper, W ~ 5eV (more precise value 4.7 eV). What’s the energy of the ejected photons? A. 0.7 eV B. 7 eV C. 70 eV D. 700 eV hc = 1.2 x 10-6 eV m Eph = h f = hc / l Ee = Eph -W Atomic theory. The next step in understanding the interaction of light with matter is the emission of light by an atom. The simplest atom is Hydrogen made of a proton and an electron. It turns out that a hydrogen atom emits (and absorbs) only particular frequencies, it has a sort of spectral fingerprint. The same is true for all elements. Explanation (Bohr). The electron orbits around the proton in some specified orbits. The atoms absorbs and emits photons when the electron jumps from one orbit to another. This explains the resulting spectrum but goes against Newtonian mechanics. There are two main problems. 1) Why does not the electron find its minimum of energy by attaching to the proton, namely why there is a lowest energy orbit?. 2) Why only certain orbits are possible? It means that new physical principles operate at distances of the atomic size. A new length scale brings in new physics! 1) Is explained by Heisenberg’s uncertainty principle. 2) Is explained by De Broglie’s particle-wave duality. In fact it is necessary to develop a new formalism called quantum mechanics. We are not going to go that far here. Uncertainty principle. In classical mechanics the state of a particle is determined by its position and its momentum (or velocity). In quantum mechanics the position and momentum cannot be determined simultaneously. If the particle is localized, the momentum is undetermined and vice-versa. The uncertainty obeys: Consider the hydrogen atom. The energy is The potential energy is minimized when the electron is on top of the proton (r=0). However localizing the electron would make the momentum completely uncertain and therefore the energy would be huge (in Newtonian mechanics, instead, we can have r=0, p=0) . We estimate We get where we defined or The energy is a function of the orbital radius and becomes a minimum for r = 2 r0. This gives a correct lowest energy (n=1) Lecture 22: De Broglie waves. Light behaves as a wave but also as particles. Can particles such as the electron behave as waves? Yes, that was the proposal of De Broglie. The frequency and wave-length are given by: We also have the usual relation between momentum and energy where we included the comparison with the case of the photon. This received truly spectacular experimental confirmation by an analogous of the two slit experiment: Electron source The interference pattern is present even if we shoot the electrons one by one!! Quantum mechanically the electron explores all possible paths. Clicker question An electron is moving at 30 m/s. What is the associated wavelength?. A. B. C. D. 2.4 x 10-5 m 2.4 x 10-7 m 2.4 x 10-9 m 2.4 x 10-11 m hc = 1.2 x 10-6 eV m Pe = me v = h / l me c2 = 0.5 MeV How can this help us understand the Hydrogen atom?. The orbits are standing waves where an integer number of wavelengths fit. Let’s see. Use Newtonian relation to get the velocity and then the momentum to find the possible orbits Spectacular agreement!! Energy: Quantum mechanics is the basis for our understanding of the microscopic world. It replaces Newtonian mechanics at the atomic level. It is fundamental to understand properties of materials (conductors, insulators, semiconductors, superconductors,…). At smaller length scales, nuclear and particle physics are based on quantum mechanics. No deviations of quantum mechanics have been found down to distances d~10-20m (LHC accelerator). Future? Quantum computers is an active topic of research and can revolutionize computations. Quantum cryptography. New material, new states of matter (Bose-Einstein condensate), etc. Although a hundred years old, quantum mechanics is present everywhere in the frontier of research and technology. Example: Lasers (and masers). It is based on the effect that an excited electron is more likely to emit a photon in a photonic state that is already occupied. It is called stimulated emission and can be used to amplify an already present wave. It can also be used for microwaves. It is the basis of atomic clocks which are used to synchronize GPS satellites. Example: Compton effect. Energetic photons can bounce form electrons losing energy in the process. The precise computation requires using special relativity to describe the electron but the result is very simple: Lecture 24: Nuclear Physics Atomic physics involve distances ~10-10 m and energies 10 eV. Nuclear physics operate at distances 10-15 m and, by the uncertainty principle we expect it involves energies 105 larger, namely E~1MeV. The nucleus is composed of proton and neutrons. Protons are positively charged and neutrons are neutral. The number of protons is called Z, the atomic number. The total number of protons and neutrons is called A, the mass number. The number of neutrons is therefore A-Z. The neutral atom has Z electrons which determine the chemical properties of the element. Therefore Z determines which type of element the atom is (Hydrogen, Carbon, Oxygen, etc.). The number of neutrons is not apparent in chemical properties and can vary for the same chemical element. The main effect is that the mass of the atom is different. Nuclei with the same Z but different A are called isotopes. It is difficult to separate isotopes because it cannot be done chemically. It can be done usually using the different densities, diffusion constants etc. Notation: A Z Z Isotope of Carbon. 