so let's start by solving the quiz and after that I will actually start the new chapter right so we are not going to be dealing with capacitors resistors circuits anymore but let's solve the squee first so I have two capacitors which have equal charge but opposite polarity so I have plus q and plus Q charge in both capacitors but the way I actually put these together I could redraw the circuit in a quite different way the polarity is like this plus Cube minus Q Plus minus Q I don't know why I'm writing with such a coar pen and here's my resistor so as time goes on what do I know I know that if I think about a surface here the net charge in the inside the surface is not going to change what's the net charge in that surface zero so it means finally in the final State my final net charge will be zero the capacitors both of them will be uncharged finally okay but let's do the following let's actually denote this I as a function of time now clearly whatever current is coming out from the first capacitor the same current must be flowing in all my circuit right so this is also going to be I this is also going to be I now I can maybe say this is going to be q1 of T this is going to be Q2 of T the charges on the capacitors but what is I of t i of T because it's leaving the plus terminal of the first capacitor must be D q1t DT what's the sign is it plus or minus if I have a positive current it's coming out of the positive terminal the charge on my capacitor must be decreasing so there is a minus sign here but this is just as well D Q2 DT right it's again the same current is actually leaving the second capacitor so I have only one equation that's the definition of my current what else do I need I actually need to use Kos voltage low for this circuit what's Kos voltage low let's say that we get a different pan color so I'll start from here and follow the circuit along this rout so the first uh jump will be across the capacitor and am I going higher up in potential or lower in potential I'm going from the minus terminal to the plus terminal right so here when I first cross my capacitor what will be the potential difference it will be plus Q2 of T ided by C so I'm already here now I cross the first capacitor if I cross the second capacitor am I going up in potential or down in potential up right I'm going from the minus terminal of the capacitor to the plus terminal of the capacitor which is at higher uh potential so I have minus q1 of T divided by C now I'm here and I'll go back to my original start starting point over the resistor here is my current am I going to go down in potential or up in potential I'll go down because I'm in the direction of the current that's the direction current is Flowing I said and current is must be flowing downhill if you want right so it must be minus I of T * R but from the first equation I can also say the initial conditions for q1 and Q2 is the same they are both related to current so q1 must be equal to Q2 at all times okay so maybe I can write okay so this is K's voltage low I'm going to write it in a different way I'll say 2 q1 of T ided C is I of T * R is there a correction a comment no but now I know what I of T is I can replace it here let me get a more visible fan so I can say D2 over RC q1 of T is minus D q1 of T DT can you use directly Capac I'm going to talk about it at so we can actually uh write this in terms of the equivalent capacitance of these two capacitors in series okay now here it's actually not going to be very hard this is the one and only differential equation we know how to solve right so what is the solution q1 of T must be some a e to Alpha T which means dqdt is Alpha * a e to Alpha T if I plug this solution in to RC a e to Alpha T is minus Alpha a e to Alpha t a is cancel I not surprisingly find that Alpha is - 2/ r C how do I find a q1 at t = 0 is small q that's where I start from which is a e to 0 which is 1 so a is Q in other words I found the charge on the first capacitor as Q E to minus 2 T over RC I think that was actually Part B so we've actually done Part B we also have to talk about the current in the circuit as a function of time so what is the current we've already found written down the formula this was minus dq1 DT so if I take the derivative minus derivative I'll have 2 Q over r c e to minus 2 t/ RC as my current and it's actually good idea to maybe write down the circulation of the current although I've shown it on my diagram here I did not really specifically stated when you solve a question it's easier to make it gradable easily gradable so I'll write this current is counterclockwise in the circuit all right and finally what's the time constant well time constant how did we Define the time constant the time constant was the in the exponential Factor e to minus t over to that sets the time scale over which my charge decays and by just comparison I can see to is RC over two the that's very important it's not 2 over RC it's RC over two and does that make sense it makes sense because as your friend suggested when I have two capacitors tied in series and here although when you maybe initially look at the figure you may think hey these guys are in parallel but I redw when I redraw the circuit you see that when I take care of the polarity they are more like in series so what's the