Lecture 27 Wave Solution, Electromagnetic Spectrum, and Radiation

hello welcome back to physics 272 I feel the same way except I'm sad today because it's our last lecture and you may have noticed that I like teaching you about physics so I'm sad I'm gonna miss teaching you about physics but anyway it's been a fantastic semester so thanks for all your contributions and coming to class and saying whoo every time I say welcome to physics 272 so you know you have a final exam all right study hard and good luck we hope you do well on the exam we always hope you do the best that you personally can and we hope that the class as a whole has learned a ton of great physics last time we completed Maxwell's equations and this was very exciting not only for the really exciting theoretical physics aspect of the set of equations together is just beautiful but also for the fact that with the four Maxwell equations in hand with including the time dependence that we've finished out now if you took that sheet of paper and went back in time you could revolutionize society so basically a lot of our modern society especially the fact that we are a global society that's enabled by Maxwell's equations and so today we are not going to get to all that stuff that's a lot so there's a lot in the notes that are online that we're not going to get to today but we will get to wave solutions of the electromagnetic waves okay so wave solutions of Maxwell's equations and we'll see that if we accelerate charges that's how to start an electromagnetic wave and we'll also if we have time for it get to why the sky is blue because you know at some point in your life at some point you're gonna have some cute little small tiles come up to you with you know blinking eyes and say you got a degree from Purdue you must know why the sky is blue so I have to tell you that right so can you imagine my career kids when they asked me this might come up give me a pencil on paper and I'll give you Maxwell's equations so you know my poor four-year-old is going mommy no more math anyway so here they are alright right there that's why the sky is blue no I'm just kidding we'll get there so these are the full story of electricity and magnetism this is everything there is to know on the right hand side I've got the integral last time we derive the differential forms okay and that was a lot of math that I don't expect you to be able to reproduce on your own I simply proved it to you so that you know that okay this set of equations which is its integral form the way I know it's an integral form is because there's integrals in it is exactly equivalent to what's the left-hand side here which is the differential forms okay so they're exactly equivalent you can use whichever one you need in particular contexts and basically the top two equations are about either in closed flux or divergence and they're about the same idea which is that if I have a point charge it'll have an electric field coming off of it and as the electric field comes off of it the field lines radiate out and so the lines diverge from each other and that means you'll you'll catch a divergence if you're using the the math that's about differentials okay you'll catch what's called the divergence term or if you're doing it in the integral form what you'll see is that the verging field has a net flux through a closed surface so same physics and also we saw that there were certain kinds that were about curling fields so you could either express that as a circulation in the integral form where I walk around a closed loop and pick up all the components of the field that are parallel to me as I walk around that closed loop that's a circulation the equivalent form for differentials is what's called a curl so it's a it's a derivative that's a kind of a curling derivative all right and I showed you the math to that last time all right and and you know the new exciting thing that we did in the past two weeks was we put in the time-dependent pieces so now that we have the time-dependent pieces the pieces that say if there's a changing electric flux I must have a magnetic field arise and then that means there's a changing magnetic field so if I have a changing magnetic flux that causes an electric field well that's a changing electric field if I have a changing electric field it causes a magnetic field to see how this kind of goes on forever right so in fact that's what I'd like to show you today is that piece of the equations that say you can get something that propagates forever and so I'm gonna start from the differential form what we're gonna do is take the differential form of Maxwell's equations and I'd like to analyze them in a region of space that's that's free space so no charges in this region of space and no currents in this region of space so what we'll assume is that somewhere else in the universe somebody wiggled some charges and started an electromagnetic wave and then I want to see what happens in the free space region so over here on the right hand side i've copied Maxwell's equations down but in my region of space there's no charge density so I set that to zero and there's no current so I set that term to zero okay so this is this is in the absence of charges these two equations are the ones let's say once a wave starts it just keeps on going and the reason is because a changing magnetic field causes a curling electric field and then that changing electric field causes a curling magnetic field and so forth so we'll see that these guys just kind of bootstrap each other and caused everything to wiggle back and forth so what we're going