Electromagnetic radiation ----------------------- In 1864 Maxwell formulated his theory of electromagnetic radiation (EMR), tying together the then separate studies of electricity and magnetism. Maxwell believed a real medium was necessary for conveying the EMR. Later, by Einstein, such requirement was proved false. EMR can travel thru empty space and it is by it that we acquire virtually all intelligence about the universe. The amount of knowledge we get by meteorites, cosmic particles, Moon rocks, solar wind, and the like is minuscule against the information from the EMR. Or, in reverse logic, with no EMR our awareness of the universe would be pretty much zilch. Frequency versus Wavelength ------------------------- We employ either wavelength or frequency to describe EMR. lambda is the symbol for wavelength; nu, frequency. Letter f is also commonly used for frequency. The two are interchangeable at will by c = lambda * nu +----------------------+ | FREQUENCY-WAVELENGTH | | RELATION | | | | c = lambda * nu | +----------------------+ The metric name for 'cycle per second' is 'hertz' but many older scientists still use cycles/second. Some works actually use 'sec^-1' to clarify this idea, like '92e6 sec^-1'. The metric unit of wavelength is a multiple of the meter like nanometer in the optical range or micrometer in the infrared zone. The micrometer is prevalently stated by its older na,e 'micron'. Because it's clumsy to write the formal symbol for 'micro-' as the letter 'mu ', the letter 'u' is sometimes used. It kind of looks like a 'mu'.. In astronomy an older unit, 1e-10 meter or angstrom is still in wide circulation. 1 angstrom is 1/10 nonometer. The choice to use is often a matter of tradition. Some regions of EMR were explored as a function of wavelength and others of frequency. Radiowaves are generally described in frequency while light is cited in wavelength. We can use either one as we like for any region of electromagnetic radiation. The EMR comprehended between two limits of wavelength (or frequency) is called a spectrum, after the visual apparition discovered by Newton in his experiments on light. He saw, when letting sunlight refract thru a glass prism, a splay, devolution, dispersion of colors. Much later these were found to be the multitude of wavelengths represented in sunlight, each refracted slightly differently thru the prism. Our eyes register each wavelength as a separate color. Irregular jargon -------------- along with the separate work in the various wavebands of electromagnetic waves comes the inconsistent jargon describing the waves. altho I here use the International system of units and terms, you will read and hear other terms, words, units of measure. The main alternative measurement system is the CGS, 'centimeter- gram-second. Irs units are still in wide use, mainly in specialized sciences. The units are mostly powers of ten multiples of SI units, making translation reasonably easy. Terminology, names, nomenclature is more varied, with terms in one waveband having different meaning in an other. You do have to study the jargon when working across wavebands.A common instance is the designation of wavebands. There are no official boundaries between the zones of wavelength, except for certain ones regulated by government agencies, such as broadcast radio. It happens that one scientists calls the band he works with 'far ultraviolet' but an other scientist in his own work calls it 'soft X-ray'. An other situation is that many combinations of units, in both CGS and SI, have honorary names, like 'joule' for 'newton.meter' and 'volt' for 'newton.meter/coulomb'. 'newton' is itself the honorary name for 'kilogram.meter/second2'. Most of these names honor scientists who worked with associated principles, effects, concepts. Electric charge ------------- The elemental unit of electric charge is the electron. Irs charge by history is 'negative' of strength one unit. The proton also carries one unit of charge, this being 'positive'. charges of like signum repel and those with opposite signum attract. In ordinary matter the protons and electrons are equal in number, netting the overall charge to zero. The protons are locked in atomic nuclei and are not easily available for useful work. The electron can move around among atoms to accumulate static charge or become an electric current. A body taking on a negative charge has an excess number of electrons. The protons in the target are too few to net out all the electrons, leaving the extra ones to give the body its net negative charge. A positively charged body has a decess of electrons, leaving the extra nuclear protons to exert a net positive charge. Transferring electrons to and from an object was primitively done by rubbing the object with silk or fur. Certain materials gained electrons to be negatively charged. Other lost electrons to be positively charged. This effect is the practice of electrostatics or static electricity, still in routine use for entertainment. A charged body creates an electric field around it that presses an force on test particles in the field. Priestley in 1766 experimented with charged metal cans. He found that within the can there is no electric field and correlated this fact with gravity. Within a hollow vessel there is no gravity field from the vessel's own mass. This effect comes from the inverse-square law. of gravity. Charges must also follow the same rule. Coulomb in 1785 built a torsion balance, like the one from Cavendish for gravity, to determine the