equivalent capacitance of two capacitors in series it's c /2 not 2 C right it's just the opposite so this c/2 here is the equivalent capacitance that which determines the time constant of the circuit okay anyal okay then I have an important question is there anyone here from manissa one person no one else from Mana no now I can see that you're all intrigued why I'm talking about Mara do you know the an ient name of Mana magnesia okay good so magnesia so Mana was magnesia and it turns out there were stones in magnesia which actually attracted each other because these manissa Stones they were called magnets Okay so there was a whole study of these Stones which was called magnetism so if you're from Mana you're uniquely suited to study magnetism okay even if uh you're not from Mana we all from Turkey well we not all from Turkey but we in Turkey uh so this is sort of the land where magnetism was discovered by the ancients unfortunately it was not explained by those ancients they sort of decided that it was some strange magic there were a lot of people claiming that they can actually communicate with magnets they believed if you take two magnets first you know they stick them to each other wait for a while say some prayers then when you separate them in different cities they still keep the connection so you can actually uh point out you can communicate over long distances using magnets the guy to disprove such communication was gilo he actually tested one of one people one of the people who claimed he can do this longdistance communication which is funny is that three centuries later when telegraph was invented that's pretty much what we do you know we have a magnet here and there's another magnet we just have to lay a cable and even now now we don't even have to lay a cable right but we do communicate over long distances using electromagnetism now so the simplest theory for magnetism is that you have these magic Stones which attract each other okay and the funny thing is they seem to have polarity so they have two poles and not people from manissa not not the ancient Greeks but ancient Chinese discovered something even more important they said hey these Stones seem to know which direction North lies so if you actually suspend them by a string they always seem to point along the north Direction all right so that's why they said the the uh direction that points along the north we call it the North Pole it shows the North Pole okay but there were so I mean people did a lot of experiments with these they were playing with these Stones one thing they tried to do is why they tried to isolate the North Pole and the South Pole of a magnet so they took a hammer cracked the stone unfortunately when you crack the stone you don't get a North Pole and a South Pole instead what you get is smaller magnets okay so this is kind of different from electrical charge I mean we'll see that there are a lot of similarities but there's a lot of difference in magnetism there is no magnetic charge there are no magnetic plus charges there are no magnetic minus charges it will always be two of them coming together so we will always have magnetic dipoles we will never have magnetic monopoles right so that's going to be one big difference and sometime later uh people were running experiments with currents now in 18th century 17th century uh Europe they also had this compasses now they knew that magnet show North one guy noticed something really interesting if you actually have a wire with a current and the compass here you could actually change the direction of the compass Okay so he made a important Discovery he said current create magnetic fields another guy he did the reverse he took two magnets so there's a lot of magnetic field here and then he actually put a wire between them and he noticed that there is a force on The Wire okay then he stated the reciprocal principle if you want magnetic field applies a force on currents so now we're going to study both of these principles we'll first we'll sort of Define the magnetic field by its action it's most important action is going to be these forces it applies on charged particles and then we'll the next chapter we'll talk about how to create magnetic fields by running currents through circuits all in all I'll tell you a little bit so we're going over the history of this Maxwell realized some more important things which we are going to study in this course that when you change magnetic field it creates an electric field when you change electric field it creates a magnetic field and what we call left or all kinds of electromagnetic wave is half electric field half magnetic field and after a while almost 70 80 years later Einstein discovered something even more important as you see electric fields are there even for static charges even if all my charges are sitting there I still have electric fields when charges start to move as in currents then I start getting magnetic fields all right so Einstein asked a very simple question he said hey you're telling me that if there's a current on a wire I get a magnetic field but what if I'm moving at the same velocity with the electrons moving in the wire in that case in that reference frame I would have no current which means I should get no magnetic field then he realize something the