Episode 39: Maxwell's Equations - The Mechanical Universe

[Music] when I was a student in college just as you are now I took a course in modern history and one day during this course the professor said and that brings us to the end of the 19th century well I typically raised my little hand and I said professor sir we were very respectful in those days I said sir you spent two weeks teaching us about the theory of 19th century warfare and you never said one word about the theories of James Clark Maxwell now the theory of 19th century warfare is a matter of no lasting significance well what Maxwell did affects every single day of our lives don't you think that's rather a bizarre distortion of what's important about history I guess maybe we weren't always so respectful after all well to his credit he was embarrassed by my question he did know who Maxwell was and he briefly explained to the class what he had done but the point is that the history of science is the stepchild of the historical profession most historians just ignore it because they don't understand it they're a little like the fellow who searched for his keys under the lamppost because that's where the light is even though he had lost them somewhere else well I'll tell you what I'm going to do I'm not going to say a word about the history of theories of 19th century warfare but I am going to tell you all about the beautiful theories of James Clark Maxwell [Music] elegant mathematical theories weren't the only things of beauty with which James Clark Maxwell surrounded himself there were his precious books of course and outside his well-appointed study there were the rolling hills of Glen lair the Maxwell family estate two days journey from Edinburgh and every morning even nearer to his heart there was Catherine morning James Oh funny last year opt they'd met in Aberdeen at Marischal College where he'd gone to teach in 1856 two years later James wrote his aunt K this comes to tell you that I am going to have a wife he predicted that a new bride certainly won't stop mathematics characteristic of Maxwell's scientific predictions this one proved accurate indeed of course Maxwell wasn't the first to make accurate predictions about the deep mysteries of natural forces two centuries earlier Isaac Newton had written a book on the subject but he'd had to express himself in terms of action at a distance and that idea would come to be applied not only to gravity but also to electricity and to magnetism [Music] and then in the 1830s on the heels of the Frenchmen who long and on para and especially on the trail of Hans Christian Oersted Along Came Michael Faraday he'd been increasingly drawn toward electricity and magnetism and by August of 1831 his path was reasonably clear picking up where erst had left off faraday confirmed that an electric current can produce a magnetic field and then going far beyond Faraday demonstrated that a changing magnetic flux can create an electric current and that a changing electric current in one circuit can induce current to flow in another [Music] thereafter especially in light of similar discoveries by America's Joseph Henry the principle of induction in the nature of electricity and magnetism was there for the scientific world to behold and certainly within the Maxwells Faraday's experimental researches in electricity was considered the definitive text but by the time he got hold of Faraday's work maxvill was ready to draw new conclusions an ability which he developed early on as a student at Cambridge Maxwell's mathematics teacher observed it appears impossible for him to think incorrectly on physical subjects soon after that as a graduate student he wrote a paper that gave the scientific community reason to agree it was called on Faraday's lines of force and in preparation for that work James Clark Maxwell had done his homework in the spring of 1846 Michael Faraday had penned a paper called thoughts on Ray vibrations in that prophetic publication he speculated that light is a type of vibration in the lines of force the lines of force he wrote are known to connect particles and also masses of matter together in other words in Faraday's imagination electric charges were linked together by lines of force in empty space however even though his idea was easy to see in the mind's eye it introduced an intriguing scientific question would a vibrating charge set the lines of force into vibration [Music] at this point Faraday might have pictured each line behaving as if it were a linked row of mechanical oscillators and in that case he would have seen a disturbance propagate along the row either as a longitudinal wave or as a transverse wave but no matter how they might have taken shape in his mind the significance of those vibrations was in the fact that Faraday had seen them and somewhere between the lines of Michael Faraday's thoughts on wave vibrations James Clark Maxwell would discover his electromagnetic theory of light however before he could read Faraday's handwriting on the wall to refine and polish it with his elegant mathematics Maxwell needed to decipher a crucial clue within the forces of electricity and magnetism he found a very significant speed anyway has a definite speed depending on what makes it go for example in deep water the speed of waves depends on the acceleration of gravity and