 Some announcements bringing updates to our course website. I think you will see the web browser view shared on your desktop. On the browser, this is the web page for our course, basically a circle circuit and then you see some announcements here which also has the recorded version of the first lecture. As we go along with the recorded lectures here, you can use these to review or interview or you miss some lecture, you can also use these things. And here you also see the class schedule and some tentative list of topics that we are going to cover in every lecture. This is only tentative, it will be revised as we make progress through the course and again here also you have links to recorded lectures as well as through the discussion forum. Now the discussion forum you can use for asking questions and I will answer them at my convenience, okay. With that let us get started with today's lecture. In the previous lecture, we looked at what a voltage and what a current is and related them to the fields. We also described why we can describe everything we want with only voltages and currents without going down to the level of details that we do in the magnetic, that is with the fields and so on, okay. And under certain conditions, we saw that there were two laws that were true for voltages and current on the Kirchhoff's voltage law, it says that the sum of voltage drops taken with the appropriate polarity around the loop is 0 and other one is Kirchhoff's current law which says that the sum of currents entering a node or the sum of currents leaving a node is 0. Now Kirchhoff's voltage law works under the assumption that the rate of change of magnetic field cutting the loop is negligible and Kirchhoff's current law works under the assumption that there is no local charge accumulation, okay. Now both these assumptions turn out to be true for a wide variety of practical circuits, that is why we are able to use them. Now there are conditions under which it's not true. Basically during the time period of the signal, if the circuit is physically so large that it takes so much time for let's say the electrical signal to go from one side of the circuit to another, then these things will not be true. A very common example is an antenna. Many of you would have seen antennas which are basically just a wire hanging in the air and but current is driven into it. Clearly current is not going off into the air. What happens is that the current alternates back and forth within the antenna and if you look at different parts of the antenna, they will have different currents, okay. KCL is clearly not true in that case but besides that KCL is true, okay. When you are not talking about such large circuits, large meaning, large compared to the electrical wave width, KCL is true and has this KBL, okay. Now before I go further, if there are some important doubts regarding what we covered in the previous lecture, I will go over them, okay. So it appears that there are no questions. So we will continue with the lecture. What we will do in this lecture is to discuss basic electrical elements that is two terminal elements and see their characteristics, okay. So before that let's see what an electrical circuit is. It's an interconnection of electrical elements and electrical elements are some things. We will see examples of that which take some voltage or current and manipulate them in some way, okay. So every electrical element is defined by at least one voltage and a current, okay. So every electrical element it turns out will have at least two terminals. I will show the terminals here and as I emphasized in the previous lecture, a voltage is applied between two terminals and a current is measured into this terminal, okay. And the same current will come out of it if it's a two terminal element. So electrical circuit will consist of interconnections of elements which can be either two terminal elements or it can have more than two terminals, okay. The initial examples that we will see will all consist of just two terminals. And so two terminals is the minimum number of terminals we need to have to have a meaningful definition of voltage or even a current, okay. Because with the single terminal we don't know the voltage with respect to what that is that we are measuring and similarly if you push in a current into a single terminal there is a question of where it will go out, okay. So we will have two terminal elements maybe more than two terminals, okay. That is more than or equal to two terminals. And in this course we will not worry about how to make these different elements but we will only take their characteristics in terms of their voltages and currents and use them to make some circuits and analyze the circuits, okay. Now the very first element that we will take is what is known as independent voltage source. And this is denoted by a symbol which has a circle with plus and minus mark. And what is given is the voltage in this polarity and that is equal to let's say some V naught, okay. So this has two terminals and what this is saying is that the voltage source will maintain a voltage of V naught between this terminal and that terminal, okay. Now one of the things I have to mention before we go further is the convention that we will take for describing the current, currents and voltages of two terminal devices. So what is done is to describe relationships between the voltage and the current. Now the voltage let's say it's defined with this polarity that is I call this terminal A and terminal B and I define voltages with A being positive and B being negative. That is I measure the voltage of A with respect to B. Then I will also specify the current that is going into the terminal A, okay. Now of course it's understood that the same current will come out of terminal B. Now we have to have this convention so that there is no ambiguity, okay. Now we know that a voltage can be measured of A with respect to B or B with respect to A. Similarly current going into A can be measured or current going into B can be measured. Now what is important is that the definition of voltage and current be consistent. So if the voltage V is defined with A