to do is take those two equations ok del cross E is minus DB by DT and del cross B is mu naught epsilon not de by DT put those two equations together and we'll see that we get one big differential equation that is the wave equation so remember I have this disclaimer about how you actually solve differential equations the way you solve differential equations is you take the difference equation you have and you knock on the door of the math department and you say I have this equation what solves it and they tell you because takes a lot of work to prove the actual solution okay so we take the answer from the math department or Wolfram Alpha you're your choice so we're gonna feed one equation into the other get a wave equation out and then we'll see what the structure of that solution looks like so let me start with del cross be back up a little bit here let me start with del cross V is buna epsilon knot de by DT by the way you may remember that Mew naught epsilon naught is 1 over c-squared so already hiding in here is the speed of light all right so I want to take another curl this is the curl of B I'm going to take a curl of this whole equation so del cross of that whole equation del cross B is mu naught epsilon not de by DT let me remind you what the del operator looks like by the so the Dell operator the Dell operator is a derivative that's in vector form so Dell looks like this D by DX comma D by dy comma D by DZ okay it means when this operator is fed something in the X component it takes an X derivative of it when this operators fed something in its Y component it takes a wider emmitt above it and so on so it's a what you call a vector operator an operator is something that you can kind of think of it as eating a function and then spitting something about sound okay so it it does something to a function so that's this operator here so I'm taking a del cross of del cross B I'll do that on both sides so the left hand side just gives us del cross del cross B the right hand side Mew not epsilon knot comes straight down now I should have had a del cross D by DT of e I written it here in the opposite order okay I've written it here as d by DT of del cross E you remember how math goes you read math right to left okay so the way math goes is this math up here says first take the time derivative of the electric field and then take the curl of that this math down here says first take the curl and then take the time derivative that is okay because time and space are independent coordinates so I switch the order and it's okay all right I'm in space don't bother each other so now I have this del cross e sitting around well I have a Maxwell equation for that right I have a Maxwell equation that says del cross E is minus DB D up by DT so I'll just feed that in and that comes right here okay so now I have this minus CB by DT sitting there and then we'll simplify that a little bit as well okay so now I'm going to use that mu naught epsilon naught is 1 over c-squared so that's going to be sitting right there and notice what happened here I now have if I look at the far left hand side I have two spatial derivatives of B and on the far right hand side I have two time derivatives of B okay so putting that all together is going to be down here I want to make one more simplification which is that this del cross del cross be two spatial derivatives of B it turns out okay so there's one cool trick that I'm not going to prove to you but it's a one cool mathematical trick to use here as this vector identity is what it's called it turns out that del cross del cross B well okay del cross del cross anything is the same as minus d squared by DX squared plus d squared by dy squared plus d squared by DZ squared and this doesn't just work for magna and it feels it works for anything so this is a vector identity it would take us a couple of pages of math to prove it so we'll just take that from Wolfram Alpha okay but suffice it to say this is two spatial derivatives and this is two spatial derivatives okay so here that gives me all together events you have a minus sign here that cancels that minus sign so equating all this together I have now d squared by DX squared plus d squared by dy squared plus d squared by DZ squared of the magnetic field is 1 over c squared d squared by dt squared b that's a lot to say all right but let's stare at that on the left hand side I have two space derivatives right del I have a del squared by Del x squared and on the right hand side I have two time derivatives okay so if you take this equation to the math department and knock on the door and say well I have a differential equation which what solves this they'll tell you it's a wave okay so this is a wave so these two equations together we fed one and to the other and we got this equation okay for the magnetic field I started remember what I did was I started from Del cross B and I took the curl of that and then fed in the other equation I could have started with del cross e taking the curl of that and then fed the other equation in so I could go either direction so these two Maxwell equations together the ones that talked about a changing field causing a curl in the other field when you feed them back in together you see that I have a wave equation in B and I have a wave equation in E okay I didn't show you the wave equation in Y but it's the same math taking a slightly different direction okay you have questions so far all right so this comes right out of the Maxell equations in a region of space where I don't have charges and I don't have currents how do you solve a differential equation