Further into Inductors | They're the magnetic analog of capacitors! | Doc Physics

drew you a pretty circuit we've got a battery here putting out some voltage and we've got a resistor there and this is just a switch right now it's open and we've got an inductor time to go a little bit further into what inductors are all about you see if I close this switch then the current will begin to flow and a magnetic field will begin to occur inside of the inductor the inductor itself will get pissed off that it's creating a new magnetic field and it will try to not allow that magnetic field that in itself is creating to exist so in order to do that it becomes a bad battery the inductor becomes as if it's a battery facing this direction that says no don't let a current don't let a current go through me and gradually the bad battery dies and we want to study how that bad battery dies gradually we want to study quickly how the bad battery gradually dies there that's a little bit better now ultimately a current will be allowed to flow because there's some resistance here and everything kind of ease is into it it's not a perfect inductor but I want you to notice that the inductor is kind of like inertia in the circuit because it tends to not allow a change to occur so this is very similar to capacitance and I think we'll see a few ways this ties in to capacitance but I like to think of an inductor as current inertia so here we are let's get ourselves a little graph I want to get a graph of current as a function of time if I close this switch it's going to be doing this it'll go from here I should get a different color on that plot right this is called infrared I bet it's not actually infrared and it will asymptotically approach some current here do you think you can figure out what that current is there I'm going to call that I final and I'm going to say that it probably is well when the bad battery is completely dead then this will be a line and this will be a line and then we'll just have V and R in here and V is IR so this current is going to be the voltage of the battery divided by the resistance of the resistor it will ultimately reach that but I guess I'm interested in this time here this characteristic time when we're up at you know what is it 37% I want to be 1 over e of that's not that's not on that axis right there I'm going to call that time tau and this is I final divided by E right here so that's when we are most of the way up but not quite I haven't looked at that and IB of being a little bit sloppy on that I'll come back and write a comment if I have so you comment right here all right but there's uh there's tau and then at 2 tau were a little bit closer and at 3 tau were a little bit closer still and we continue to get closer and closer but this is a characteristic time for every shape that looks like this you know the shape of this it's a decaying exponential that's kind of flipped over an inverse exponential dying and so I'm going to say that the current as a function of time this doesn't mean I times T but current as a function of time well it's going to be I final times 1 minus something that's going to start at 1 and then gradually die because well this looks like that flipped over so we'll say e to the minus T over tau and you're wondering what makes this take longer I suppose if tau were a bigger number then it would take longer to get up to a significant fraction so I can draw a a it taking longer kind of thing it might be a kind of graph like that if it took longer to get near to the to get near to the final current and so well we hopefully you've played with these equations lot but if you haven't that's the kind of feel that I'm going for so tau has to have units time and I'm going to define tau as oh man I suppose it will take longer if the inductor is better so I think that tau has to depend linearly on the inductor and it will take longer now this is maybe a little bit counterintuitive it will take longer if there's less resistance because the resistance eases us in to the the resistance eases us into getting a current through the inductor if there's no resistance at all here that we suddenly try to have an enormous we try to have an enormous current going through this inductor and the inductor says no and it takes a very long time so it's actually inversely proportional to the resistance so a bigger resistor ends up taking a longer know a shorter time really really counterintuitive I want to I want to point out a little a little relationship you remember when we were charging a capacitor we said that tau was one note I was just R times C and and then we had the resistance a larger resistor slowed down the charging capacitor turns out a larger resistor speeds up the energizing of an inductor all right now let's go on and investigate what sorts of things are needed here to charge an inductor we need some voltage and the needed in voltage needed voltage is the inductance times the current that we want to change divided by time and that's just going to be this well the change in current is going to be I final minus zero and the change in time is just going to be the time that it takes to charge so this this voltage that we need is the inductance times the final current divided by the time so my plan here is to increase the current from zero to I final in some time T and I want to think about what kind of power is required to do that so in order to do that I need some power which is current average times voltage and so I'm going to say that the average current is one-half I final times the voltage mm-hmm which is well I know what the voltage needed is it's this business right here so I'm going to plug that in and I'm going to find that it's one-half the inductance times I square divided by the time C I'm plugging in this the voltage I'm plugging in the voltage right here and I get that relationship right there and this is a very interesting relationship what if I what if I say I'm interested not in so much the power but what about energy used energy used to energize oh boy that sounds repetitive and inductor well that's going to be the power times the time that it takes to energize it in that case it's going to be power times this capital T that I'm using here all sloppy like