being positive and B being negative, then the current I is measured with measured as going into the terminal A, okay. And this is known as the passive sign convention. And for every element we are going to use this, okay. That is the current I goes into positive terminal of V. What is meant is the terminal which is defined as plus while defining this voltage, okay. So for the voltage source I have to define the current as going into the positive of this V, okay. Now this plus and minus sign inside the voltage source only denote the definition. The value of V naught itself could be positive or negative. Again this is some source of confusion. I don't necessarily mean that because I put plus on top and minus on bottom. This V naught is always positive, okay. I could define it any which way. These signs are used so that I can use a consistent direction for voltages and currents, okay. While defining them. So what this says is this terminal A and terminal B. An independent voltage source maintains terminal A above terminal B by an amount V naught. The voltage at terminal A is maintained above the voltage at terminal B by an amount V naught. And the current can be anything, okay. It is not restricted by the voltage source. That's fine. So that's like a source of voltage and an analogy could be made to an infinitely large reservoir which will maintain its level, okay. Now you can take water out of this reservoir but it is not going to change its level because there is infinite amount of water in it. Similarly, voltage source is a source of infinite amount of current. You can draw any current from it but it will still maintain one of the terminals to be V naught above the other terminal, okay. Now it is very common to draw graphs of I versus V to obtain a graphical encryption of the same characteristics, okay. Now I would like answers from the participants. What would this look like? The I versus V curve for a voltage source, okay. There is a raised hand which I have approved. Please ask your question. I think somebody raised their hand in order to ask your question. Please go ahead. So a lot of people have answered this and it is pretty obvious. V equals V naught. That is all that is there to it and I can be anything. So this will be a vertical line on the IV plane at V equals V naught, okay. So it is the characteristic of a constant voltage source. Now the next element we will look at will be a constant current source. So this symbol means that a current I naught is flowing from A to B, okay. And if we have to measure the voltage across this, we will do it with this polarity, okay. Again there is option. Please ask. You are not able to hear any questions. I mean somebody raised their hand and I approved the request. Okay. Let us go ahead with the lecture. So as per our passive sign convention, the voltage will be measured like this and the current I would be flowing that way. And in the case of a cross, it will maintain a current of I naught. Please go ahead. Please go ahead. This is an independent current source whose value is a constant. Sir, we are talking about constant current source or independent current source. What do you mean by constant? So we should write it, see last. Yeah, so that means that it is a constant current. Because we have to define what it is independent with respect to. In the initial part of the course, we will deal with things that are constant with time, okay. So that is one definition of what is constant and not. Here what is meant by independent is that it is independent of any other quantity in the circuit, okay. So that is what is meant. No, no. You can write both. The constant usually refers to constant with time. That is what we are talking about. That means independent of other quantities, other electrical quantities in the circuit. And soon we will come to dependent sources, okay. So now, so this is a constant current source which is of course independent. If I plot I versus V for this, it will be a horizontal line with a value I naught, okay. So what the symbol means is that it will maintain a current of I naught through it regardless of its voltage, okay. So the voltage can be anything but the current value will be I naught. Now these are useful idealizations of some components that we actually encountered. And these are also the sources that will stimulate our circuit, okay. Now in case of electromagnetic fields, you know that charges are the primary source. You place the charge somewhere and it will create an electric field and then if the charge is moving and accelerating it will also create a magnetic field and so on, okay. But in our case we will not go down to the level of fields but we will deal with these voltage sources and current sources as stimuli for our circuit, okay. It appears that some people are having some difficulty with the bandwidth so I am going to pause the camera for some time. Now the next basic element that we will consider will be a resistor, okay. And as usual we define the voltage in some way, okay. Now I have chosen A to be positive and B to be negative but you could easily do it the other way. But what is important is once you have chosen the sign of V, you should be looking at I going into the positive terminal, okay. This is the sign convention that universally followed so that is what you must use. And I think all of you are already familiar with the relationship between V and I for this particular element. V would be equal to I times R, okay. And this is the famous Ohm's law and we can also write I to be G times V, okay. Where this R is the resistance and this G is the conduction, okay. If we plot I versus V for a resistor, what is the kind of plot that we are going to get? I would like some answers from participants on what is the plot that we will get if we plot I versus V for a resistor. Yes I can, okay. I am not able to hear the question. Somebody commented that there is no video because of bandwidth constraints for some users. I am now not sharing my video. I am only sharing the board on which I am writing. And many people answered this question as well. The answer is pretty obvious. It is a straight line passing through the