Maxwell Equations And Electromagnetic Radiation Pdf
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- Maxwell's Equations and Electromagnetic Waves
- James Clerk Maxwell
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Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism , classical optics , and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges , currents , and changes of the fields. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.
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They were the mathematical distillation of decades of experimental observations of the electric and magnetic effects of charges and currents, plus the profound intuition of Michael Faraday. It made evident for the first time that varying electric and magnetic fields could feed off each other—these fields could propagate indefinitely through space, far from the varying charges and currents where they originated.
Previously these fields had been envisioned as tethered to the charges and currents giving rise to them. The integral of the outgoing electric field over an area enclosing a volume equals the total charge inside, in appropriate units. The first term is integrated round a closed line, usually a wire, and gives the total voltage change around the circuit, which is generated by a varying magnetic field threading through the circuit.
Ampere discovered that two parallel wires carrying electric currents in the same direction attract each other magnetically, the force in newtons per unit length being given by. We are using the standard modern units SI. That's not quite fair—it has dimensions to ensure that both sides of the above equation have the same dimensionality.
Of course, there's a historical reason for this strange convention, as we shall see later. Anyway, if we bear in mind that dimensions have been taken care of, and just write the equation. However, after we have established our unit of current—the ampere — we have also thereby defined our unit of charge , since current is a flow of charge, and the unit of charge must be the amount carried past a fixed point in unit time by unit current.
Therefore, our unit of charge—the coulomb —is defined by stating that a one amp current in a wire carries one coulomb per second past a fixed point. To be consistent, we must do electrostatics using this same unit of charge. We must first establish the unit of charge from the unit of current by measuring the magnetic force between two current-carrying parallel wires.
Second, we must find the electrostatic force between measured charges. In fact, the ampere was originally defined as the current that deposited a definite weight of silver per hour in an electrolytic cell. We represent these small areas as vectors pointing outwards, because we can then take the dot product with the electric field to select the component of that field pointing perpendicularly outwards it would count negatively if the field were pointing inwards —this is the only component of the field that contributes to actual flow across the surface.
Just as a river flowing parallel to its banks has no flow across the banks. The second Maxwell equation is the analogous one for the magnetic field, which has no sources or sinks no magnetic monopoles, the field lines just flow around in closed curves.
Thinking of the force lines as representing a kind of fluid flow, the so-called "magnetic flux", we see that for a closed surface, as much magnetic flux flows into the surface as flows out. This can perhaps be visualized most clearly by taking a group of neighboring lines of force forming a slender tube—the "fluid" inside this tube flows round and round, so as the tube goes into the closed surface then comes out again maybe more than once it is easy to see that what flows into the closed surface at one place flows out at another.
The first two Maxwell's equations, given above, are for integrals of the electric and magnetic fields over closed surfaces. The other two Maxwell's equations, discussed below, are for integrals of electric and magnetic fields around closed curves taking the component of the field pointing along the curve.
These represent the work that would be needed to take a charge around a closed curve in an electric field, and a magnetic monopole if one existed!
However, we know that this is only part of the truth, because from Faraday's Law of Induction, if a closed circuit has a changing magnetic flux through it, a circulating current will arise, which means there is a nonzero voltage around the circuit. The equation analogous to the electrostatic version of the third equation given above, but for the magnetic field, is Ampere's law. We must now consider whether this equation, like the electrostatic one, has limited validity.
In fact, it was not questioned for a generation after Ampere wrote it down: Maxwell's great contribution, in the 's, was to realize that it was not always valid. A simple example to see that something must be wrong with Ampere's Law in the general case is given by Feynman in his Lectures in Physics. Suppose we use a hypodermic needle to insert a spherically symmetric blob of charge in the middle of a large vat of solidified jello which we assume conducts electricity.
Because of electrostatic repulsion, the charge will dissipate, currents will flow outwards in a spherically symmetric way. Question : does this outward-flowing current distribution generate a magnetic field?