force relation directly. Coulomb verified Priestley's work and found that force = K * charge1 * charge2 / distance^2 charge1 and charge2 are the two charges, with their positive or negative signa. If their product is positive (+ times + or - times -) the force is repulsion. A negative product (+ times -) is an attraction. The unit of charge was is primitively the electron but this is a in infinitesimal unit. The SI unit is the Coulomb, equal by experiment and measurement to 1 coulomb = 6.4518e18 electrons K is the Coulomb constant like the Newton constant.It was a single number under older systems of units. In the SI it is diffracted into K = 1 / (4 * pi * epsilon) to coordinate with other electromagnetic units. epsilon is a property of the neduyn carrying the electric field. Today we start with the definition of electric current, with unit ampere. The electric current flowing thru an electrical device is marked on device's nameplates on electric devices as a flow of electricity thru the device. The ampere is 1 ampere = 1 coulomb/second Batteries are sometimes specified by thee ampere.hours of charge they hold. With 3600 seconds per hour, 1 ampere.hour = 3,600 coulomb Before the electron was discovered as the sole charge carrier in electricity, we spoke of the electric fluid flowing from the positive charge to the negative charge. This convention embedded into the nascent 19th century electrical industry and persists today. Electricity flows from the positive end of a battery to the negative end. The flow of electrons ended up by luck being the opposite flow, from negative to positive. This is part of electronics, developed in the early 20th century. Electric field ------------ An electric charge has around it an electric field. It is similar to the gravity field around a mass. The electric field of a point charge follows Coulomb's inverse-square law. force = (1 / (4 * pi * epsilon)) * charge1 * charge2 / distance^2 In a way analogous to Newton's gravity field strength, a charge q1 sets up an electric field of strength E at distance r (field strength) = (1 / (4 * pi * epsilon)) * charge1 / distance^2 epsilon for a vacuum has the smallest value of all media and is given the symbol epsilon0. Its value is epsilon0 = 8.854e-12 C2/N.m2. The 4*pi factor is an effort to simplify calculations like for illumination. It didn't work as hoped but we're stuck with it. +-----------------------------------------------+ | COULOMB'S INVERSE SQUARE LAW | | | | F is force, E is field strength, q is charge, | | r is distance | | | | F = (1 / (4 * pi * epsilon)) * q1 * q2 / r^2 | | | | E = (1 / (4 * pi * epsilon)) * q1 / r^2 | | | | epsilon0 = 8.854e-12 C2/(N*m2) | +-----------------------------------------------+ Magnetic poles and fields --------------------- It turns out that there is little need for a distinct parameter of a magnetic pole strength. This would be symmetrical with the unit of electric charge or of mass. Because wo can not actually have separate magnetic poles and the whole purpose of magnets is to build a magnetic field in a given volume, work is performed with just the magnetic field strength. An other reason is that magnetic fields can be created via electromagnets. There generally are no 'poles' in the ordinary sense. In Ampere's law the magnetic field surrounds th wire in closed loops. There is no actual 'poles'. In history, Michell in 1750 did discover the inverse- square law for magnetic poles. His equation parallels those for gravity and light. (Priestley and Coulomb were still in the future.) magnetic force = M * pole1 * pole2 / distance^2 where M is the Michell constant. No value is standardized for it because we hardly ever use this inverse-square formula. pole1 and pole2 are magnetic poles, north or south. Like poles repel; opposite, attract. From the absence of separate magnetic poles, there is no standard value for a pole strength. It could be feasible to experiment with Michell's law with a long bar magnet and keeping close to each end of it. This dilutes the influence of the other end on the force. The magnetic field strength is found by the action of the field on a test electric current. In a lab setup of specified geometry and operation within the magnetic field. The force on the apparatus is measured. We have (magnetic field strength) = func(force, current, geometry) The result of such an experiment is that the magnetic field strength is in newton/(ampere.meter). Note this incorporates the main factors defining field strength: newton for force, ampere for current, meter for geometry. This is hardly symmetrical with the newton/coulomb units for the electric field strength. Given the lack of separable magnetic poles we are compelled to go thru this round-about definition. The units newton/(ampere.meter) is the tesla. The CGS unit, gauss, is still widely use. 10,000 gauss equal one tesla. The Earth's magnetic field strength at ground level is 1/2 to 1 gauss or 5e-5 to 1e-4 tesla. Aren't we engaged in circular reasoning? We first used a magnetic field to define the ampere, the coulomb/second. Then we use the ampere to define the magnetic field! In a way, yes, and philosophers debate this feature of physics endlessly. For one thing it leaves us with no explicit quantity for the strength of a magnetic pole! Ampere and Fraday --------------- In the early 1800s there were hints that electricity and magnetism are somehow linked. Oersted and also Ampere in 1820 discovered that an electric current can produce a magnetic field. Connect a wire