origin. The slope of this would be the slope of this would be G, the conductance G, okay. Is there a problem with audio? Yes, I can hear you. What is the question? Okay, let's deal with that later, okay. Okay, some people said there is no audio. If this is the case, please type it in the chat window. And I will disable the video because some people had a very low bandwidth and with the camera on it was taking a lot of their bandwidth. But if there is no audio, if you are not able to hear me then please type it into this. Okay, it appears that the audio is fine. So the I1SV characteristics for a register would be a straight line passing through the origin and that's quite important. And the slope of that would be G, okay. Now let's quickly look at how a register is made. This is not the main focus of the course but this you can think of as some general knowledge. The easiest way to imagine a register that is made, although it may not actually be made this way, is to have a slab of some material. And you can have a contact at this end and the other end, okay. And this will serve as the terminals to which you can apply either the voltage or the current, okay. So you can apply voltage here and you can measure the current there. And as we said earlier, we would be related to I by this relationship, okay. So now what happens in a register is that electrons will be accelerated by the field created by the voltage applied between the two but of course it's not moving in a vacuum. So it will accelerate a little. It will collide with some fixed charges in the lattice of the material. Then it will start accelerating again and so on. And depending on the material, it will acquire a certain average speed and that will give you a certain average current, okay. So you need the knowledge of electromagnetics and materials to calculate the resistance for a given material, okay. But we will not go into those issues. We will take the resistance value for granted, okay and then use that. Now for a slab of this type, let's say this dimension is W and this dimension is L and the thickness of this is T. I think many of you already know that the resistance R will be the resistivity rho times the width of the material times thickness that is the cross section area across which the current is flowing, the area of that divided by the length that is the length along which the voltage drop is measured, okay. So this is the formula for the resistance and it is quite simple for a slab like this. But the resistance itself could be made in any number of ways. You could have resistance which is coiled into a spiral and so on. In those cases, the formula will be complicated but what is important for us is the relationship between I and V and it will always be simple like this, okay. We will not worry about how to calculate the resistance for a given physical structure but we will take the resistance as granted and we will use this model. We will use what is given here V equals IR or I equals GV and sometimes we will use this grad, okay. Now there is some comment in the chat window. Somebody said that it is a straight line passing through the origin with 45 degree slope, okay. Now here we have to understand that the x-axis and y-axis have different dimensions. So measuring the slope as 45 degrees has no meaning, okay. That will be meaningful only if they have the same dimensions. Then you can say the slope is unity. In this case, the slope is in conductance which has some dimensions, okay. So as you know the resistance is measured in ohms with this symbol and the conductance is measured in Siemens. We use the symbol S, okay. So that is the definition of a resistor, okay. That is once we define the relationship between the voltage across the resistor and current through the resistor, our job is done. We have described the resistor. If I connect the voltage source and the resistance like this, it is pretty obvious that the voltage across the resistor equals the voltage applied by the voltage source, okay. So this is V naught and V naught is applied across the resistor. Actually we use a very trivial application of Kirchhoff's voltage law. If you count the drop from here to there and there to there in the resistor, the sum will be equal to 0, okay. So under these conditions, V will be equal to V naught. This is the KVL and this is so obvious that we do not think of it as application of Kirchhoff's voltage law but that is what we are doing here, okay. Now because voltage V naught is applied across the resistor, a current I which is V by R which is V naught divided by R will flow through the resistor, okay. And that of course has to come from the voltage source. So the voltage source will supply a current of V naught by R, okay. Now whether you apply a voltage or apply a current, the resistor behaves in the same way. You can think of it as the current generated by an applied voltage or a voltage generated by an applied current. So I could also do this. Let us say I equals I naught that is what you are saying is this will maintain a current flowing from top to bottom equal to I naught through the current source. So that means that the current flowing here will be equal to I naught. This again is a trivial application of Kirchhoff's current law. The current entering this node here equals the current leaving that. That is the current supplied by the current source equals the current flowing in the resistor, okay. And in this case, the voltage across the resistor it has to be measured like this because the current is flowing downwards through the resistor and that will be equal to I times R which is I naught times R, okay. So there is a question from Subashish. Please go ahead. Subashish, go ahead. Okay. Nothing else that he has dropped off. Now, so I think all of you would already be familiar with these cases. That is if a voltage is applied across the resistor, we will have a current V naught by R flowing through it or if my current is applied, current source is connected to a resistor, current I naught will flow through it. In this case, you will get a voltage drop of I naught R and in this case, you will get a current of V naught by R, okay. Now, this