you know the answer okay these differential equations are famous they are the wave equation so this is the wave equation and you read off the the velocity right here I'm sorry the speed you read off the speed right here C is the speed of light so this tells you that when an electromagnetic wave is propagating in a region of free space it always travels at the speed of light okay and it's it's waving and it's also this is the lovely piece here this is what enables all of wireless communications all wireless communications are are built on these wave equations so again if I handed you the sheet of Maxwell equations and you had a time machine and traveled back in time 300 years you'd have all the intellectual information you need to start doing things like thinking of wireless communications like cell phones like satellites okay like radio and TV all that old-school stuff all right okay this is a cat that's waving cats don't travel at the speed of light so this the cat is not a solution of the equation but the cat's waving and it was late at night and I thought the cat was cute okay so this is a wave equation with speed C the solution to what we have is waves let me show you now that the solution to this equation is waves right that's how we do this we we know the solution should be a wave I'm going to write down a wave and we'll just show that it actually works so you need the equation the wave equation all right and I'm also going to need to take some derivatives of a wave okay the standard wave form we put in as a sine or a cosine so let me remind you how to take derivatives of sines and cosines so D by DX of sine is cosine D by DX of cosine is minus sine all right if you're like me you have trouble remembering where that darn minus son goes so well you know the risk of being undignified let me help you remember this okay I made up a little rap for this you ready cosine is negated when the sine is integrated before okay cosine is negated when the sine is integrated so when the sine is integrated to integrate a sine and cosine is negated okay cosine is negated when the sine is integrated never forget it right now you get the signed up okay so if you translate this translate this into derivatives michaei what this is going to tell you is that the derivative of cosine is minus sine so that's what I have here all right derivative of cosine is minus sine so I'm gonna use these trig identities basically well trig differentials in order to show you that a wave a sine wave solves that equation so we're gonna use this one cool trick which is I'm going to need two spatial derivatives of a sine wave so d squared by DX squared of sine is d by DX of well the first derivative of sine is cosine now I need to take a derivative of that and the derivative of cosine is minus sine so here's the key two derivatives of sine is minus sine that's that's the key here so two derivatives of sine is minus sine so you can already kind of see that if I feed in a sine wave on the left hand side I'll have two derivatives of it which gives me a minus sine wave right and on the right hand side the same thing will happen I'll take two derivatives of a sine wave which will give me minus that sine wave so it must work out so we'll show you how that works out let me try this waveform I'm gonna try an electric field of a traveling wave form so that looks like this he not sine of KX minus Omega T times Z hat e not just means there's some number in front that's got the units of electric field okay the sine part sine of KX minus Omega T all right so when you see a sine of X you know what it does it's a function that does this the K tells you the wavelength alright so so you know the how fast it waves is going to be related to K K is actually 2 PI over lambda by the way so if I have a really long wavelength then I act have a small cake so the fact that I have a KX minus Omega T what that means is as time goes on the coordinate that the sign gets fed is changing all that's gonna do is take your sine waveform and shift it and shift it in in space basically as time goes on your sine waves gonna travel this this is set up so that the sines work out I've got a minus sign in the middle such that this sine wave travels forward along the X direction in time so I know that because there's a X there the Z hat okay there's a lot of geometry going on here the Z hat means that the direction of the electric field is in the Z direction so let's let's choose some axes let's say that that's the x axis coming out to you as the Y axis and up is going to be the z axis so this tells me I'm looking at an electric field that's that's along the z axis okay so it's it's pointing up or pointing down depending where I am in space but it's a sine wave so it's pointing up here and then down there and up here and down there that's what that sine wave looks like and then that whole pattern is shifting forward in time along the x axis to have any questions about what the waveform means okay all right so that's what that waveform means and now I want to show you that mathematically it fits the wave equation okay it's a traveling wave solution of the wave equation so it better work so d squared by DX squared plus d squared by dy squared plus d square by DZ squared of this waveform I've only written down one term here what happened to my other terms what happened to him it's okay to speak up I kind of heard some whispers of it yeah the other terms are zero okay so if I look at this function there's an X there so I will get a derivative with respect to X okay so I know that term is gonna survive if I think of the