definition of electric field and magnetic field depend on the velocity of the Observer okay so electric field and magnetic field can if you want turn into each other under relativity so if you have a friend who is moving at half the speed of light he would measure different electric and magnetic fields and you would measure different electric and magnetic fields so that's why maybe we should view this object as a more compound object not separate the electric field or magnetic field that's why we say we talk about the electromag magnetic field okay so this was the first unification in physics two things which look like separate entities are actually sort of two sides of the same coin Einstein was so happy about this result he spent the rest of his life trying to unify gravity with electromagnetism which didn't work doesn't work still to this day but nonetheless it's quite interesting and let me tell you let me come back to the to our original Stones right they're not very good magnets these stones from Mana right now we have much better magnets actually for you know for five lit you can buy extremely strong magnets they are so strong that you can actually break your finger bones with them if you get two strong neum magnets you can break your finger bones some guy in our lab did that with permanent magnets and it's very interesting that still the study of what makes materials magnetic is very hard there is no fully complete Theory so we cannot we can do a lot of things about materials but but it's very actually hard to uh come up with some compound when crystallize will give you a strong magnet much of that is still uh trial and error okay and one thing that's really interesting if you really want to understand magnetism you need to learn quantum mechanics I mean I can tell you and I will tell you all these cartoon Mickey Mouse pictures of what magnetism is I'll let me give you a spoiler I'll tell you that all magnetic field is created by current so how come a stone from manissa how does it create a magnetic field and I say in these manissa Stones there are circulating currents inside each atom so these currents inside the atom create small magnetic fields they add up that's the special thing about Mono stones or a piece of iron those currents can add up so they can create big magnetic fields that's how permanent magnets are made but to fully appreciate to fully understand or to fully calculate what goes on in let's say iron you need to use you need to learn quantum mechanics so that's a even harder job so let's come back let's actually Define magnetic field and I'll do this a little bit differently from your book I'll start by forces on a moving charge so how do we know that magnetic field exists because it actually creat some forces I said it creates forces on permanent magnets or it creates forces on currents but maybe the purest definition of magnetic field is that it actually creates a force on moving charges okay so here is the thing so the observation is the following if I have a charge Q moving with velocity V under some magnetic field B There is a force on this church the force on the charge is quite interesting the force on this charge is always perpendicular to the velocity of this object it actually is also perpendicular to the magnetic field and it's proportion so if this is charged there's a force if I reverse the charge of the object the force also reverses so it's proportional to the charge so an uncharged object does not see such a force how do we generate vectors which are perpendicular to two vectors cross product great right that's why we learned cross product so the force the magnetic force is q v cross B all right hm what does that mean let's let's talk about let's say that I have a magnetic field a uniform magnetic field and I'm going to show it by these circles and here is my notation you probably know this what do these crosses mean a circle with a cross is into the page a circle with the dot is out of the page so here I'm assuming there's a uniform magnetic field into the page questions so what happens if I take a particle let's say of positive charge and give it velocity V let's find the direction of the current on this direction of the force on this particle so what do we need to do Force I know is going to be q v cross P this is a time to remember how we did cross products so B is into the page V is like this how did we do the V crossb product we take our four fingers Point them along the first Vector I still have the freedom to orient my palm but I Orient it in the direction of the second Vector which is into the page B so when I close my fingers from the first Vector to the second Vector my thumb shows the direction of the force so which way is the force here up okay so q v cross B gives me a force which is up so if this particle feels a force what is it going to do is its speed going to change H let's think how much work does the magnetic force do on the object do you remember the power transferred to the object if I have an object with velocity V what's the power transferred by a force F do v f do V so this is going to be q v cross B do V what's the dot product of two vectors which are perpendicular to each other the dot product is zero why because force is perpendicular to V which means the kinetic energy of