comes from very trivial applications of KVL and KCL. I am not going to describe that. If you want to, you let me know and I will do that. But it is pretty obvious that the same current is flowing in the voltage source and the resistor here and in the current source and the resistor on the right hand side, okay. Now, one of the properties of a resistor, now before we go ahead, let me look at what happens when voltage sources are connected in series. Let us say I have a voltage source of voltage equals V1 and another one whose voltage equals V2 and they are connected in series. Now, what does series connection mean? Series connection means that for two terminal elements, you pretty much connect one on top of the other, that is, there will be one common terminal and the important criterion for whether things are connected in series is whether the same current I is flowing through them, okay. Now, because the bottom terminal of the upper voltage source and the upper terminal of the top terminal of the lower voltage source are connected together, by a trivial application of Kirchhoff's current law, the current flowing downwards from the upper voltage source has to flow into the lower voltage source, the same current is flowing, okay. So, this is what is implied by a series connection. Now, again, if you measure with respect to this, the voltage drop here is V2 and if you measure with respect to this, the voltage drop here is V1, that is what is given by, not in this polarity, sorry, in this polarity. So, again by a trivial application of Kirchhoff's voltage law, if I measure from here to there, the voltage will be V1 plus V2, okay. So, if you connect voltages in series, the voltages will be summed together. Please go ahead with the question. Yes. Hello. Yes, please go ahead. Hello. Yeah, please go ahead. Yeah, please go ahead. Yeah, that is correct. Okay. So, again, this is a confusion. I would like to clear right at the beginning. The question is, in this circuit, the current is flowing in this direction, okay, in this direction, that is, it is flowing from bottom to top in the voltage source and then top to bottom in the resistor, okay. Now, when I showed the definition of the independent voltage source, I marked I like this. Now, like I emphasized then, this I does not say that the current is flowing this way, okay. It is only measured this way for the sake of plotting and for the sake of convention, okay. If you look at this graph, what it says is this value of I can be anything. It can be any positive value or any negative value. Now, what it means in this particular circuit, let me show the two parts separately. I will measure the voltage across the voltage source in this direction and I will measure the current through the voltage source in that direction, okay. And for each element, I will follow its convention. I will measure the voltage across the resistor in that direction and the current through the resistor in that direction, okay. Now, clearly by KVL, VR equals VV which is equal to the voltage source value V naught and by KCL applied at this node, IV plus IR will be equal to 0 or IV equals minus IR and IR itself is given by the property of the resistor which is V naught by R, okay. IR will be VR divided by R and that happens to be V naught by R and IV happens to be consequently minus V naught by R, okay. That is for this particular circuit and there is no contradiction here because this only shows the direction in which I have measured the current through the voltage source but this graph tells you that it can be either positive or negative, okay. Now, in that particular circuit, it happens to be negative and it happens to be equal to minus V naught by R. I hope that clears the doubt. Is that okay? Please go ahead with the question. Pumit, are you still there? Yes, sir. Yeah. Sir. Yeah, please go ahead with the question. In the previous explanation, you said that the voltage is minus, so current is I equal to minus V naught by R. The current as measured in the voltage source as per the convention, that is minus V naught by R, yes. Sir, shouldn't we follow a single convention? Shouldn't we follow a single convention there, either take it from the positive side or negative? We are following two conventions for resistors different and for voltage sources. No, no, I have used exactly the same convention for resistors and voltage sources and I will use it for any two terminal element, okay. So, for each element, for all elements, if I define the voltage like this, okay, with the upper terminal positive, I will measure current flowing into the positive terminal of that V, okay. That is current flowing into the upper terminals and that is exactly what I have used for the current source here, and the resistor here, okay. I R is flowing into this plus sign of V R and I V is flowing into plus sign of V V, okay. So, that is just a convention for denoting the relationship between voltage and current, okay. It has nothing to do with which way the currents are flowing. Now, depending on the, depending on the circuit connections and element characteristics, these currents could be positive or negative. In case of a resistor, it says that if you have V R defined like this and I R defined like that, V R will be I R times I R, V R, I R times R, okay. That is Ohm's law. Similarly, for a voltage source, it says that if I have V V like this and I V defined like that, V V will be some fixed value V naught and I V can be anything either positive or negative, okay. So, I have used the uniform convention. Sir, if any circuit is given, then we have to independently take the elements and mark them positive, negative and then we have to go conventionally rather than looking at the circuit as a whole. No, no. First of all, if it's a small circuit, you may not have to do it systematically, but when analyzing last circuit, this is what you will do, okay. So, this is the convention that is used for showing the characteristics of the element, okay. Now, whether the current is positive or not depends on the circuit. Like I said in the very first class, I can show a wire with A to B and I can mark I equals 1 ampere and