derivative with respect to Y there's no y in here so that derivative has to go away what about the derivative with respect to Z I have a Z hat there what's what's what's the deal with that right Z hats just a unit vector that never changes so it's it's not changing there's no do you know the derivative of Z hat is zero so the only term here that's left is a d squared by the x squared of E and two derivatives of a sine-wave gives me minus the sine wave okay there's also two powers of K that come out so altogether this is minus K squared e naught sine of KX minus Omega T in the Z hat direction right now when I take the set was the left-hand side of this wave equation the right-hand side is 1 over c squared d squared by DT squared of e 2 time derivatives of the sine wave will also give minus the sine wave okay so once again I get minus the sine wave what came out is I had two powers of Omega come out because of the time derivative actually two powers of minus Omega so minus Omega times minus Omega is plus Omega squared right here the minus sign is from taking two derivatives of the sign so if minus Omega squared over C squared e naught sine that C squared was already in the wave equation here so I've almost shown you that it works basically if we know that this part is equal to that part then it works then our sine wave solve the wave equation okay oh go figure a wave solves the wave equation so if Omega is equal to CK then this works and in fact that's correct that's how we know how Omega and K are related is that this Omega must be related to that K by the factor of C do you have any questions so far or complaints yes please Oh Oh may go okay so mmm Oh Oh make a is um so if I talk about the frequency of the way even hurts okay so you've heard of Hertz right so like for example the the electric current that's being delivered by the the power company inside the building you know is jiggling at 60 Hertz okay 60 Hertz is the frequency F Omega is 2 pi times that so that's what Omega is Omega is 2 pi F good question okay other questions all right so now this B wave 2 what I've shown you is that I have the wave equation for e and I showed you that yes a sine wave solves the wave equation for e alright but I also given that that's e I'd like to see what the magnetic field is as a result of that due to Maxwell's equations right so Maxwell's equation says all I need to do is take the wave that I already have an e u--'s del cross e is minus DV by DT and that'll tell me what's the form of the magnetic field ok so will feed it in so let's take del cross e of what we have and the way you set up del cross is an e cross product you can set up as this determinant of the matrix X hat Y hat Z hat the Dell vector okay remember how the Dell vector went the Dell vector is a vector operator but you just follow the same script that you always follow which is that the pieces of whatever this vector are go right here so Dell itself was d by DX that goes there d by dy that goes there and D by DZ that goes there okay and then e has only a Z component so when I write down the components of eat out here it's zero comma zero comma easy ok so that's how we set that up and now we just take that determinant and you use the algorithm you like for taking determinants I like to think of taking the diagonal lines down in to the right and then diagonal lines down I'm sorry up in to the right so the diagonal line down and to the right gives X hat with a D by dy with an easy okay so that means I'll have a D by dy of easy right here in the X hat direction and then going up into the right I'll catch an easy and a D buddy x and a y hat so that means I'll have a term that's minus D by DX easy Y hat now one of those two terms is 0 now's the time and lecture when we play which term is zero so one of those two terms is zero which one zero what do you think is the first term zero or the second term zero in the way you can answer that question is you go back to easy here's the form of e okay everything in front of the Z hat is easy so does this thing have a does this thing have a derivative of respect to X or does it have a derivative with respect to Y yeah it's got a derivative with respect to two x okay so if I take D by dy of this function that has no y dependence it's zero so this term is zero okay so I'm left with minus D by DX easy in the Y hat direction okay so now I need to take one spatial derivative of e so the the e naught comes down one derivative of sine is cosine so I have the the K comes out and I have a cosine of KX minus Omega T it's all in the Y hat direction notice that Y hat snuck in there right the Y hat is coming in because of the crossed derivative we've started off in an electric field pointing in the Z hat direction Maxwell's equation tells us to take a cross derivative to figure out what the magnetic field is and that cross derivative is the thing that left us with the Y hat okay so the Y hat here and that's equal to minus DB y dt okay you have any questions so far all right okay so continuing on then what I need to do now is look at this differential equation and figure out well what's the form of B that's going to that's going to do that alright and in this case I don't need them the math department for this one okay we can we can think about this one and and try something and show that it works so let me let me see what happens here if I take this equation here and and think of putting in a particular magnetic field so think of putting in magnetic field equals minus e knot K over Omega sine of KX minus Omega T all right if I take the time derivative of this guy