across a battery, source of electric current, with a small lamp bulb, like fro a pocket torch. The lamp shows that current is flowing and prevents the battery from discharging by a short circuit thru the wire. Put a scouting or hiking compass near the wire. The compass deflects away from its normal alignment to magnetic north. Every where along the wire the compass reacts this way. The electric current generated around the wire a magnetic field that attracts the compass needle. When the circuit is opened, the current stops, the magnetic field vanishes. The compass returns to its natural alignment. In words Ampere's law is (electric current)_ -> (magnetic field) where '->' means 'can produce or generate'. Faraday in 1838 found that a varying magnetic field can produce an electric current. Connect the wire to the lamp bulb this time with no battery..Sweep one end of a bar magnet over the wire.The motion of the magnet changes its field along the wire. As long as the magnetic field is varying, bu continuing to move the magnet. an electric current flows in the wire and lights the bulb. The bulb may glow weakly but a stronger glow is obtained by more vigorously sweeping the magnet. . When the magnet is removed or held still, the current stops. The galvanometer shows zero flow. A verbal statement of Faraday's principle is (varying magnetic field) -> (electric current) Note carefully that Ampere and Faraday laws are not symmetrical. While a steady electric current produces a magnetic filed by Ampere, it takes a changing magnetic field to make an electric current for Faraday. A steady field does not yield electricity. Faraday's principle quickly leaded to the invention of the electric dynamo or generator. Until then elecctricity was made by chemical batteries or by static charge. In a dynamo a coil of wire is rotated within the poles of strong magnets. As the coil rotates it 'sees' a varying magnetic field and a current is impressed in it. This current is drawn off to run machinery. Such dynamos were built by the 1860s for on-site production of electricity, like in factories and ships. The coil was rotated by steam engines or water wheels. By way of history, Edison did not 'invent electricity'. Edison was the first to commercialize electricity for anyone to obtain its benefits. In doing so he invented the electric service industry. Edison's original company from 1882 is today's Consolidated Edison Company of New York. The discoveries by Faraday and Ampere was elaborated into mathematical expressions, skipped here, as new laws of physics. Lenz effect --------- If under Faraday's law a magnetic field induces an electric current, doesn't that current in turn under Ampere's law set up a magnetic field? If so, can these two fields combine to produce more current? And we could bleed off some of this current to run our machinery without further input of energy to the wires! Is this a perpetual motion machine in the rough? No. The resultant field set up by the electric current opposes the stimulant field to temper the rate of change. The fields are antagonistic, not sympathetic. You may experience the Lenz effect with the wire and lamp. With the circuit open you can freely sweep the magnet over the wire. When the circuit is closed you feel resistance against the magnet. This is the Lenz field opposing the magnet. This is way a dynamo needs a continuous external force to keep the magnet-coil moving and maintain the output of electricity. Lines of Force ------------ A line of force associated with a field is not a real physical feature in nature. Older texts sometimes spoke of lines of force as if they were actual traces thru a field. None the less they are an excellent aid for visualizing fields and they in fact do have a physical foundation. A line of force is the trajectory of a test particle placed in the field and released to move under the force of the field. It's like a freefall trajectory in astrodynamics. There is no initial velocity of the particle. The particle is gently let go in the field, not or injected or propelled. For an electric filed we use a positive charge, like a proton. When released in the field it is repelled from other positive charges and attracted to other negative charges. The path of this particle is a line of force. In principle you can actually map out the field. The direction of motion of the test particle is commonly indicated in diagrams by arrows along the line. We can speak of a line 'going from positive to negative' as a jargon, not a statement of real movement of the line itself. The number and density of lines of force qualitatively show the strength of the field. More and denser lines indicate a field stronger than where the lines are spread apart. The number of lines emanating from or cascading into a charge qualitatively is the strength of the charge. Stronger charges have more lines associated with them. We can not actually place a magnetic pole into a magnetic field and watch how it moves. There is no isolated pole, like an electric charge. Poles always come in paris, north and south together, they cancel out in the field and don't move under its force. In thought experiment we place a north pole in the field and trace out its trajectory. This is the magnetic line of force. Arrows along it in diagrams show the motion of the test pole. Gauss's Law for charge -------------------- Imagine a spherical shell enclosing some electric charges, a mix of positive and negative ones. They may be scattered inside the shell, not bunched