The answer must be no , because since we have a completely spherically symmetric situation, it could only generate a spherically symmetric magnetic field. But the only possible such fields are one pointing outwards everywhere and one pointing inwards everywhere, both corresponding to non-existent monopoles. So, there can be no magnetic field.
Obviously, the left hand side of Ampere's equation is zero, since there can be no magnetic field. It would have to be spherically symmetric, meaning radial. On the other hand, the right hand side is most definitely not zero, since some of the outward flowing current is going to go through our circle. So the equation must be wrong. Ampere's law was established as the result of large numbers of careful experiments on all kinds of current distributions.
So how could it be that something of the kind we describe above was overlooked? The reason is really similar to why electromagnetic induction was missed for so long. No-one thought about looking at changing fields, all the experiments were done on steady situations. With our ball of charge spreading outward in the jello, there is obviously a changing electric field.
Imagine yourself in the jello near where the charge was injected: at first, you would feel a strong field from the nearby concentrated charge, but as the charge spreads out spherically, some of it going past you, the field will decrease with time.
Recall, however, that we defined the current threading the path in terms of current punching through a soap film spanning the path, and said this was independent of whether the soap film was flat, bulging out on one side, or whatever. With a single infinite wire, there was no escape— no contortions of this covering surface could wriggle free of the wire going through it actually, if you distort the surface enough, the wire could penetrate it several times, but you have to count the net flow across the surface, and the new penetrations would come in pairs with the current crossing the surface in opposite directions, so they would cancel.
Once we bring in Maxwell's parallel plate capacitor, however, there is a way to distort the surface so that no current penetrates it at all: we can run it between the plates! The question then arises: can we rescue Ampere's law by adding another term just as the electrostatic version of the third equation was rescued by adding Faraday's induction term?
Ampere's law can now be written in a way that is correct no matter where we put the surface spanning the path we integrate the magnetic field around:. Notice that in the case of the wire, either the current in the wire, or the increasing electric field, contribute on the right hand side, depending on whether we have the surface simply cutting through the wire, or positioned between the plates. Actually, more complicated situations are possible—we could imaging the surface partly between the plates, then cutting through the plates to get out!
In this case, we would have to figure out the current actually in the plate to get the right hand side, but the equation would still apply. Maxwell referred to the second term on the right hand side, the changing electric field term, as the "displacement current".
This was an analogy with a dielectric material. If a dielectric material is placed in an electric field, the molecules are distorted, their positive charges moving slightly to the right, say, the negative charges slightly to the left. Now consider what happens to a dielectric in an increasing electric field. The positive charges will be displaced to the right by a continuously increasing distance, so, as long as the electric field is increasing in strength, these charges are moving: there is actually a displacement current.
Meanwhile, the negative charges are moving the other way, but that is a current in the same direction, so adds to the effect of the positive charges' motion. Maxwell's picture of the vacuum, the aether, was that it too had dielectric properties somehow, so he pictured a similar motion of charge in the vacuum to that we have just described in the dielectric.
The picture is wrong, but this is why the changing electric field term is often called the "displacement current", and in Ampere's law generalized is just added to the real current, to give Maxwell's fourth—and final—equation. Our mental picture here is usually of a few thin wires, maybe twisted in various ways, carrying currents. More generally, thinking of electrolytes, or even of fat wires, we should be envisioning a current density varying from point to point in space.
The question then arises as to whether the surface integral we have written on the right hand side above depends on which surface we choose spanning the path. Obviously, in a situation with steady currents flowing along wires or through conductors, with no charge piling up or draining away from anywhere, this is zero. Because two different surfaces spanning the same circuit add up to a closed surface. As a preliminary to looking at electromagnetic waves, we consider the magnetic field configuration from a sheet of uniform current of large extent.
Think of the sheet as perpendicular to this sheet of paper, the current running vertically upwards. It might be helpful to visualize the sheet as many equal parallel fine wires uniformly spaced close together:. The magnetic field from this current sheet can be found using Ampere's law applied to a rectangular contour in the plane of the paper, with the current sheet itself bisecting the rectangle, so the rectangle's top and bottom are equidistant from the current sheet in opposite directions.