alright when I take the time derivative of this guy derivative of sine is cosine I'll come out with another minus Omega so the minus signs will cancel the Omega will cancel and I'll be left here with an e knot okay over a see you never see sine KX minus Omega T alright so this alt all comes together here all right and if I take the time derivative of this guy I'll see that it gives me this right so feed in the time derivative of that and I get what I already had so here's the deal this electric field that was in then the electric field we started off with is a sine wave where E is in this Z hat direction okay I'll just remind you what II meant and then we'll figure out what's the B that went along with it so we set up an electric field that points in the Z Direction Z's up okay and yet as I walk along in space it's going up and then down it's following a sine wave it's up and then down okay and that whole thing is traveling as well all right now the B component that came out okay Maxwell's equations told us how the magnetic field had to go the the the magnetic field is pointing in the y axis which is towards you guys okay so the B component is is a sine wave it's coming out and then back and then out and then back and it's varying as I walk along the coordinate X okay and the whole thing is traveling in the X direction the KX minus Omega T means shift this thing in time forward all right so all together then it means the wave right has to be not just electricity and not just magnetism it has to be both in it has to be what we call an electromagnetic wave and those equations together look like this if left hand is electric field on the right hand is magnetic field it's doing this it's going out and then negative and then this way and then negative and who knew that disco came out of Maxwell's equations yes okay so that's what that looks like do any questions alright good cuz I'm not doing that dance again alright okay so this is a self-sustaining wave that can go on forever remember we said we had these pieces of Maxwell's equations that said a time derivative in one field caused a curl in the other and then a time derivative of that field caused a curl on the other so that feedback effect back and back and back and back is this is this it's it's a wave so it's a self-sustaining wave that keeps self-propagating and it can go on forever and this is actually light alright so and and it propagates at the speed of light the way you know it's the speed of light is because the speed is Omega over K and Omega over K we said has to be C for these equations to work so this is propagating at the speed of light questions so far okay all right this is what that dance looked like that disco dance I did looks like this graphically that the electric field is in in one axis alright and then the form of it follows a sine wave right the magnetic field is along a different axis and it follows a sine wave and then the whole shape travels in time so E and B are perpendicular to each other and they're also perpendicular to the direction of propagation okay and in fact V the the velocity of the travelling wave is parallel to each receive so if I look at this this graph here I'd see that EE cross B gives me this direction so that waves traveling this way or when I was doing this wave here right would had electric field going up and down a magnetic field coming back and forth e cross V meant that wave had to travel this direction okay so what's the UH what's the magnitude of V what's that what's the value of V if I just said what's the magnitude of V yeah it's C it's the speed of light so the magnitude of V is the speed of light and no kittens can't travel at the speed of light sorry kitten not unless you're a photon it's a photonic Katz there's no all right so electromagnetic radiation can been at several different wavelengths so remember what I said is that there's that there let's go back to the diagram there's a wavelength to this right the wavelength is go up go down and come back that distance is the wavelength or this is another graphical representation of it go up go down come back that distance is the wavelength even though that way it's gonna travel by you can still measure the wavelength so what's written down here is different wavelengths of electromagnetic radiation and how we experience them as humans okay is it tiny sliver in the middle which is visible light right there so visible light is happening oh it's it's basically you know in this in this region that's it should be something like hundreds of nanometers okay so small right so right here is ten to the six you can't see a micron all right micron is about the width your hair but you can see about a tenth of that so if you take a tenth of the with your hair that's the wavelength of the photons that are going into your eye and exciting something that allows you to see light if I go a little bit longer wavelength that gets to be what's called the infrared and four just means longer okay so infrared would be this is showing us that it starts around around microns okay so things like that big or so our infrared light now there's a lot of infrared then infrared infrared light in the room that your eyes can't detect but if you had night-vision goggles on okay if we turn off all the lights pitch-black and everyone had night-vision goggles you would see that every person in this room is glowing and the glow that your night-vision goggles are picking up is the infrared radiation that people put out because they're at a particular temperature everything has a particular glow according to its temperature and