similarly, this is exactly the same as saying the current is flowing from B to A which is minus 1 ampere, okay. Similarly, if I have two terminals, I can say this is 1 volt and that is minus 1 volt, okay. So, these two are exactly the same. So, I can say A is above B by 1 volt or B is above A by minus 1 volt, okay. I will go ahead with the lecture. So, now if I have voltage sources in series, the total voltage will be V1 plus V2. This is again by some trivial application of Kirchhoff's voltage law, okay. Now, let's take three cases. I apply V1 across a resistor R. The current flowing would be V1 by R. Similarly, if I have a voltage V2 applied across a resistor, the current will be V2 by R. If I have V1 and V2 in series applied across the resistor, the voltage across the resistor will be V1 plus V2 and the current through the resistor will be V1 plus V2 divided by R which is V1 by R plus V2 by R, okay. The voltage across the resistor here is V1. The voltage across the resistor here is V2. Now, if we think of the voltage as cause and current as effect, it could be both ways for the resistor but if we think of the voltage as a cause and current as the effect, you see that here I have a cause equal to V1 and I have some resulting current. Here I have cause equal to V2, some resulting current and here I have a cause that is the sum of these two, okay. That is the cause. And if you look at the result, that also happens to be the sum of the individual results, okay. That is if I apply voltages V1 and V2 separately to a resistor, I get currents of V1 by R and V2 by R respectively. If I apply a voltage V1 plus V2, I will get the sum of individual currents, okay. Now, this property is known as linearity. In general, if you have any system which has a certain cause and a certain effect, I can apply different values of cause individually and I can measure the different effects. If I apply all the causes together, all the results, all the effects will be summed together, okay. So that is the meaning of linearity. That is if you have stimulus or stimuli which are combinations of certain values of stimulus and the responses will be sum of individual responses, okay. And this is known as linearity and this will be exploited very heavily in our analysis of circuits. Linear circuits form a very large class of useful circuits and mainly in this course, that is what we will be analyzing, okay. And any element that follows this is known as a linear element and obviously a resistor is a linear element because if you apply a cause that is the sum of causes, you will get an effect that is the sum of effects, okay. Now the question I have is we have discussed three elements, an independent voltage source and independent current source and a resistor. Now please let me know if the voltage source and the current source are linear or not. I got some responses and some said they are linear, one person I think said they are not linear. Again the way to evaluate this is to see whether the effect is proportional to the cause or if you have individual causes will the effects be summed up, okay when you sum the stimuli, okay. Now let us consider a voltage source or perhaps let me take a current source because that will be analogous to this example. Now let me copy over this whole thing. Instead of a resistor let me have current sources, okay. Let me have a current source I0 and here I have I0 and here I have I0. Now clearly the current in this case is I0, the current in this case is also I0 and the current in the third case is also I0, okay. That is here I apply V1, the current anyway is independent of what is applied to it. So it will be I0, here also it will be I0 and here also it will be I0. Although I apply a voltage that is a sum of the individual voltages here that is here I have applied V1 and here I have applied V2. Now if this principle that is if combining the causes will give you a combination of the effects if that is true then this current should have been 2 I0 but it is just I0, okay. So in fact it is not equal to the sum, right. The resulting current is not equal to the sum of currents in individual cases. So this says that the current source is not linear in the sense that a resistor is linear, okay. So what says if something is linear or not? If it obeys the principle of superposition and what is the principle of superposition? The response to a combination or sum of inputs is a linear combination that we are talking about is the sum of responses to individual inputs, okay. If this is the case, this is a super resistor then it means that it obeys the principle of superposition and if it obeys that it implies linearity, okay. So a resistor is linear, a voltage and current source are not linear, okay. Now this has nothing to do with the characteristics being a straight line, okay. Sometimes that is what is called linear but here for us what linearity means is whether it applies, whether it obeys superposition or not, okay. So now I have another question for the participants. We draw these IV characteristics and I showed you that the current source is not linear and in a similar way voltage source is not linear also, okay. You can apply two individual currents to a voltage source, you will get some voltage and if you apply the currents together you will get the same voltage. So that is also not linear and we have these IV characteristics that we have drawn. So what is the, what feature of the IV characteristic guarantees that the element is linear, okay. That is my question. Now I have a number of responses. Basically if we have a two terminal element then if the IV characteristic passes through the origin that is linear, okay. And clearly this is true for a resistor but for a voltage source or a current source it is not true, okay. So resistor is linear and voltage source and current source are not linear in that they do not follow superposition. Now just some more details of what is superposition. Let us say you have some system, okay. I will show it in an abstract way like this and here we are talking about two terminal elements whose stimulus could be a voltage and the response current or vice versa. So let us say you have