the length of the wave in air the speed of sound depends on the air pressure as well as the density of the medium and four linked oscillators the speed is given by the spring constant the mass and the distance between oscillators [Music] so it was time for Maxwell to get down to determining the speed of waves in Faraday's lines of force from Maxwell's mathematical perspective Faraday's lines were an expression of the 1 over R squared nature of the forces of gravity of electricity and of magnetism at the same time each of those has a specific constant and in the right scientific hands those facts got around to this since the fundamental forces of electricity and magnetism aren't independent the electric constant k sub e and the magnetic constant K sub M should be related but the question is how the ratio K sub e divided by K sub M has the unit's meters squared over second squared in other words the square of a speed but what speed it's square is 9 times 10 to the 16th meters squared per second squared divide the electric constant by the magnetic constant and there it is the speed of light occasion by separate it was an amazing discovery after fifty years of the closest scientific scrutiny elsewhere James Clark Maxwell saw the light and found its feed in the forces of electricity and magnetism but the fundamental forces weren't all that occupied Maxwell's mind around 1857 an academic prize was offered to anyone who could explain the solid nature of the rings of Saturn and their steady movement around the planet Maxwell's answer was right on the mark the only structure that could account for such stability Maxwell Road was one made up of clusters of disconnected particles those in which there is Maxwell's essay not only won the cambridge prize it won the praise of the scientific community at large Sir George Airy Britain's Royal astronomer called it a most remarkable application of mathematics perhaps because one good turn deserves another Maxwell immediately set his sights on moving particles other than those of Saturn's rings along a given path he developed a kinetic theory of gases in which gas molecules are small elastic particles that collide with one another as they move about with various velocities Maxwell believed all physics research should advance on in his words strict mechanical principles that belief led not only to Maxwell's theory of gases it led to his great accomplishments in electricity and magnetism to pursue his research the Maxwell's moved to Kensington London in 1860 for five years at King's College he was productive in kinetic theory and electricity elec and here at London's Royal Institution Maxwell finally met Faraday Ferriday remember had seen electricity in mechanical terms not nearly as an idea in keeping with newton's action at a distance but as a thing of physical lines a solid structural force that radiated through space Maxwell would state with great admiration that Faraday had seen a medium where the strict Newtonian 's saw nothing but distance perceptively Maxwell added that Faraday sought the seed of the phenomena in real actions going on in the medium of course as it would later be seen the medium is the electromagnetic field and that's where the action really is but before Maxwell could get the message from the medium he had to determine whether waves could travel through the electric and magnetic fields at the speed of light if so they would have to obey the four laws of electricity and magnetism the electric flux through any closed surface is equal to 4 pi K sub e times Q or Q over epsilon not where q is the total charge enclosed by the service and the magnetic flux through any closed surface is always equal to zero these are Gauss's laws for electricity and magnetism the circulation of magnetic field around any is equal to nu nought times the electric current passing through that path this is on Pierce law the circulation of electric field around any closed is equal to minus the rate of change of magnetic flux through that this is Faraday's law so suppose that there can be a wave in the intensity of the electric field this one's a transverse plane and it travels of course at the speed of light but consider the field at any one instant since it's up in one region and down the next it has a circulation about this back according to faraday's law that means that there must be magnetic flux through that same path and furthermore that it must be changing in time you and that can only mean one thing the magnetic way has tagged along with the electric wave everywhere it goes but and here's the crux of the problem that in turn means at any instant there's a circulation of the magnetic field about this period and according to amperes law that can only be true if there's current flowing through but in empty space there's no matter and no electric currents so there's no way around the fact that according to the laws of electricity and magnetism in Maxwell's time there could be absolutely no electromagnetic waves in space [Music] the point was this if the laws of fist stated that waves couldn't exist in empty space could it have been that as fundamental as they were those laws were somehow incomplete Maxwell's task was to examine the fundamental laws of his era and his triumph was to find the missing piece and he did so by looking into a simple and