based on your temperature you are radiating in the infrared ultraviolet light okay these are the UV things that cause us to use sunscreen so the Sun puts out a very broad spectrum okay and it peaks in the visible and the ultraviolet components are the ones that well they're very tiny wavelengths okay you see how the ultraviolet stuff is getting to be like nanometre all right very tiny wavelengths actually it turns out as you go along this chart the smaller wavelengths the tinier stuff actually end up being higher energy photons okay when you think of it in terms of Einsteins little packets of energy it turns out that the packets come in in higher energies for the smaller wavelengths but you can also kind of imagine that if I get a really really really small wavelength that starts to be able to flick atoms around and what happens with the UV radiation is it gets into your skin and it flicks your DNA around causes cancer so wear sunscreen x-rays if you've had an x-ray at the doctor okay those are even tinier waves but higher energy and gamma rays okay don't don't don't get involved with those now going the other direction the longer wavelength stuff here's kind of some fun things okay some radio and television over the waves I don't know how many people do this anymore but anyway way back in the day we would get TV just over the airwaves it was by electromagnetic radiation same thing with the radio bands the FM band and the AM Ben does anybody know what this band is the amateur band anybody happen to have an amateur radio license anybody here a licensed broadcaster really okay well I'm an amateur radio broadcaster I have a ham radio license now the reason I have one is because I love my husband all right so when I got so you know you do things when you get married because you know you love your spouse and and and and you got to play along so I married into this wonderful family of ham radio operators and basically they looked at me and they said what you're a physicist go get a license so okay right so I took the test I got a license so I'm authorized to put out transmissions on the amateur band okay so these are all electromagnetic radiation but there are different wavelengths okay so these are rather long wavelengths these guys go from like a meter up to a kilometer okay so some radio waves are basically the size of this room so think of that all right so microwaves this is the fun one this is actually the your microwave oven is using waves that are in this range here all right so about a millimeter about a centimeter maybe about you know a few centimeters you can actually find out you can actually find out the wavelength of microwave radiation that your microwave uses would you like to know how so here's how to do it don't make a mess if you do this okay but you need a big bar of chocolate so take your microwave oven take the turntable out take the biggest bar chocolate you can find and put it in there and and hit go now watch very carefully because you don't want to melt the whole bar that'll make a mess and your roommate will be very mad at you okay and then don't say I told you to do it but if you watch very carefully watch for when it just starts to melt what you'll find is that there'll be some little dots it'll start to melt at a few distinct locations those are the places where the wave is really wiggling the highest and the distance between those is half the wavelength okay so you can measure the wavelength of radiation in your microwave oven by slightly melting a chocolate bar okay questions about any of that stuff that's just sort of physics in your life that's all this electromagnetic radiation and it's all the same phenomena it's just that we experience it differently at different wavelengths and really to measure the entire spectrum requires some sophisticated equipment all right now physics in your life color vision all right let me tell you a little bit about why it is that you perceive colors and how you do that so your your eyes have a few different types of receptors in them back in the retina you have rods and cones the rods are things that are just looking for intensity of light is there some light present or not they don't care about which color and they're very sensitive to small intensities so at night when you know you're you're basically using your rods but if there's enough light intensity around and then the light excites your cones okay this is why you can see in color during the daytime but at night you really can't perceive color so the cones tell you the different colors and what you have basically is you have some cones and you're either tuned to red some are tuned to green and some are tuned to blue so the the blue ones peak around 440 nanometers the green ones peak around 540 nanometers and the red ones peak around 560 nanometers and that roughly corresponds to their little specific organic molecules okay whose job is to pick up that frequency alight so basically when that frequency of light comes why they grab it they send a signal to your brain so your brain isn't measuring all so your eyes aren't measuring all colors your eyes are just measuring three specific wavelengths of colors and then your brain interprets all of that as green or orange or purple or whatever okay and here's another kind of fun fact unfortunately not everybody sees in all three of those colors right there's a phenomenon called color blindness which can knock out one of those cones and so if you if you're color blind it typically means that your your one of your cones is defective and you're not seeing all three colors you're really