some input and it has some output, okay. I am here talking very generally. I am not saying what system it is and what input output it receives and so on. So first of all for linearity y must be 0 if x is 0, okay. Otherwise you can show that it cannot be linear. Now if x1 gives you y1 then if it is linear alpha times x1 will give you alpha times y1, okay. And similarly if x1 results in y1 and an input x2 results in an output y2 and input of x1 plus x2 will result in an output of y1 plus y2, okay. Now these are not independent statements that I am making. Some of them imply the other, okay. So these are all necessary conditions. Now this says that if you have the IV characteristic it has to be passing through the origin, okay. Now let us consider some other simple two-dimensional elements. This we will just look at the characteristic now and we will analyze circuits with capacitors and inductors later. But if I look at a capacitor like this and define the voltage Vc in this as you will I have to look at the current in this direction. So again it does not mean the current will always be flowing this way. This is the current that I will measure, okay. You have to measure current. I have to choose some direction and this is the direction that I am choosing. And you know that IC will be some parameter of the capacitor C times the time derivative of the voltage across the capacitor, okay. And if I invert this I can write the voltage across the capacitor as 1 over C integral IC dt, okay. Now one of the things about the capacitors that distinguishes it from a resistor is that the value of the voltage across the capacitor depends on the current not at the present time but all of the past time, okay. Whereas for a resistor the voltage across the resistor at any instant is dependent only on the current at that instant, okay. V equals IR is true even if V and IR varying with time. For each instant V will be equal to IR that is the current at each instant will depend only on the voltage at that instant or the voltage will depend on the current at that particular instant. Whereas the same is not true for a capacitor. The voltage across the capacitor depends on not only the current at the present time but also the history of the current, okay. Now this is the definition of the IV relationship of a capacitor and because it is related by this derivative we cannot plot I versus V, okay. We can do that only when the element is memory less that is the current and voltages are related only at that instant of time. That is current in that instant is related only to the voltage at that instant of time, okay. But this gives you the relationship just like we did for the resistor except it is slightly more complicated it is a derivative. My first question to the participants is is a capacitor linear or not? I have a number of responses a majority of them say that it is not linear or a couple of people said it is linear and somebody asked if for I versus V is a straight line like I said we cannot plot I versus V for this because if you think about it that is possible only when the voltage at this instant of time is related to the current at this instant of time, okay. In this case that is not true. The voltage at this instant of time depends not only on the current at this instant but also on all the previous instance of time, okay on the history of the current. So we cannot plot I versus V but we can state this for linearity. How do we do that? Let us say I apply V1 across the capacitor. There will be a certain current I1 I which will be C times the time derivative of V1, okay. Now first of all I am assuming that this voltage is varying with time. Now if it is not varying with time the formula still applies but the current will be 0. That is if you apply a constant voltage across the capacitor there will be no current flowing through the capacitor. So the more interesting thing is when the voltage across the capacitor is varying with time if that is the case the current will be related to or it will be proportional to the time derivative of the voltage, okay. Now instead if I apply V2 the current the new value of the current will be C times dV2 by dt. And finally I could apply the two together, okay. Exactly the same experiment I did with the resistor earlier V1 plus V2. In this case the new current would be C times d by dt of V1 plus V2 but the time derivative of sum of two quantities is the sum of individual time derivatives, okay. So clearly the responses are sum of individual responses and linearity is indeed true, okay. So this is how you test for linearity. I applied V1 from the current. I applied V2 from the current. I applied V1 plus V2. What I say is that the current will be the sum of the individual cases and the capacitor is therefore a linear element also, okay. The IV characteristic may be more complicated than that of a resistor but it is still a linear element, okay. Now there were some comments. Somebody said that it is not a dissipative element. That is why it is not linear. But it has absolutely nothing to do with whether it is dissipative or not. It only has to do with whether the IV relationship is linear or not and it is a time derivative and it happens to be linear, okay. Okay, let us look at a simple construction of a capacitor. The way this is made. Again this is just a depiction. The practical capacitor does not have to be made like this. It just has to have two conductors which are separated from each other and two terminals are connected to it. A, B, A and B, okay. I described the capacitor by its IV relationship but I think you already know what it is. If I apply a voltage across the capacitor, what happens is, so let us say this is certain V on the upper plate where this V is applied. The positive terminal of V, the plate connected to the positive terminal of V, there will be a positive charge. There will be a charge, I mean that whether it is positive or negative depends on the value of V. The charge depends on this voltage V and it is given by C times V. And similarly on the other plate there will be an equal and negative charge and that will be minus C, okay. If I call this terminal A, this is QA and