and that is simply one-half L I score and wait a second if I had to use that much energy to get a current going through an inductor at me because it's got some inertia I am pushing pushing pushing to try to get a current to go through that sucker then that means that that's the energy that's being stored in the conductor and I want you to reference reference the energy stored in a capacitor which is 1/2 C times V squared look at this for just a moment you've got a half right because of some average that comes in there and you've got the value of the thing this is the inductance and this is the capacitance and you've got here you've got current and that's the way the inductor stores energy and here you've got voltage and that's the way the capacitor stores energy so this beautiful symmetry here so it is incredibly increased to each other one of them is about static and electrical energy and the other one is about static magnetic energy this one is about moving electrical energy and this one is about moving whoa let's not even go there but you know what I'm thinking I know I'm gonna keep going here I know that the inductance of a well what am I trying to say oh the inductance of a solenoid a simple solenoid we showed in the previous video is Mew naught times the number of turns per unit length square times the area times the length we found that that was the volume also so I'm going to plug this in right obviously not there I'm going to plug it in right there and I'm going to switch colors because that's getting boring so I'm going to say the energy in an inductor the energy in an inductor is wait for it 1/2 L I Square and I'm about to plug in all this business right here I've got 1/2 and I've got mu naught n square ay L I score oh my goodness but I want to point out that the magnetic field inside an inductor well I guess the magnetic field inside a solenoid is the result that I'm using right here is simply mu naught times the number of turns per unit length times the current that's going through there so I see this here you see that I've got Mew not and N and I in fact I've got this score except the square over mu naught equals mu naught times N squared times I squared you see this in there mu naught n square I square so I'm going to put B square mu naught to replace that and here we go I'm going to get 1/2 B Square over mu naught and what do I have left the I squared is gone the N squared is gone and I've got a they'll and just as we did for a capacitor we can investigate not just the energy that's stored in an inductor but let's look at the energy density stored in an inductor so this is the energy in an inductor and then lower U is going to be the energy of an inductor divided by volume because it's energy density and if we take it why this is volume so if we're investigating the energy density of an inductor that has a magnetic field inside of it and that's the way that it's storing its energy then we find that that is one-half B square over mu naught and I want you to remember that the energy density of a capacitor is one-half epsilon knot e square so look at the similarity between these equations yeah we got something about muna P and then a denominator and epsilon naught being in the numerator but look we're squaring the value of the field that's how fields store energy as the score of the value of the field and we're multiplying by 1/2 ding oh yeah one final thought this is the energy density of any magnetic field anywhere this result is actually completely general so it doesn't have to be in an inductor if there is a field that exists in space then the energy per unit volume is that and if there's an electric field that exists in space the energy per volume is that so I'm going to say that this is the energy of an electric field and to demonstrate some of the symmetry I'm going to show you that this is the energy of a magnetic field yep see purple orange great all right I want to I mean it may have occurred to you but maybe not I want to point out that inductors don't just have a problem getting current up to a certain level imagine that this has been going for a long time you've got the voltage of the battery here and you've got some resistance here you've got a closed switch and you've got an inductor and this had been closed for long enough so that a steady current would be going through her and the inductor becomes very happy now if I open the switch though you can bet that starting from some initial current which is probably let me point out that I naught is probably going to be well it's as if there's no inductor here remember so it's just V / R but the current will gradually slow down in fact rapidly at first and then rapidly at first and then tapering off because there's less and less of an impetus so this is a standard dying exponential and I find that the current as a function of time is my initial current times e to the negative T over tau and you saw this performance here for the rising to a certain level of current and and I realized that I got the YZ mixed up silly silly silly prepping right that's very important so it's at this point here that I can define tau let's look at it a little more carefully when T equals tau then the current at T equals tau is I naught times e to the negative tau over tau and e to the negative tau over tau is simply e to the negative one and that means that the current at T equals tau is I naught divided by e that's where my 1 over e is supposed to be this current is I naught divided by e so what I should have done up here and I'll have it corrected in the video also is that this is not AI final over e it's AI final times 1 minus 1 over e and 1 minus 1 over e well that's your favorite right we can put this as e over e so it's e minus 1 over e really and this is just well it's 1 minus 1 over e ok so that's that value right there so not only do inductors not like to start having a current they also don't like to lose a current so that in that way they are extremely like electrical inertia and frankly I'd like to call them electrical inertia devices

Channel: Doc Schuster

Views: 32,424

Rating: 4.9267015 out of 5

Keywords: tutor, problem, help, solve, understand

Id: 0IVJaXO_0XA

Length: 15min 40sec (940 seconds)

Published: Fri Feb 01 2013