at the center. For simplicity, each charge has the same strength and is assigned one line of force. The force line from a positive charge aims outward. That from a negative charge points inward. These are the directions of motion of a positive test particle placed on the line. All lines of force pass thru the enclosing sphere, some inward, some outward. Tally the in and out lines, algebraicly, over the sphere. Each inward line offsets a outward line, leaving leftover lines of one polarity. Toss the offfset pairs and keep the leftover lines. Each pair of lines represents a positive-negative pair of charges inside the sphere. Any left over lines, with no paired opposite ones, are attached to extra positive or negative charges in the shell. Gauss's law for charges states that on an enclosing sphere with charges within it, the algebraic count of lines of force, the electric field, all over its surface equals the net electric charge within the sphere. Gauss worked this out in 1835 for both electric charges and magnetic poles. Gauss's law itself can not tell the separate number of charges, only the residual unpaired ones. You could have 20 gazillion electrons and 20 gazillion and two protons. You 'see' only the two extra PROTONS. In words: (electirc lines thru sphere) = (net charges inside sphere) The sphere does not have to be a sphere. It may be any shape so long as it is closed, ensuring that lines of force from inside must pass across its surface and not thru a rip, tear, other opening. It may have folds! A line passing thru one ply of a fold in one direction, passes an other ply in the opposite direction. Eventually it leaves the enclosure and is then counted. The spherical shape is routinely used because of its simple geometry and maths. Gauss's Law for poles ------------------- Repeat the setup of the enclosing sphere with magnets inside. Let each north or south pole emit one line of force. The north pole lines point outward; south pole, inward. Tally the lines passing the sphere. pair off the in and out lines and toss them and keep any left over unpaired lines. Some north poles pair with south poles and their lines cancel in the tally. We find that ALL lines papir with NO left over ones! No matter what mix of magnets are in the sphere, every un line is offset by an out line, leaving none as a net excess. Gauss's law of poles says that the net count of lines, the magnetic field, is always zero over the sphere. The reason is that so far as we know there are no lone magnetic poles. EVERY pole has its mate of equal strength but opposite polarity. EVERY enclosing sphere contains EXACTLY equal numbers of north and south poles and the net magnetic field on the sphere is ZERO. Like for Gauss's law for charges the separate number of poles inside the sphere is not determined. Only the excess of north or south poles can be found, which is always none. This in words is: (magnetic lines thru sphere) = (zero net poles inside sphere) Maxwell's Equations ----------------- In 1864 Maxwell issued his theory of electromagnetism, uniting the two separate disciplines into a new single construct, electromagnetic wave or radiation. He demonstrated that all of the phaenomena of electricity and magnetism can be reduced to four fundamental equations. They are elaborations of Gauss's law for charge, Gauss's law for poles, Faraday's law, Ampere's law. Maxwell's equations in words are: +----------------------------------------------------------------+ | MAXWELL'S FOUR EQUATIONS OF ELECTROMAGNETISM | | | | electric field on Gauss sphere = net electric charge in sphere | | | | magnetic field on Gauss sphere = zero magnetic poles in sphere | | | | changing magnetic field around wire -> elec current in wire | | electric current in wire -> magnetic field around wire | + ---------------------------------------------------------------+ These equations divide into two groups. The Gauss equations relate to poles and charges at rest. They cover electrostatics and magnetostatics.They deal with each field separately without commingling them. The Ampere and Faraday equations relate to charges in motion or magnetic fields in motion. They deal with electrodynamics and magnetodynamics. Gauss's Law for mass ------------------ We divert to examine an other Gauss sphere and the field on it. This is sometimes left out of home astronomy dialog on cosmology but it is one of its most fundamental principles. Same Gauss sphere with some particles of mass inside. Each particle has a single gravity line of force. Because gravity is an attracting force, a small test mass moves inward along the line. Tally the lines over the sphere, netting the inward oneS with outward ones. We find that ALL lines are inward! There are no outward lines. Gauss's law for mass states that the count of lines, the gravity field, over the sphere is the TOTAL mass within the sphere. Not a net of 'positive' and 'negative' mass but ALL of it as only 'positive'. Entirely unlike Gauss spheres for charges and poles, we can add only mass of the one kind and increase the count of lines on the sphere. There is no way to add mass that cancels out some lines, like for charges, or all of the them, like for poles. Cosmology is often a part of relativity. In such work gravity is the force that governs the behavior of the universe on the larger scales of volume. There are vast magnetic and electric fields in space. They are crucial for the study of nebulae, stars, galaxies. Astrophysicists need a solid grounding in electromagnetic