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## Principi e Metodi della Fisica

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**Principi e Metodi della Fisica**Elettromagnetismo: dalla carica elettrica, ai campi, alle equazioni di Maxwell ed alle onde elettromagnetiche Relatività: dal mistero della velocità della luce, allo spazio-tempo, alla equivalenza di massa ed energia. La gravitazione come geometria Meccanica Quantistica: la crisi della Fisica Classica, comportamenti corpuscolari ed ondulatori di materia e radiazione, i quanti di energia. Principio di indeterminazione. Statistiche quantistiche e struttura atomica della materia Bibliografia:Robert H. March, Fisica per Poeti, Ed. Dedalo, 1994 Website:http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html LT in Comunicazione Scientifica, a.a. 2004-05 Mauro Anselmino**Electric Charge**The unit of electric charge is the coulomb. Ordinary matter is made up of atoms which have positively charged nuclei and negatively charged electrons surrounding them. Charge is quantized as a multiple of the electron or proton charge: The influence of charges is characterized in terms of the forces between them (Coulomb's law) and the electric field and voltage produced by them. One coulomb of charge is the charge which would flow through a 120 watt lightbulb (120 volts AC) in one second. Two charges of one coulomb each separated by a meter would repel each other with a force of about a million tons! The rate of flow of electric charge is called electric current and is measured in ampères. In introducing one of the fundamental properties of matter, it is perhaps appropriate to point out that we use simplified sketches and constructs to introduce concepts, and there is inevitably much more to the story. No significance should be attached to the circles representing the proton and electron, in the sense of implying a relative size, or even that they are hard sphere objects, although that's a useful first construct. The most important opening idea, electrically, is that they have a property called "charge" which is the same size, but opposite in polarity for the proton and electron. The proton has 1836 times the mass of the electron, but exactly the same size charge, only positive rather than negative. Even the terms "positive" and "negative" are arbitrary, but well-entrenched historical labels. The essential implication of that is that the proton and electron will strongly attract each other, the historical archtype of the cliche "opposites attract". Two protons or two electrons would strongly repel each other. Once you have established those basic ideas about electricity, "like charges repel and unlike charges attract", then you have the foundation for electricity and can build from there. From the precise electrical neutrality of bulk matter as well as from detailed microscopic experiments, we know that the proton and electron have the same magnitude of charge. All charges observed in nature are multiples of these fundamental charges. Although the standard model of the proton depicts it as being made up of fractionally charged particles called quarks, those fractional charges are not observed in isolation -- always in combinations which produce +/- the electron charge. One of the fundamental symmetries of nature is the conservation of electric charge. No known physical process produces a net change in electric charge.**Coulomb's Law**Like charges repel, unlike charges attract. The electric force acting on a point charge q1 as a result of the presence of asecond point charge q2 is given by Coulomb's Law: Where ε0 = permittivity of spaceNote that this satisfies Newton's third law because it implies that exactly the same magnitude of force acts on q2 . Coulomb's law is a vector equation and includes the fact that the force acts along the line joining the charges. Like charges repel and unlike charges attract. Coulomb's law describes a force of infinite range which obeys the inversesquare law, and is of the same form as the gravity force. Electric Force Example (1 Kg-peso = 9.8 N)**How many electrons in matter?**If such enormous forces would result from our hypothetical charge arrangement, then why don't we see more dramatic displays of electrical force? The general answer is that at a given point in a wire, there is never very much departure from electrical neutrality. Nature never collects a Coulomb of charge at one point. It might be instructive to examine the amount of charge in a sphere of copper of volume one cubic centimeter. Copper has one valence electron outside of closed shells in its atom, and that electron is fairly free to move about in solid copper material (that's what makes copper a good electrical conductor). The density of metallic copper is about 9 grams/cm3 and one mole of copper is 63.5 grams so the cubic centimeter of copper contains about 1/7th of a mole or about 8.5 x 1022 copper atoms. With one mobile electron per atom, and with the electron charge of 1.6 x 10-19 Coulombs, this means there are about 13,600 Coulombs of potentially mobile charge in one cm3 of copper. A mole (abbreviated mol) of a pure substance is a mass of the material in grams that is numerically equal to the molecular mass in atomic mass units (amu). A mole of any material will contain Avogadro's number of molecules. 