6 protons, 8 neutrons. Most abundant is Carbon 12, it has 6 protons and 6 neutrons. The mass of the proton and neutron are close to each other and to the value m~1.6x10-25 Kg The mass of the nucleus is approximately equal to the sum of the masses of protons and neutrons that it contains. However precise measurements determine a mass defect: This is actually related to the binding energy by the famous Einstein formula: This is always true but the energies involved in nuclear physics are large enough (and the mass can be measured with enough precision) that Einstein formula can be verified. We need however more precise data for the masses of protons and neutrons: The binding energy comes from the nuclear force. Protons and neutrons attract each other with a force larger that the electric repulsion (between protons). Nuclei can either spontaneously or by external influence, decay, namely be converted into other nuclei. 1) Alpha decay. A nucleus of Helium is a particularly stable nucleus called an alpha particle. Some nuclei decay by emission of alpha particles: Example Radium (Ra) 2) Beta decay. Free neutrons are actually unstable and decay into protons emitting an electron and a neutrino. Electrons are called beta particles in this context. Inside a nucleus this is usually not energetically favorable so it does not happen. However some nuclei are unstable Notice that one neutron was converted into one proton. This is the effect of the weak force. In this particular case it is used for Carbon dating. Other cases: Electronic capture: An orbital electron is captured by the nucleus. A proton becomes neutron. X-rays are emitted when other electrons fill the hole left by the captured electron. Finally we can have inverse beta decay where a proton becomes a neutron and emits a positron: 3) Gamma decay. This is a standard electromagnetic transition where, for example, a nucleon changes orbit emitting a photon, the same as for atoms. The photon in this context is called a gamma ray and has E~ 1 MeV. These photons are actually the most dangerous products of nuclear decays and are responsible for the “danger radioactivity signs”. Alpha particles and electrons are easily stopped by matter. Gamma rays are not, for example lead bricks are used to (partially) stop them. The dose of absorbed radiation is measured in Gray (Gy). 1 Gy = 1 J/ Kg = (energy absorbed in Joules)/(mass in kilograms) For any of those decays, we start with a number N of nuclei. After a while a fraction of those decayed. The time it takes for half of them to decay is called the half-life T1/2. The number of nuclei in the original states decreases as For example the half-life of is 5730 years. The ratio of to is the same as in the environment as long as a organism is alive. After death, the amount of begins to decrease and can be used to date the (archeological) remains. Another way to characterize how fast a sample is decaying, and therefore know for example the intensity of the radiation that it can produce is by counting the number of decays in a second. This is known as the activity of the radioactive source and depends on the half-life of the nucleus and also on how many nucleus are present. It is measured in Curies or Becquerels: The other two important processes we need to consider are fission and fusion. Fission is the split of the nucleus into two nuclei of approximate half the size. This can be spontaneous but most commonly occurs by neutron bombardment. Example: The most common form of Uranium is actually which does not undergo fission. The only way to get a sustainable rate of fission (or explosive) is by increasing the concentration a process called uranium enrichment. This is a source of energy Both, in nuclear reactor and nuclear explosives. For example 1g of Uranium completely undergoing fission releases 9x1010J to be compared with 1g of TNT which produces 4200 J. Finally we have fusion where two nuclei combine to form a larger one. This is actually how the elements in the Universe were produced (in stars). Example: These are different isotopes of hydrogen actually. They are called deuterium and tritium. Fusion also releases large quantities of energy and is the energy source of the Sun and the basis for fusion weapons. It very difficult to control in a way that it produces more energy than the one used to produce it. If that is achieved it will be the source of energy of the future. There is a large international collaboration attempting this (ITER). See also National Ignition Facility, Livermore, CA (DOE).