This is the magnetostatic analog of the electrostatic result that the electric field from an infinite sheet of charge is independent of distance from the sheet. In real life, where there are no infinite sheets of anything, these results are good approximations for distances from the sheet small compared with the extent of the sheet.
We will assume that sufficiently close to the sheet, the magnetic field pattern found above using Ampere's law is rather rapidly established. Let us now apply Maxwell's equations to this guess to see if it can make sense. We are forced to conclude that for Maxwell's fourth equation to be correct, there must also be a changing electric field through the rectangular contour. Let us now try to nail down what this electric field through the contour must look like.
First, it must be through the contour, that is, have a component perpendicular to the plane of the contour, in other words, perpendicular to the magnetic field. In fact, electric field components in other directions won't affect the fourth equation we are trying to satisfy, so we shall ignore them. Therefore, the right hand side of the equation must also be zero. Note that, unlike the magnetic field, the electric field must point the same way on both sides of the current sheet, otherwise its net flux through the rectangle would be zero.
It's spreading both ways, hence the 2. But we have another equation linking the field strengths of the electric and magnetic fields, Maxwell's third equation:. This contour is all on one side of the current sheet. This is how Maxwell discovered a speed equal to the speed of light from a purely theoretical argument based on experimental determinations of forces between currents in wires and forces between electrostatic charges. This of course led to the realization that light is an electromagnetic wave, and that there must be other such waves with different wavelengths.
Hertz detected other waves, of much longer wavelengths, experimentally, and this led directly to radio, tv, radar, etc. Here are the equations: 1. A Sheet of Current: A Simple Magnetic Field As a preliminary to looking at electromagnetic waves, we consider the magnetic field configuration from a sheet of uniform current of large extent. It might be helpful to visualize the sheet as many equal parallel fine wires uniformly spaced close together
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They were the mathematical distillation of decades of experimental observations of the electric and magnetic effects of charges and currents, plus the profound intuition of Michael Faraday. It made evident for the first time that varying electric and magnetic fields could feed off each other—these fields could propagate indefinitely through space, far from the varying charges and currents where they originated. Previously these fields had been envisioned as tethered to the charges and currents giving rise to them. The integral of the outgoing electric field over an area enclosing a volume equals the total charge inside, in appropriate units. The first term is integrated round a closed line, usually a wire, and gives the total voltage change around the circuit, which is generated by a varying magnetic field threading through the circuit. Ampere discovered that two parallel wires carrying electric currents in the same direction attract each other magnetically, the force in newtons per unit length being given by. We are using the standard modern units SI.
The physicist James Clerk Maxwell in the 19th century based his description of electromagnetic fields on these four equations, which express experimental laws. The most compact way of writing these equations in the metre-kilogram-second mks system is in terms of the vector operators div divergence and curl. Maxwell's equations Article Additional Info. Print Cite verified Cite. While every effort has been made to follow citation style rules, there may be some discrepancies.
(x,t)ˆk. Figure Electric and magnetic fields at a few selected points along the x -axis associated with a plane electromagnetic wave.
Maxwell's Equations and Electromagnetic Waves
Войдя, Дэвид увидел мигающую лампочку автоответчика. Слушая сообщение, он выпил почти целый пакет апельсинового сока. Послание ничем не отличалось от многих других, которые он получал: правительственное учреждение просит его поработать переводчиком в течение нескольких часов сегодня утром. Странным показалось только одно: об этой организации Беккер никогда прежде не слышал. Беккер позвонил одному из своих коллег: - Тебе что-нибудь известно об Агентстве национальной безопасности.
James Clerk Maxwell
Он напал на. - Что. Этого не может .
- За счет заведения. Превозмогая шум в голове, Беккер представил себе грязные улицы Трианы, удушающую жару, безнадежные поиски в долгой нескончаемой ночи. Какого черта. Он кивнул.