the this particle is the same so magnetic force does no work on the particle oh that's also a important principle because but what does it do the speed of the particle is the constant but is the velocity a constant no it's not if I if I'm moving towards you and there's a force that's actually to the right what will I do I'll turn I'll start turning but the force is always turning with me so what kind of motion do I expect from this particle it will go around in circles okay so before we do that let me talk about the units here what's the unit of magnetic field clearly from this formula it must be the unit of force divided by unit of charge the unit of velocity so it's going to be Newtons per Kum per m/ second Newton Second per Kum meter not a very good unit right you don't say Newton seconds per K meter all the time it's given a name just like any other unit in electromagnetism it's given the designation Tesla so I think the car company was very clever they get free advertising in physics 102 that's not a bad joke come on I can do better than that okay so so Tes but so I mean let me give you a sense of this unit Tesla it turns out one Tesla is quite a lot of magnetic field so what's the order of magnitude of Earth's magnetic field there's actually a separate unit of magnetic field not an SI unit Earth's magnetic field on the surface of the Earth the Earth is called more or less one GA okay that's another unit which is 10 to minus 4 Tesla so this is one Tesla is 10,000 times larger than Earth's magnetic field so to create one Tesla in a small area you need electromagnets of pretty much this size okay to create five Teslas of magnetic field even in a small region you need huge electromagnets with probably cooling the highest magnetic field we can generate in the lab is a 100 Tesla that's our current capability we do not have machines to create 1,000 Tesla so Tesla is kind of a big unit okay generally when you go into the lab it's m Tesla M Tesla is a is a big magnetic field all right now let's go back let's talk about the motion of this charged particles the what does this charge particle do is there a question no so my charged particles is moving with velocity V feeling the force F so what will it do it will actually do uniform circular motion all right let's try to find the radius of this motion so turn this into an example what's the radius of the uh Circle my particle is going to trace I also would like to find out what's the period of the motion well it's not too hard if I'm doing uniform circular motion with velocity V what what's my acceleration come on physics 101 v² over R so this v² over R acceleration M * a must be supplied by the magnetic force m v² / R I have the them in the same direction so F magnitude is q v * B why am I so sure of that because V and b v is in this plane B is into the page so they are perpendicular to each other cross products gives me just their magnitudes right so good so the radius is m v over uh 1 power of V cancel Q * V does that make sense let's think about this if I have more velocity I trace out a bigger radius it kind of makes sense I mean it's it's hard to turn if you uh have a lot of momentum and that's actually good what I have is MV right in the new at I have MV so I have actually a lot of if I have more momentum then it's the radius would be larger it's it's harder to make a turn how about if I increase the magnetic field I go in tighter circles does that make sense that also makes sense good because I'm I have more Force now how about the period of the motion what's the period time to complete one Circle right so the total length I have to travel is 2 pi r my velocity is V so it's going to be 2 pi MV / V QB V is canceled so it's going to be 2 pi over QB / m wow it's kind of strange it doesn't depend on the velocity for example if I have a particle an elementary part or composite particle it suddenly explodes okay let's let's say that there is a chemical reaction or some nuclear reaction so many charged particles come out of it with different velocities if I'm under a magnetic field I'll see something very interesting I'll have let's say a particle with charge Q an electron with velocity V another electron with velocity V Prime they will both Trace out circles but they will come back to the same point at the same time so if there's an explosion one period later all the particles will come back to the same point so that's very strange that's very interesting actually so I can write this as 2 pi over Omega and I'll call it Omega C Omega C is QB over M the fre quency of motion is independent of velocity this is called cyclotron frequency do you guys know who cyclotron is come on you don't P Transformers Megatron Optimus Prime cyron some of you are not sure even if this is a joke okay so okay obviously it's a joke right okay so the cyclotron is the first particle accelerator and let me and you know the particle accelerator at Sun LHC right how big is that it has a 4 kilomet radius so it's a it's a pretty big object the first particle accelerator the cyron could fit in your hand and it cut to the electrons to 0.9 of light velocity okay so it's a pretty nice device let's take a 10minute break when you come back I'll tell you how this cyclotron works and we'll talk about other things about magnetic field