QB, okay. So what a capacitor does is to store charge and store charge Q equals C times V, okay. Now first of all you have to interpret properly what this store charge is. This means that on one plate there is plus C times V and on the other plate there is minus C times V, which is plus and which is minus. That depends on which is the plus terminal of V and which is the minus terminal of V, okay. And also this relationship Q equals C V, you see that this is somewhat similar to V equals R times I, okay. And this makes linearity even more obvious, okay. Now how do we get the relationship between I and V? We know that the current is rate of change of charge. What happens is as this voltage changes, the charge on the upper plate changes. And how will it change? The way it will change is by drawing current from this wire or pushing current into that wire, okay. This charge has to come from somewhere and it has to come from this wire. That means that the current in this wire will be dependent on change of voltages, okay. And the current will be the rate of change of charge, which will be C times the rate of change of voltage, okay. So that is how it comes about. And from this relationship if you plot Q versus V, it will be a straight line passing through the origin and that will be a linear relationship. And the I-V relationship is simply the time derivative of this and time derivative is a linear operator. So that relationship is also linear, okay. The capacitor stores charge, Q equals C-V and that is a linear element. Any questions about the capacitor? Okay. Now my next question is can the voltage across the charge change instantaneously? The question is can the voltage across the capacitor change instantaneously? I think one person said yes and a number of people said no. My question was can the voltage across the capacitor change instantaneously? That is can it be one value and then suddenly change to another value, okay. Oh, it is possible. I mean normally you hear that a capacitor holds its voltage and it cannot change instantaneously but we have to qualify that, okay. If it changes instantaneously, what it means is that if not I versus C it will be 0 and here it will go to infinity and that is denoted by an impulse and then it can go to 0. So what we really mean is that if the currents are finite the capacitor cannot change its voltage instantaneously, okay. An instantaneous change in the capacitor voltage will demand an infinite current and that is will demand an infinite current. So if the currents are restricted to be finite then it cannot change its voltage instantaneously. So if you allow an infinite current which is not practical but in theory it is possible then the voltage can change instantaneously. And one way is if you have a voltage source across the capacitor and the value of the voltage source changes instantaneously then the voltage across the capacitor has to be exactly equal to the voltage across the voltage source and it will also change instantaneously, okay. But in this case what happens is the amount of current will be drawn at the instant of the voltage change, okay. Now finally the last of the simple elements that two-terminal elements is the inductor, okay and the symbol for an inductor I think you are all familiar with it is like a coil. Again I will define V and I in this consistent direction. Now in terms of that the inductor and capacitor are dual elements of each other. What does the capacitor do? We already discussed that it stores charge and the amount of charge is c times the voltage across the capacitor and in case of an inductor it also stores something it stores energy in the form of magnetic field. A capacitor stores energy in the form of the electric field between the plates. This side it stores magnetic flux, okay. We will not worry about the details of construction of the inductor and how much flux it stores and so on. But just like a capacitor stores charge it stores flux and the amount of flux linkage that is stored is given by some number L which is called the inductance of the inductor times the current through the inductor, okay. So this c the constant of proportionality is called the capacitance for a capacitor and it is measured in parrots. One parrot will store one coulomb charge one coulomb of charge when one volt is applied across it and similarly this is the inductance and it is measured in Henry's, okay. And it turns out that one Henry is the inductance which will store one rubber of flux when one ampere of current is applied through the inductor, okay. Now a capacitor had a current which is basically the rate of change of charge and which turned out to be c times the rate of change of voltage, okay. And the role of this holds true for the inductor. The voltage across the inductor will be the rate of change of flux linkage through the inductor and that will be L times dI by dt, okay. So you can see it is the exact counterpart of that. There is a question from Priyanka, please go ahead. Priyanka, okay. So the inductor is the counterpart of the capacitor. You can see that the role of voltage and current are reversed if you compare it with a capacitor. In a capacitor, the current is the time derivative of the voltage and here the voltage is the time derivative of the current. Now the question is, is the inductor linear? This is a question for all the participants, okay. So as I think many of you guessed it correctly this time and it is linear, okay. Because a current I1 will give a voltage L times dI1 by dt and a current I2 will give a voltage L times dI2 by dt. I am assuming that you are thinking of current source I1 applied to the inductor, current source I2 applied to the inductor and if I apply a current of I1 plus I2 I will get L times d by dt I1 plus I2 and that will be LdI1 by dt plus LdI2 by dt, okay. So inductor is also a linear element. If you apply two currents individually, you get two voltages and if you apply the two currents together that is if you apply the sum of the two currents then you will have voltage that is the sum of the individual voltages, okay. So this is also linear. Now the only one point of confusion that can arise sometimes