1 mole contains ≈ 6 1023 molecules,Avogadro’s number**Electromagnetic forces bind atoms in molecules**Strong forces bind protons and neutrons (nucleons) in nuclei u,d are quarks**Nucleus contains protons with charge +e and uncharged**neutrons 10-10m 10-14 m**“Empty” atomic matter**Nuclear matter Composite nucleons No evidence yet for further structure of electrons and quarks**Electric field is defined as the electric force per unit**charge. The direction of the field is taken to be the direction of the force it would exert on a positive test charge. The electric field is radially outward from a positive charge and radially in toward a negative point charge. The electric field of a point charge can be obtained from Coulomb's law:**Electric Potential Energy and Electric Potential**Potential energy can be defined as the capacity for doing work which arises from position or configuration. In the electrical case, a charge will exert a force on any other charge and potential energy arises from any collection of charges. For example, if a positive charge Q is fixed at some point in space, any other positive charge which is brought close to it will experience a repulsive force and will therefore have potential energy. The potential energy of a test charge q in the vicinity of this source charge will be: where k is Coulomb's constant The potential energy of a point charge can be evaluated by calculating the work necessary to bring a test charge q in from an infinite distance to some distance r. The zero of potential is chosen at infinity. From the knowledge of the electric potential energy one can compute the electric force. In electricity, it is usually more convenient to use the electric potential energy per unit charge, just called electric potential or voltage.**Magnetic Fields**Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field B is defined in terms of force on moving charge in the Lorentz force law. The interaction of magnetic field with charge leads to many practical applications. Magnetic field sources are essentially dipolar in nature, having a north and south magnetic pole.**Electric and Magnetic Monopoles**The magnetic field of a bar magnet The electric field of a point charge is radially outward from a positive charge Magnetic sources are inherently dipole sources - you can't isolate North or South "monopoles". Electric sources are inherently "monopole" or point charge sources.**Magnetic Force**The magnetic field B is defined from the Lorentz Force Law, and specifically from the magnetic force on a moving charge: The implications of this expression include: 1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B. 2. The magnitude of the force is F = qvB sinθ where θ is the angle < 180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero. 3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector product. From the force relationship above it can be deduced that the units of magnetic field are Newton seconds /(Coulomb meter) or Newtons per Ampere meter. This unit is named the Tesla. It is a large unit, and the smaller unit Gauss is used for small fields like the Earth's magnetic field. A Tesla is 10,000 Gauss. The Earth's magnetic field is on the order of half a Gauss.**Lorentz Force Law**Both the electric field and magnetic field can be defined from the Lorentz force law: The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the force is given by the right hand rule.**Magnetic Field of Current**The magnetic field lines around a long wire which carries an electric current form concentric circles around the wire. The direction of the magnetic field is perpendicular to the wire and is in the direction the fingers of your right hand would curl if you wrapped them around the wire with your thumb in the direction of the current. An electric current acts with a force on a test small magnet (Oersted) The magnetic field of an infinitely long straight wire can be obtained by applying Ampere's law. The expression for the magnetic field is µ0 = permeability of free space r = distance from wire**Magnetic Field of (Atomic) Current Loop**Electric current in a circular loop creates a magnetic field which is more concentrated in the center of the loop than outside the loop. at center of loop Magnetic Dipole Moment B = (μoi) / (2r) (Notice: µ of dipole is not the same as the permeability µ0 )**Magnetic Force on a Current (Faraday)**F = i L x B**Energy in Electric and Magnetic Fields**Both electric fields and magnetic fields store energy. For the electric field the energy density is This energy density can be used to calculate the energy stored in a capacitor. For the magnetic field the energy density is**Electric Field Energy in Capacitor**The energy stored on a capacitor is in the form of energy density of an electric field and is given by This can be shown to be consistent with the energy stored on a charged parallel plate capacitor E = V/d**Capacitance of Parallel Plates**The electric field between two large parallel plates is given by The voltage difference between the two plates can be expressed in terms of the work done on a positive test charge q when it moves from the positive to the negative plate. it then follows from the definition of capacitance C = Q/V The Farad, F, is the SI unit for capacitance, equal to a Coulomb/Volt.