just seeing two typically so about 10% of men have color blindness it's it's because it's carried on its carried on the X chromosome so if you only have one X chromosome you can't compensate women are much less likely to have colorblindness they need to have two of those X chromosomes that have the the colorblindness on them in order to be colorblind so one roughly 1% of women are colorblind but here's another fun fact right and again I'm not a biologist so you really want to check me up on this but I've been told by biologists that women have two X chromosomes so think about this if there's a mutation that gives you a slightly different cone color all right if you only had one X chromosome you probably wouldn't notice but if you had two X chromosomes and one of your mutations said I have a slightly different cone color Wow now you'd have four kinds of cones right okay so there are some women out there who have this mutation and they rather than having three different colors of cones they have four different colors of cones they're called tetrachromats okay how do you know if you're a tetrachromats what you'll find is that RGB screens really frustrates you because RGB screens are expecting those three cones that most people have if you're a tetrachromats you have an extra cone that's a different frequency and you'll just be forever frustrated by RGB screens they just don't give the right colors so that's how you know if you're a tetrachromats questions it's kind of fun physics in your life if you happen to be a tetrachromats let me know I don't know just never met one before so questions so far all right now I need to teach you about accelerated charges and what we showed so far was that there is this phenomenon called electromagnetic radiation waves that can that can self propagate question is how do you get them started so here's how you get them started you need to take some charges and shake them so if you take some charges and wiggle them back and forth they put out electromagnetic waves okay now how can we think of what that looks like I got to confess I spent all week trying to read arrived this stuff that's in the book so let me tell you kind of a cheaters version of how to understand this and again it took me a while to work out this math because your your book left out an important piece in my mind so let's do the following experiment let's say that there's a charge in the middle of the classroom right over here so let it be whatever charge you like say a positive charge okay and it's putting out an electric field all right and I the observer over here can observe there's an electric field radiating out of that will coming out of that chart now see the charge all of a sudden moves to a new position if the charge moves to a new position in order to do that it had to accelerate alright in the new position I'm gonna measure a different electric field right I was measuring electric field pointing this way the charge moved I'm now measuring an electric field pointing that way but the information about the charge moving takes time to get to me right so the charge moves and I don't find out until the electric field has had time to adjust and it adjusts at the speed of light so at the speed of light coming from that accelerated charge they'll be basically a sphere of information right beyond that sphere nobody knows yet that the charge move if you're inside the sphere you know the charge move okay so there's a sphere of information coming out all right and the sphere travels at what speed speed alike right so all electromagnetic disturbances travel at the speed of light so this is a graph of that sphere of new information which and the new information is the change in the electromagnetic field okay so an observer here at one already knows that the charge moved the observer here at two doesn't know yet and then as that sphere propagates out at the speed of light now observer two gets the message but here's what it looks like to you The Observer okay we see that chart and we see the electric field pointing this way the charge moves and now the electric field is going to change but during the change I see a big you know I see a big effect that's it whoa it changed when it changed direction there's a large what we call transverse component transverse meaning here transverse to the spheres the spheres coming out and I can think about well what kind of disturbances are happening the main thing I'm going to notice is that sphere goes by me is I'm gonna notice changes alright so there'll be a change in the electric field on that sphere some of the changes now I'm gonna break it up into a radial component okay which is radial to the sphere and then I'm gonna break it some of it up into parallel components on the sphere and we'll call that transverse okay so there's the radial component and then there's the transverse components so I want to know what all those changes look like it turns out that the radial disturbances fall off like 1 over R cubed this is what I've finally calculated this week so the radial disturbances fall off like 1 over R cubed the transverse disturbances okay so if I see that charge move I'll detect a changing electric field 1 bits radial 1 bits transverse the radial bit is small and it just falls off like 1 over R cubed so it's very hard to measure so the main thing you're going to measure when this disturbance goes by is a transverse electric field because it's large it falls off like 1 over R so at large distances okay at large distances since 1 over R cubed falls off so much faster than 1 over R all you're going to measure