with inductors is that in elementary physics textbooks you will see that the current will oppose the change in flux and it is minus LdI by dt and here we have said it is LdI by dt. Now there is no contradiction at all here in physics textbooks usually they do not show the sign convention what they mean is that if there is a current going here and it tries to change then their convention is to measure the voltage like this okay with plus on the bottom and minus on the top where i is flowing that way and we will use this convention that is the passive sign convention that is consistent for our purposes. So in this polarity it will be minus LdI by dt and in this polarity it is plus LdI by dt okay. Also another thing is I showed you the simple construction of resistor and that of capacitor but I will not do that for the inductor because it is rather complicated. But whatever it is it will consist of some coils of wire and the measurement of inductance depends on the geometry of how the coil is wound and what material is built and so on. Again we do not have to worry about that. In fact part of what you learn in this course is that we can describe the element by the voltage at the terminals of the element and the current through the terminals of the element do not worry about the internal details that is for another place and time. For us as long as we know the voltage the relationship between the voltage across the terminals and current through the terminals we can describe them with some relationship and with that we can do circuit analysis okay. Now this is how this is like a hierarchy of explanations and this is how we have to build up right. We cannot go down to the level of fields every time so we describe all we want with voltages and currents and we can make complicated circuits based on that okay. So we have discussed now independent sources which are a voltage source and the current source and we also discussed a certain number of linear elements a resistor which follows Ohm's law a capacitor whose IV are related by and an inductor whose IV are related by okay. And these have similar quantities that have this proportionality relationship okay. There are no voltages and currents in this case it is charged and in the case of the inductor it is flux linkage okay. And these make it even more obvious that elements are linear and the derivative operator itself is linear so all these are linear elements okay. Another question is can the current and an inductor change instantaneously again many people said no it cannot change instantaneously we have to qualify this statement okay. Now it can change instantaneously if you allow for an infinite voltage across the inductor okay. If you restrict the voltage to be a finite value which of course is a practical case but in theory it can be infinite so if you restrict the voltage to a finite value then the current cannot change instantaneously okay. Okay. So now brings us to the end of this lecture. Now in the next lecture we will discuss what is known as the mutual inductor. It turns out that you can have two coils close to each other again we will not worry about the physical construction of these things where the flux induced by one coil can link with the other coil so the voltage across one coil depends on current not only through itself but also on current through another coil we will discuss that one and a number of other things. So what we did was first of all we established what is known as the passive sign convention and discussed several two terminal elements the voltage source, current source and resistor, capacitor and the inductor we also looked at issues of what is meant by linearity resistor, capacitor and inductor are linear but voltage source and current source are not. Now for us what is meant by linearity is that if it obeys superposition then it means it is a linear element and so what is superposition? Superposition says that if you have two individual inputs you will get certain outputs. Now if you combine the inputs and apply the output should be the combination of the individual outputs that is meant by that is what is meant by superposition and from this it will be implied that if you apply a zero input you should get a zero output. If you do not get that then it cannot be a linear element. By that criteria by that criterion the voltage source and current source are not linear whereas the resistor, capacitor and inductor are not linear. So that you should be able to recognize and also you should be able to apply the passive sign convention without confusion. That is only the way only the polarity with which voltages and currents are measured. It does not say anything about whether the actual voltage and current that are present are positive or not. So I will take some last questions and end this lecture. Yes. Okay it appears that there are no more questions. I just have some announcements basically whatever I mentioned earlier I am going to repeat now. What is the question? Yes. Yes please go ahead. Yes. I mean if you want to check for linearity you have to check for superposition principle. The question is how do you test for linearity and to test for linearity you have to apply superposition and then see if it is valid or not okay. Now of course based on elementary math you will know I mean if it is a proportional relationship if V equals IR that will follow superposition and similarly if it is proportional to derivative and so on it will also obey superposition because the derivative is a linear operator okay. So after some experience you will be able to simply look at the relationship and see if it is linear or not okay. Yeah from the IV relationship the current the voltage is sorry the current is C times dv by dt you can say that it is linear okay. If you already know the know that the time derivative is a linear operator and so on. Now if you are not sure you can test it and see okay. The derivative is linear operator what it means is if you take the if you differentiate the sum of two functions you will get the sum of derivatives of the individual functions okay that is linear okay. So with that we come to the end of the lecture.