theory. look again at the three Gauss's laws. On the scale of many millions of lightyears, like within a galaxy cluster, there are inside a Gauss sphere immense numbers of charges, poles, particles. Because every magnetic pole has a matching opposite pole, there is no imbalance of poles inside the sphere. no net magnetic poles. The force lines thru the sphere cancel to zero. Altho locally, within stars or nebulae, magnetic forces are important, on the grand universal scale they are absent. There are individual charges, such as electrons and protons, in space. Within stars and nebulae we have humongouss flows of protons or electrons separate from each other. A Gauss shell around these local volumes can contain an unbalanced number of charge. On the larger scale it is excedingly tough to sustain charge segregation. The forces between opposite charges is incredibly strong, overcoming other natural forces that try to keep the charges apart. A Gauss sphere tends to enclose less and less imbalance of charge, more and more equal numbers of positive and negative charges. The electric field and its force tend to zero. Looking at enclosed mass we have a whole different situation. There being only one kind of mass, a Gauss sphere of ANY size will contain an imbalance of mass, all 'positive'. The larger the sphere the more mass it embraces as the sphere ropes in ever larger volumes of space. The gravity field on its surface also increases, this being the ever greater count of gravity lines of force from the interior atoms. On the scale of the whole universe the only governing force is gravity. Cosmology is truly driven by the action of the mass, and not the poles and charge, within the universe. Maxwell's waves -- --------- When Maxwell studied the four cardinal formulae, he found that they are properties of single new entity, a wave of perpendicular interacting electric and magnetic fields. We examine some major features of this new wave. A general wave function, such as a mechanical wave,, looks like d^2(amplitude) / dx^2 = (dt^2 / dx^2) * (d^2(amplitude) / dt^2) The amplitude of the wave is the displacement of the wave perpendicular to the direction of the wave's motion. It is a math expression, like a sine function. The 'd's are part of the symbols for 'derivative' in calculus, which we won't actually handle as such. The left side relates amplitude to the downrange place along the direction of wave motion, or the x-axis. The right side relates amplitude to the time elapsed since the wave started from the zero point of the x-axis. In his investigations of the interaction of electric and magnetic fields Maxwell developed an expression for the strength of the electric field d^2(strength) / dx^2 = mu * epsilon * d^2(strength) / dt^2) He compared this with the general wave equation and matched the terms. The amplitude in the general case becomes the electric field strength. The dt^2/dx^2 becomes (mu*epsilon). These two parameters are the ability of the medium, in which the wave travels, for sustaining a magnetic (for mu) or electric (for epsilon) field. Some substances let a magnetic field penetrate them well while others are good shields against the field. This permeability of the medium is expressed by mu. A similar reasoning applies to an electric field. epsilon is the permittivity of the medium. Before Maxwell, these parameters were experimentally measured for a huge assortment of materials as basic properties of these materials. When the material was a vacuum the mu and epsilon are denoted epsilon0 and mu0 is used. mu0 and epsilon0 -------------- The Maxwell's wave formula we see the matching of terms and letting the medium be a vacuum. dt^2 / dx^2 = epsilon0 * mu0 = (dt / dx)^2 = 1 / speed^2 speed^2 = 1 / (epsilon0 * mu0) speed = sqrt(1 / (epsilon0 * mu0)) When Maxwell plugged in the a;ready known values for mu0 and epsilon0 he came up with a speed of his electromagnetic waves. It matched virtually exactly the a;ready known speed of light! From his other work with electromagnetic waves, he already explained optical effects, including refraction of light across media. In 1876 Maxwell revealed that light was not a distinct phaenomenon in nature. Light was an electromagnetic wave behaving according to his theory. we have here three basic parameters, all historicly determined by experiment by three different schools of physics. The speed of light was mechanicly measured by physicists dealing with light. mu0, the permeability of a vacuum, was measured by scientists working with magnets. epsilon0, the permittivity of a vacuum, was measured by scientists in the electricity discipline. The values of all three were published in the physics litterature. Values for epsilon0 and mu0 are now folded into the metric system of measures as +----------------------------------------------+ | ELECTROMAGNETIC CONSTANTS | | | | c = 2.998e8 meter/second | | | | mu0 = 4*pi*1e-7 newton/ampere2 | | = 12.567e-7 newton/ampere2 | | | | epsilon0 = 8.854e-12 coulomb2/newton.meter2 | +----------------------------------------------+ Where's the medium? -----------------In Maxwell's time light was believed to travel thru a physical medium, like other waves then known. altho a vacuum contained nothing, it just had to be filled with a substance, the aether, that conveyed light thru outer space. c, the speed of light in vacuum, was the speed of light thru this aether. Attempts to detect the aether were tried in the 1880s. The first was the Michelson-Morley experiment to measure the speed of light