is the transverse field this is so this is what took me all week to figure out this is why your book focuses on the transverse component ok so when your book is talking about the radiative field they focus in only on the transverse piece because at long distances that's actually all you can measure so one falls off like 1 over R cubed 1 falls off like 1 over R do you remember how the electric field of a point charge falls off 1 over what excellent oh man lots of people participated in that one excellent everybody gets a star today so 1 over R squared so this transverse radiative field falls off like 1 over R but the static field falls off like 1 over R squared so this is a very long range thing all right what that means is that if I'm far away from a charge let's say the charge in the next room and it's oscillating up and down and I try to measure the radiative field coming off of it which is this transverse field that if the thing keeps wiggling then I'll keep noticing that and I'll keep seeing this transverse field if I try to compare the magnitude of that transverse field to the magnitude of the Bayer field coming off of it the one that falls off like 1 over R squared at long distances I'm gonna find that the by far the dominant effect is this transverse field so this is actually so for example when we're measuring we can actually measure electric fields that come off of stars right ok they're very very far away we don't measure the direct electric field that's falling off like 1 over R squared from all those charged particles ok nevermind the fact that the star is actually net neutral okay but the charges are widdling and they put off electromagnetic radiation okay and then that wiggly bit falls off like 1 over R so that's that's this is part of what allows us then to to measure that stuff at large distances to have any questions so far about that general idea ok all right now I need to tell you a little bit about the geometry so we'll focus in on the transverse radiative field because that's the dominant component and that's what's denoted here in the diagram is that as you shift this electric field visit there's a large transverse bit that runs by you so the transverse pulse propagates at the speed of light okay because there's a what I would observe is that I'd observe oh the electric field changed in my region of space a changing electric field has to have a magnetic field with it and you already know how this goes a disturbance in the electromagnetic field propagates at the speed of light and it has the following properties e is perpendicular to B is perpendicular to V so this change radiating off of that particle if I sense a transverse electric field this way the magnetic field must be perpendicular to both the electric field and the velocity of propagation so there's only one direction left right if E is this way and the wave is propagating that way the B must be this way so tells you the directions e is perpendicular to B and it's all perpendicular to the direction of propagation of this sphere coming out at the speed of light now we also want to understand a little bit more about the magnitude now I'm not going to go through this calculation for you because it sits in the back yearbook it takes several pages but let me tell you who the calculations do too many years ago there was a young man sitting right here okay name is Edward Purcell and he got his Bachelor of Science degree in in electrical engineering at Purdue he went on to win the Nobel Prize this argument is due to him so basically the idea is the following we got the direction to get the magnitude you can do a Gauss's law analysis so the Gauss's law would say you know find yourself the gaussian box and very carefully draw the geometries out and figure out which way the fields go so the end result of that two or three page calculation is this that the radiative electric field by which we mean the transverse component because it's dominant the transverse component of the radiative electric field is 1 over 4 PI epsilon nought times minus Q a perpendicular over C squared R let me tell you what that means ok so first of all notice it falls off like 1 over R so it's a very long range phenomenon also there's something called a perpendicular so what's that here's that sphere of disturbance propagating out from the charge that moved here's our observation point and I'm looking directly at the charge that has moved the charge accelerated let's say it accelerated in this direction but according to my vantage point according to my observation point the only component of acceleration that I could see is the transverse acceleration so this diagram here defines that symbol a perpendicular it's the transverse component and now B what the equation tells me is that if I looked over here the charge accelerated this way I only notice the transverse component the electric field is in the opposite direction the electric field is in the minus Q a perpendicular direction so that's what I measure do have any questions about how to interpret that ok all right so unfortunately we just ran time and I didn't get to tell you where the sky is blue so I'm sorry about that but I do have to end class because we're out of time and if you want to know why the skies will come forward I'll fill you in

Channel: Prof. Carlson

Views: 5,044

Rating: 4.9384613 out of 5

Keywords: iMovie, Physics, Electricity, Magnetism, Electricity and Magnetism, Phys 272, Purdue, Matter and Interactions, Waves, Maxwell, Equations, Erica Carlson

Id: jbHgG1iHoY0

Length: 46min 7sec (2767 seconds)

Published: Wed Dec 07 2016