**Storing Energy in a Capacitor**The energy stored on a capacitor can be expressed in terms of the work done by the battery. Voltage represents energy per unit charge, so the work to move a charge element dq from the negative plate to the positive plate is equal to V dq, where V is the voltage on the capacitor. The voltage V is proportional to the amount of charge which is already on the capacitor. If Q is the amount of charge stored when the whole battery voltage appears across the capacitor, then the stored energy is obtained from the integral: Element of energy stored: This energy expression can be put in three equivalent forms by just permutations based on the definition of capacitance C = Q/V:**Motional EMF**The magnetic force exerted on the charges in a moving conductor will generate a voltage (a motional emf). This motional emf is one of many settings in which the generated emf is described by the Faraday’s law. The motional emf expression is an application of Faraday's Law, as can be seen from: Φ = BA is the flux of magnetic field: it is given by B times the perpendicular area that it penetrates. ΔΦ/Δt gives the variation with time of the flux.**Faraday's Law**Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.**AC Generator**The turning of a coil in a magnetic field produces motional emfs in both sides of the coil which add. Since the component of the velocity perpendicular to the magnetic field changes sinusoidally with the rotation, the generated voltage is sinusoidal or AC. This process can be described in terms of Faraday's law when you see that the rotation of the coil continually changes the magnetic flux through the coil and therefore generates a voltage.**Circular Path from Magnetic Field**If a charge moves into a magnetic field with direction perpendicular to the field, it will follow a circular path. The magnetic force, being perpendicular to the velocity, provides the centripetal force.**Electric Shock**The primary variable for determining the severity of electric shock is the electric current which passes through the body. This current is of course dependent upon the voltage and the resistance of the path it follows through the body. An approximate general framework for shock effects is as follows: One instructive example of the nature of voltage is the fact that a bird can sit on a high-voltage wire without harm, since both of its feet are at the same voltage. You can also see that the bird is not "grounded" -- you will not be shocked by touching a high voltage if there is no path for the current to reach the Earth or a different voltage point. Typically if you touch a 120 volt circuit with one hand, you can escape serious shock if you have insulating shoes which prevent a low-resistance path to ground. This fact has led to the common "hand-in-the-pocket" practice for engineers and electrical workers. If you keep one hand in your pocket when touching a circuit which might provide a shock, you are less likely to have the kind of path to ground which will result in a serious shock.**Will the bird on the high voltage wire be shocked?**Electric current flow is proportional to voltage difference according to Ohm's law, and both the bird's feet are at the same voltage. Since current flow is necessary for electric shock, the bird is quite safe unless it simultaneously touches another wire with a different voltage. Want a scary job? Maintenance on high voltage transmission lines is sometimes done with the voltage "live" by working from a platform on a helicopter, sitting on a metal platform! The helicopter must make sure it doesn't touch neighboring wires which are at a different voltage. Power: P = VI = RI2**Maxwell's Equations**Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships. ρ, J = charge, current density**The Wave Equation**Maxwell's Equations contain the wave equation for electromagnetic waves: In 1-dimension with the same form applying to the magnetic field wave in a plane perpendicular the electric field. The wave equation for electromagnetic waves arises from Maxwell's equations. The forms of a plane wave solution for the electric and the magnetic fields are To be consistent with Maxwell's equations, these solutions must be related by**Electromagnetic waves carry energy as they travel through**empty space. There is an energy density associated with both the electric and magnetic fields. The rate of energy transport per unit area is described by the vector which is called the Poynting vector. This expression is a vector product, and since the magnetic field is perpendicular to the electric field, the magnitude can be written (B = E/c)**Traveling Wave Relationships**A single frequency traveling wave will take the form of a sine wave. A snapshot of the wave in space at an instant of time can be used to show the relationship of the wave properties: frequency, wavelength and propagation velocity. The motion relationship "distance = velocity x time" is the key to the basic wave relationship. With the wavelength as distance, this relationship becomes λ= vT. Then using f = 1/T gives the standard wave relationship v = f λ Waves change periodically in space and time k = 2π/λω = 2π/T**Transverse Waves**For transverse waves the displacement of the medium is perpendicular to the direction of propagation of the wave. A ripple on a pond and a wave on a string are easily visualized transverse waves. Electromagnetic waves are transverse waves. Longitudinal Waves In longitudinal waves the displacement of the medium is parallel to the propagation of the wave. Sound waves in air are longitudinal waves.**Sound Waves in Air**A single-frequency sound wave traveling through air will cause a sinusoidal pressure variation in the air. The air motion which accompanies the passage of the sound wave will be back and forth in the direction of the propagation of the sound, a charateristic of longitudinal waves.