at various directions from the Earth's orital motion. The thinking was the light would travel faster or slower as Earth moved into or away from a beam of light, like a boat's speed is found by tracking a floating object thrown forward and then rearward. All attempts failed to yield changes in c with orientation. c was the same with one nasty interpretation being that Earth did not move, but in fact stood still in space. We don't take up relativity here but we can appreciate how lightspeed can be a constant for every one. It comes from two physical properties of nature, epsilon0 and mu0. c is an algebraic combination of these parameters, tehmselfs constants. The properties of a vacuum don't change with motion of the observer. Physicists wrestled with this situation with many curious explanations. In 1905 Einstein showed that light did not need an actual medium and c was the same for all observers. From this realization he developed the theory of special relativity. Refraction of Light ----------------- Maxwell found that he could explain all the phaenomena of light which were previously known only thru experiment and empirical rules. For rxample, Snellius in 1621 figured the law of refraction. Until thrn trgtavtion wwas a tabulated function. Ptolemaeus in about 130 AD first compiled tables of refraction. Ptolemaeus, lacking good instruments and trigonometry, could not come up with a proper analysis. Snellius's law is n1 * sin(a1) = n2 * sin(a2) a1 is the angle of the light from medium 1 into medium 2. a2 is the angle of the light in medium 2. Both are measured from the orthogonal on the boundary of the media at the entry. point. n1 and n2 are empiricly determined properties of medium 1 and medium 2, the index of refraction. It expresses the 'strength' of a medium for bending the path of light thru it. A larger index implies that light is more steeply bent than a medium with a smaller index. The index is a relative one, banking off of a base medium as unity. For most earthly work, the base medium is air, with n - 1.000. For optics nuilt for use in outer space, the vacuum is the medium as 1.000 and air is a bit more. The refraction index varies with wavelength, which makes possible prism spectrometers, but here we ignore this feature. In the diagram here the boundary between the two media is the vertical line '#'. The horizontal line is the perpendicular at the entry point of the light ray. The ray comes from the left in medium 1, index n1, at an angle of incidence a1. It enters medium 2, index n2. It continues in medium 2 with angle of refraction a2,. # / (n1) # / ) # / a2 ) - - - - - -#- - - - - - - - ( a1 /# ( / # (n2) / # Maxwell thru his EMW theory worked out that refraction across two media is sqrt(1 / (mu1*epsilon1))) * sin(a1) = which is the same form as the empirical Snellius law. Matching terms from Snellius and Maxwell, we see that sqrt(1 / (mu1*epsilon1))) = n1 sqrt(1 / (mu2 * epsilon2)) = n2 From the wave equation sqrt(1 / (mu1*epsilon1)) = speed1 sqrt(1 / (mu2*epsilon2)) = speed2 These are the velocities of light in each medium. speed1 * sin(a1) = speed2 * sin(a2) Because medium 1 is prevalently vacuum or air, speed1 is c itself. c * sin(a1) = speed2 * sin(a2) c / speed2 = sin(a2) / sin(a1) speed2 / c = n1 / n2 = 1.000 / n2 -> 1 / n2 The indices of refraction are the ratios of the speed of light in vacuum (or air) to that in medium2. n1 for air or vacuum is set to unity, so the ratio is always 1/n2. The sense is that in a medium other than a vacuum light travels SLOWER than c. Maxwell by this analysis showed that refraction index is a direct feature of electromagnetic waves. And it applies to EMW of any wavelength, not just those exciting vision. Cherenkov effect -------------- The diminished speed of light within media other than a vacuum leads to a peculiar effect described by Cherenkov in about 1950. In nuclear experiments particles are accelerated to very nearly the lightspeed. When they are sent into a detection medium, such as liquid helium, they emit light in a pattern resembling the wake of a boat in water! The light LAGS the particle and is left behind! The nuclear particle is moving faster than light! This is the Cherenkov effect. The particle is moving faster than the REDUCED lightspeed within the helium. The light radiates away at the actual speed of light for that medium and slower than particle itself. No Einstein rules are broken. It remains true that no matter can excede the speed of light in vacuo, but it is also true that it can excede the slower speed of light within the medium it passes thru. Propogation of EMW ---------------- Altho Maxwell's theory explained optics and radiation with electromagnetic eaves, these waves were only a maths model. Was light the only electromagnetic wave? Can other kinds exist? Can we deliberately make an electromagnetic wave? Hertz in 1886 was the first to generate artificial electromagnetic waves. He built an electric circuit that oscillated electric and magnetic fields. he rigged up an other circuit to receive them. This was the birth of the electromagnetic industry. Today we produce RMW by electronic devices, like at broadcast transmitters. They produce oscillating electric and magnetic fields which are released into the air as EMW thru an antenna. In the sketch here '#' is the transmission house, '=' is the antenna. ++ A -- <- B <- -- C ++ -> D -> +===#-==- ===#=== -===#===+ ===#=== >>>>>>> ....... <<<<<<< xxxxxxx >>>>> ........ <<<<<<< >>>>>>> ....... >>>>>>> Scene A has the base oscillator sending electrons to the right end of the antenna and sucking them out of the left end. The positive charge there comes from the atomic protons that no longer have electrons to offset them. The separated charges produce an electric field pointing right, '>'. Scene B has the base 1/4 oscillation cycle later when it stops pumping electrons. The electrons at right end of the antenna racing left. '<-'. This current sets up a magnetic field pointing out, to you, '.' suggesting an arrow point. It pushes away, sown in the sketch, the electric field from the previous 1/4 cycle. Scene C is at the 2/4, 1/2, cycle when the base pumps electrons from the right end. This separated charge yields an electric field pointing left , '<'. It pushes the previous electric and magnetic fields away. Scene D is at 3/4 cycle of the oscillator. Pumping stops and electrons race right along the antenna, '->'. They generate a magnetic field pointing away from you, 'x', alluding to arrow feathers. This field shoves away the earlier electric and magnetic fields. After scene D the transmitter circuit begins a second cycle back at scene A. The fields travel away at lightspeed to constitute the electromagnetic wave. We now stand downrnge from the broadcast base and watch the wave pass by. The base sends out in alternation an electric and then a magnetic field in a sequence of repeating cycles. The fields are 1/4 1/4 cycle out of step. As one is cresting to its maximum strength the other is passing thru its zero strength. Stand alongside an EMW passing in front of us, we see something like this, where the wave is traveling to the right along the x-axis at speed c. This is a snapshot of several wavelengths of the EMW showing the direction of the two interacting fields. The base is off-page at the left and wave travels by us left to right. Prior segments move to the right farther downrange. ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx | | | | |<-wavelength-->| |<-wavelength-->| ^ = positive pointing electric field (arrow up) . = north pointing magnetic field (arrow point) v = negative pointing electric field (arrow down) x = south pointing magnetic field (arrow feathers) The distance between corresponding points on successive cycles is the wavelength of the EMW. This distance is NOT only that between peaks or zero points of successive waves, as some books assert. The number of wavelengths passing by us per second, traveling at speed c, is the frequency of the wave. We ,au designate this wave by wither its wavelength or frequency according as local practice. For waves sent out by AM radio stations, the frequency is in hundreds of kilohertz. In the FM band it's tens of megahertz By standing within the wave and facing into it, we experience a cyclical succession of electric and magnetic fields, shown here in steps of 1/4 cycle. ^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>> 0 1 2 3 4 5 6 7 8 9 10 11 |<----one cycle---->| |<----one cycle---->| ^ = positive pointing electric field (arrow up) < = north pointing magnetic field (arrow left) v = negative pointing electric field (arrow down) > = south pointing magnetic field (arrow right) 0, 1, 2, ... = successive quarter cycle interval. Energy of a Field ---------------- An electric field carries energy coming from the energy spent to create the field.. Without going thru the derivation, we note that given a field of strength E, the density of energy, joule/meter3, is densityE = (epsilon0 * E^2) / 2 where E is the electric field strength A parallel formula gives the energy density of a magnetic field densityB = ((1(1 / mu0) * B^2) / 2 also in joule/meter3. The electric field strength is measured in newton/coulomb, parallel to the gravity field strength in newton/kilogram. The magnetic field strength is in (newton.second)/(coulomb.meter) or newton/(ampere.meter). Recall that coulomb/second = ampere. This odd unit comes from the way we define magnetic field strength by way of an electric current. Energy in EMW ----------- As an EMW passes over an observer he sees the electric and magnetic fields in alternation. The energy densities of these fields are added together to give the total energy density of the EMW. The two fields alternate in cresting and zeroing. The energy producing the fields is exchanged between them as each field rises and falls in strength during a cycle. The sum of the two portions is constant, equal to the input energy of the generator. We have We have for an electromagnetic wave densityEMW = densityE + densityB = ((epsilon0 * E^2) / 2) + (((1 / (mu0) * B^2) / 2) = (((epsilon0 * E^2) + ((1 / mu0 ) * B^2)) / 2 This does not mean the two individual densities are always the same. Each varies as the amplitude of its field, 1/4 cycle offset from te other field. The SUM of the two is constant over the entire cycle. Also this equation deals with the energy content of the fields, NOT strengths. The two fields have entirely different field strengths and their sum over a cycle is not constant. Root Mean Square -------------- The energy equations are based on the instantaneous strengths of the electric and magnetic fields. In an EMW the two fields vary in strength from a maximum positive value to a maximum negative value in a given cycle. It is normally not practical or important to measure the fields at a given instant and work out the energy densities. If the wave has any considerable frequency, the oscillation of fields is so rapid that it looks like a steady field of equivalent strength as the real one. For