Maxwell's Equations for. Electromagnetic Waves. Vector Operations. Any physical or mathematical quantity whose amplitude may be decomposed into.
Халохот быстро осмотрел стодвадцатиметровую башню и сразу же решил, что прятаться здесь просто смешно. Наверняка Беккер не настолько глуп. Единственная спиральная лестница упиралась в каменную камеру квадратной формы, в стенах были проделаны узкие прорези для обозрения, но, разумеется, никакого выхода он не. Дэвид Беккер поднялся на последнюю крутую ступеньку и, едва держась на ногах, шагнул в крошечную каменную клетку. Со всех сторон его окружали высокие стены с узкими прорезями по всему периметру.
В центре помещения из пола торчала, подобно носу исполинской торпеды, верхняя часть машины, ради которой было возведено все здание. Ее черный лоснящийся верх поднимался на двадцать три фута, а сама она уходила далеко вниз, под пол. Своей гладкой окружной формой она напоминала дельфина-косатку, застывшего от холода в схваченном морозом море. Это был ТРАНСТЕКСТ, компьютер, равного которому не было в мире, - шифровальная машина, засекреченная агентством. Подобно айсбергу машина скрывала девяносто процентов своей массы и мощи под поверхностью. Ее секрет был спрятан в керамических шахтах, уходивших на шесть этажей вниз; ее похожий на ракету корпус окружал лабиринт подвесных лесов и кабелей, из-под которых слышалось шипение фреоновой системы охлаждения.
Сзади его нагоняло такси. Он смотрел на приближающиеся огни центра города и молил Бога, чтобы он дал ему добраться туда живым. Беккер проехал уже половину пути, когда услышал сзади металлический скрежет, прижался к рулю и до отказа открыл дроссель. Раздался приглушенный звук выстрела.
Четыре. Три. Эта последняя цифра достигла Севильи в доли секунды. Три… три… Беккера словно еще раз ударило пулей, выпущенной из пистолета.
Четыре года назад конгресс, стремясь создать новый стандарт шифрования, поручил лучшим математикам страны, иными словами - сотрудникам АНБ, написать новый супералгоритм. Конгресс собирался принять закон, объявляющий этот новый алгоритм национальным стандартом, что должно было решить проблему несовместимости, с которой сталкивались корпорации, использующие разные алгоритмы.
Сьюзан понимала: как только Хейл заподозрит, что она искала что-то в его компьютере, то сразу же поймет, что подлинное лицо Северной Дакоты раскрыто. И пойдет на все, лишь бы эта информация не вышла из стен Третьего узла. А что, подумала Сьюзан, если броситься мимо него и побежать к двери. Но осуществить это намерение ей не пришлось.
Сигналы продолжались. Источник их находился где-то совсем близко. Сьюзан поворачивалась то влево, то вправо. Она услышала шелест одежды, и вдруг сигналы прекратились. Сьюзан замерла.
Нет, а-а… нет, спасибо, сэр. - Ему трудно было говорить - наверное потому, что он не был уверен, что его появлению рады. - Сэр, мне кажется… что с ТРАНСТЕКСТОМ какая-то проблема. Стратмор закрыл дверцу холодильника и без тени волнения взглянул на Чатрукьяна. - Ты имеешь в виду работающий монитор.
На завтрашний день, пожалуйста. - Ваш брат Клаус приходил к нам? - Женщина вдруг оживилась, словно говорила со старым знакомым. - Да. Он очень толстый.
Она инстинктивно отпрянула назад, застигнутая врасплох тем, что увидела. Из-за решетчатой двери кухни на нее смотрели. И в тот же миг ей открылась ужасающая правда: Грег Хейл вовсе не заперт внизу - он здесь, в Третьем узле.
Или это ненависть. Они буквально пожирали ее тело. Новая волна паники охватила Сьюзан. Хейл всей тяжестью своего тела придавил ее ноги, холодно следя за каждым ее движением.