altogether conventional apparatus the capacitor the magnetic circulation around any path depends on the electric current linked through the path the only way to be sure that the current goes through the path is to imagine a membrane bounded by the path [Music] and the same current must pass through the membrane no matter what shape the membrane takes [Music] but that's no longer true if the current flows into a capacitor then even though current is flowing there may be none through the membrane if so what is the magnetic circulation equal to seeking an answer to that question Maxwell took a page from Faraday's book changing magnetic flux creates electric circulation and then as usual he looked at things the other way around good changing electric flux he wondered create magnetic circulation the answer promised to solve the capacitor problem as current flows into the capacitor charge builds up which creates an increasing electric field between the plates the electric flux through the membrane can be deduced from Gauss's law by imagining a closed surface all the flux goes through the dome-shaped membrane and it's equal to the charge on the capacitor plate over epsilon not [Music] the rate of change of electric flux can be found by differentiating both sides of the equation it's given simply by the current flowing in the wire in other words epsilon-not times the rate of change of electric flux through the dome-shaped membrane is the same as the electric current through the flat membrane [Music] this was Maxwell's crucial discovery the precise manner in which changing electric flux can generate a magnetic field as if it were a kind of electric current in fact Maxwell himself called this mathematical term the displacement current in other words the magnetic circulation around a closed path is given not only by the electric current through it but also by the rate of change of electric flux through this was how James Clark Maxwell completed the laws of electricity and magnetism [Music] she would hand with these laws Maxwell could show that electromagnetic waves of every frequency and wavelength lab would propagate through empty space at the speed of light Maxwell's discovered which seemed purely theoretical at the time would lead to the sights and sounds of the 20th century radiating through space not only as visible light but also as radio waves microwaves and the entire electromagnetic spectrum it was a discovery that would lead far beyond the earth into the deepest reaches of space where a wave and the intensity of the electric field implies the existence of a changing magnetic flux [Music] that means a wave in the magnetic field and that's made possible by changing electric flux and it's made clear only because Maxwell understood that electric and magnetic fields create each other if they move in a way together at the speed of light and so it goes in lockstep through eternity both fields propagating throughout the universe and in that light always at the speed of light one can finally see the power and the glory of Maxwell's equations just as Newton was born the Year Galileo time maxwell was born in 1831 the year Faraday discovered electromagnetic induction and he died in 1879 the year of Albert Einstein's birth what could be more fitting for between Newton and Einstein no men in scientific history made a deeper mark Maxwell went ahead to refine and redefine the fundamental laws of nature and in the process to build a bridge to modern physics the fact is that Maxwell was endowed with one of the finest minds that the human race produced during the 19th century and that combined with his superb Cambridge education made it possible and to do absolutely anything he wanted to I haven't told you about his sense of humor although once a long time ago if you remember I read to you one of his comic poems about the Atlantic telegraph company well I would like to read to you a little bit today from another one of his comic verses the poem is in this book which is very old and rare and valuable and I don't want to handle it too much in class so I'll just tear out a few pages so that I can read it to you this is a homework problem and in the first verse he sets out the problem an inextensible heavy chain lies on a smooth horizontal plane an impulsive force is applied at a required the initial motion at K what he's talking about is there's a chain lying on a table like that and somebody jerks it at one end and the question is how does the other end of the chain begin to move that is absolutely typical of the kind of extremely difficult and utterly useless problem that Cambridge students have traditionally been trained on well having set out the problem he then shows us how to solve it in great detail so I'll tell you what I'd like you to do I'd like you to go home tonight and work out your homework problems inverse and bring them back with you that way when you come here next time I'll see you then there [Music] Annenberg media [Music] for information about this and other Annenberg media programs call 1-800 lerner and visit us at

Channel: caltech

Views: 203,866

Rating: 4.9114428 out of 5

Keywords: Caltech, science, technology, research, physics, lecture, David Goodstein, The Mechanical Universe, Maxwell's Equations

Id: SS4tcajTsW8

Length: 28min 47sec (1727 seconds)

Published: Mon Dec 19 2016