most motors, heating, lighting, household applications don't need a detailed profile of the wave. They can be built to accept an equivalent steady energy flow. From long experience in engineering a fair and overall good equivalent wave energy density is sqrt(2)*(maximum). In decimals this is (0.707)*(maximum). For rough work you may use 0.7. This is the 'root mean square', or 'RMS', field energy treated as tho it were a steady, not varying, field. We have densityRMS = densityErms + densityBrms = epsilon0 * Erms^2) / 2 + ((1 / mu0) * Brms^2) / 2 = ((epsilon0 * Erms^2) + ((1 / mu0) * Brms^2)) / 2 Just about all uses of EMW work with the RMS value of its energy density, often without actually stipulating it as such. If you really mean the peak value you have to deliberately say so. By removing, for practical purposes, the varying values of electric and magnetic energy density, replacing them with the level RMS value, we have densityRMS = densityErms + densityBrms = (epsilon0 * Erms^2) / 2 + ((1 / mu0) * Brms^2) / 2 = ((epsilon 0 *Erms^2) + (((1 / mu0) * Brms^2)) / 2 = ((epsilon0 * Erms^2) + (epsilon0 * Erms^2)) / 2 =*2 * epsilon0 * Erms^2 / 2 = epsilon0 * Erms^2 We can simply add the two rms values and then equate them because the RMS value is a levelizing of the field variation into a steady uniform field. Electric Field Overwhelms ----------------------- Recall that the RNS electric and magnetic energy densities are equal, we have (epsilon0 * E^2) / 2 = ((1 / mu0) * B^2) / 2 epsilon0 * E^2 = (1 / mu0) * B^2 epsilon0 * E^2 / B^2= 1 / mu0 E^2 / B^2 = 1 / (epsilon0 * mu0) E / B = sqrt(1 / (epsilon0 * mu0)) = c This is an extraordinary result! The electric field strength is some 300 million times greater than the magnetic field strength in an electromagnetic wave. Any authors scramble this statement really good. The energy densities of the fields are the same in an EMW. The field strengths themselfs in that EMW are vastly different. What this means is that we routinely depict in textbooks an EMW as a SINGLE field, the electric field, orthogonal to the direction of propagation. In optics, for instance, a lightwave is a single oscillating field, the electric field This one-field model does help explain polarization easier than with the both fields. To see how vastly larger the electric field strength is, make a scale drawing of an EMW with the two intersecting fields. Let the peak magnetic field strength be one millimeter. What is the height of the electric field strength on the same scale? E?M = 300 million and M = 1mm, making E be, hold your hat, 300 KILOMETERS tall! You need a hell of a large paper for the drawing. Irradiation --------- When an electromagnetic wave is intercepted by a target it delivers its energy to the target. This energy can be wasted as heat or utilized to produce work. 'Work' is a very general term meaning any action useful to humans. One action crucial for astronomy is exciting the eye to produce vision. The eye pupil is the target. As the EMW flows over the target, a certain volume of it passes thru the target per second. The energy within this volume is imparted to the target as energy per unit time, or power. time. The irradiation on the target is this energy/meter3 times the speed of the wave, which is c the speed of light. irradiation = densityEMW * c Irradiation is in joule/meter2.second, or watt/meter2. The joule/second has the special name 'watt', the unit of energy flow, or power. The target has an area facing the EWMW in meter2. The total power captured is the irradiation times this area, power = irradiation * area This power is converted by the target into work. In this way electromagnetic waves allow us to accomplish useful work from a remote station without wires. Sun's irradiation --------------- the Sun's irradiation is nearly enough 1,350 watt/meter2. This is from satellite observations at the Earth's distance from the Sun. It varies with Sun distance from Earth, sunspot cycle, solar eruptions, secular changes. It is attenuated by the atmosphere to around 1,3200 W/m2 at the ground. What is the strength of the Sun's electric field? The energy density of the EMW is the irradiation divided by speed of light densityEMW = irradiation / c = (1,350 W/m2) / (2.988e8 m/s) = 4.518e-6 W.s/m3 = 4.518e-6 J/m3 The electric field strength is densityEMW = epsilon0 * E^2 E^2 = densityEMW / epsilon0 = (4.518e-6 J/m3) / (8.854e-12 C2/N.m2) = 5.103e5 J.N.m2/m3.C2 = 5.103e5 N.m.N.m2/m3.C2 = 5.103e5 N2.m3/m3.C^2 = 5.103e5 N2/C2 E = sqrt(5.103e5 N2/C^2) = 7.144e2 N/C -> 714 newton/coulomb A more familiar statement is, with the definition of the 'volt' as one joule/coulomb is newton/coulomb = newton.meter/coulomb.meter = joule/coulomb.meter = volt/meter The energy density of the electromagnetic radiation is in joule/meter3. One joule m the unit of energy or work,is itself one newton.meter. We have joule/meter^3 = newton.meter/meter^3 = newton/meter^2 which is a force over a unit area, or pressure. An incident EMW of a given energy density exerts on a target surface a pressure equal in value to that energy density. If we have the irradiation of the wave, in watt/meter for the Sun at Earth distance away the radiation pressure is pressureEMWW = densityEMW = 4.685e-6 N/m2 +-----------------------------------+ | IRRADIATION PARAMETERS OF SUN | | | | irradiation = 1.350 W/m2 | | | | elec field strength = 7.144e2 N/C | | | | energy density = 4.518e-6 J/m3 | | | | radiation pressure = 4.5185e-6 N/m2 | +------------------------------------+ -