**Interference and Phase**If a mass on a rod is rotated at constant speed and the resulting circular path illuminated from the edge, its shadow will trace out simple harmonic motion. If the shadow vertical position is traced as a function of time, it will trace out a sine wave. A full period of the sine wave will correspond to a complete circle or 360 degrees. The idea of phase follows this parallel, with any fraction of a period related to the corresponding fraction of a circle in degrees. waves add with their phases**Diffraction**This is an attempt to more clearly visualize the nature of single slit diffraction. The phenomenon of diffraction involves the spreading out of waves past openings which are on the order of the wavelength of the wave. The spreading of the waves into the area of the geometrical shadow can be modeled by considering small elements of the wavefront in the slit and treating them like point sources. If light from symmetric elements near each edge of the slit travels to the centerline of the slit, as indicated by rays 1 and 2 above, their light arrives in phase and experiences constructive interference. Light from other element pairs symmetric to the centerline also arrive in phase. Although there is a progressive change in phase as you choose element pairs closer to the centerline, this center position is nevertheless the most favorable location for constructive interference of light from the entire slit and has the highest light intensity The first minimum in intensity for the light through a single slit can be visualized in terms of rays 3 and 4. An element at one edge of the slit and one just past the centerline are chosen, and the condition for minimum light intensity is that light from these two elements arrive 180° out of phase, or a half wavelength different in pathlength. If those two elements suffer destructive interference, then choosing additional pairs of identical spacing which progress downward across the slit will give destructive interference for all those pairs and therefore an overal minimum in light intensity.**AM Radio Band**TheAmplitude Modulated (AM) radio carrier frequencies are in the frequency range 535-1605 kHz. The frequencies 30-535 kHz are used for maritime communication and navigation and for aircraft navigation. Carrier frequencies of 540 to 1600 kHz are assigned at 10 kHz intervals. f = 500 kHz = 5 105 s-1 T = 1/f = 2 10-6 s λ = cT = 3 108 m/s 2 10-6 s = 600 m E = h f = 6.6 10-34 J s 5 105 s-1 = 33 10-29 J = 33 10-29 6.25 1018 eV = 2 10-9 eV TV and FM Radio Band h = Planck constant = 6.6 10-34 J s1 eV = 1.6 10-19 J Rays with quantum energy of 1 eVcorrespond to λ≈ 12 10-7 m,E = hc/λ**Visible Light**The narrow visible part of the electromagnetic spectrum corresponds to the wavelengths near the maximum of the Sun's radiation curve. White light may be separated into its spectral colors by dispersion in a prism. 1 nm = 10 Å**X-rays and gamma-rays**X-ray was the name given to the highly penetrating rays which emanated when high energy electrons struck a metal target. Within a short time of their discovery, they were being used in medical facilities to image broken bones. We now know that they are high frequency electromagnetic rays which are produced when the electrons are suddenly decelerated - these rays are called bremsstrahlung radiation, or "braking radiation". X-rays are also produced when electrons make transitions between lower atomic energy levels in heavy elements. X-rays produced in this way have definite energies just like other line spectra from atomic electrons. They are called characteristic x-rays since they have energies determined by the atomic energy levels The term gamma ray is used to denote electromagnetic radiation from the nucleus as a part of a radioactive process. The energy of nuclear radiation is extremely high because such radiation is born in the intense conflict between the nuclear strong force and the electromagnetic force, the two strongest basic forces. The gamma ray photon may in fact be identical to an x-ray, since both are electromagnetic rays; the terms x-ray and gamma rays are statements about origin rather than implying different kinds of radiation. In interactions with matter, both X and gamma rays are ionizing radiation and produce physiological effects which are not observed with any exposure of non-ionizing radiation, such as the risk of mutations or cancer in tissue. gamma-rays X-rays**Gamma radioactivity is composed of electromagnetic rays. It**is distinguished from x-rays only by the fact that it comes from the nucleus. Most gamma rays are somewhat higher in energy than x-rays and therefore are very penetrating. It is the most useful type of radiation for medical purposes, but at the same time it is the most dangerous because of its ability to penetrate large thicknesses of material.**Microwave Interactions**The quantum energy of microwave photons is in the range 0.00001 to 0.001 eV which is in the range of energies separating the quantum states of molecular rotation and torsion. The interaction of microwaves with matter other than metallic conductors will be to rotate molecules and produce heat as result of that molecular motion. Conductors will strongly absorb microwaves and any lower frequencies because they will cause electric currents which will heat the material. Most matter, including the human body, is largely transparent to microwaves. High intensity microwaves, as in a microwave oven where they pass back and forth through the food millions of times, will heat the material by producing molecular rotations and torsions. Since the quantum energies are a million times lower than those of x-rays, they cannot produce ionization and the characteristic types of radiation damage associated with ionizing radiation. E = (10-5 – 10-3) eV →λ = (12 – 0.12 ) cm