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In the previous lecture, we discussed in detail opamp circuits, we looked at a couple of example opamp circuits and also I emphasize that opamp has to be used in negative feedback and I also mentioned how to find the signs of the opamp in any circuit such that it is in negative feedback. Okay, an opamp is basically a voltage controlled voltage source with a very large gain because the gain is very large and it is a negative feedback, it forces the input signal of the opamp to be a very small value which can be considered to be 0 and if the gain of the opamp goes to infinity that is the idealized version of this which we call the ideal opamp in that case the input difference voltage of the opamp can be is assumed to be 0 okay and there is no input current into the opamp. So based on these things you can analyze any opamp circuits that you have okay. So if there are any questions regarding what we did in the last lecture please go ahead now otherwise we will start with today's topics. So far we have looked at circuits that have resistors and control sources and so on and all these are known as memoryless elements. What is meant by this is that the current at instant T depends only on voltage at instant T that is it does not depend on the voltage at any other instant okay or similarly if you have a voltage to current relationship the relationship is exactly the same and for all instants and more importantly the voltage at some instant is related to current at that instant only okay similarly for control sources the control quantity is related to the controlling quantity instantaneously only at that instant. Now if we have capacitors and inductors we know this these properties are not true if you have a capacitor the voltage across the capacitor depends on not only the current at that instant but on the history of the current. Similarly if you have a current through the inductor it depends not only on the voltage across the inductor but on the history of the voltage across the inductor okay. So capacitors and inductors are elements with memory okay so what we will do in this lecture is to try and analyze circuits which have inductors and capacitors in addition to resistors is this okay many questions so let me take a circuit that we are all familiar with a resistive divider okay now let's say I call this V s and the voltage is 0 okay what will be the output voltage please try to answer this what is the output voltage when I say output in this case I have defined the output to be this particular voltage okay what is that voltage going to be what will be the voltage here when the input voltage V s is 0 anybody what is the voltage going to be so this voltage clearly from the voltage divider formula is V s times r2 by r1 plus r2 okay and if V s is 0 this voltage is also 0 okay this is a simple this is a simple resistive divider let's say it jumps from 0 to some voltage V p okay what happens then the output voltage jumps from 0 to V p times r2 by r1 plus r2 okay so you can see that the variation of the output voltage with time the horizontal axis here is time this is time and here also it is time the variation of the output voltage with time is exactly the same as the variation of the input voltage with time with a scaling factor okay now let's say we do something we have V s and I replace r2 with a capacitor and I have R and C okay what happens in this case let's say initially the capacitor voltage is at 0 and the input is also 0 clearly in that case what happens is that the capacitor voltage is 0 and because V s is 0 no current flows through the resistor which means no current through the capacitor so if there is no current through the capacitor its voltage cannot change okay so in this case the voltage remains at 0 if the input is 0 now let's say the input jumps from 0 to V p okay what happens to the voltage please try to answer this let me call this V c V c when V s jumps from 0 to V p okay there is some of you attempt to answer this what is the voltage V c going to be when V s jumps from 0 to V p okay we have an answer it says that 1 over C integral I of t V p okay this is correct but the only issue we have now is that we don't even know what this I of t is okay so that also we have to solve for I mean we have done this for the simpler circuit before the current in this is V s by r1 plus r2 okay whereas here we have to find out and the current itself depends on the voltage of the capacitor okay so now what I would like to do is to write the equation for this circuit so that I can solve it I will do what I will do is this I will write kcl at this node so the current through this I will express everything in terms of the capacitor voltage V c so the current through this is C times dV c by dt we know that that is the current through a capacitor which has a voltage V c across it okay now that has to be equal to the current coming from the resistor so what is the current through the resistor in terms of V c please try to answer this okay that is correct so the capacitor current which is C times dV c by dt equals the current in the resistor which is V s minus V c divided by r okay what I will do is I will group all of the terms containing V c to the left hand side and I will also multiply both sides with r so here rc dV c by dt plus V c equals V s okay now this is a feature of circuits containing capacitors or inductors or both that the equation governing the circuit will be a differential equation okay whereas previously they could write if I still call this V c there is no capacitor here I will just call that voltage V c V c equals V s r2 by r1 plus r2 okay so it is an algebraic equation whereas in this case it is a differential equation if you have resistors or capacitors okay so we have to solve the differential equation to find the solution okay so let us try and do that so before that we will try and solve a couple of cases intuitively and then go on to finding the solution of the differential equation okay any questions so far so let me rewrite the circuit and the equation again okay this is the circuit and for arbitrary inputs it is not very easy to solve for these things so we will take specific kinds of input so initially in particular I will take an input that goes from 0 to V p at a particular time instant okay so this is t and let us say this is t equals 0 okay and we also know that the differential equation governing this is rc dV c by dt plus V c equals V s okay so now first let us forget the differential equation and try to sketch the output based on what we know about resistive capacitor and the voltage okay I already said that initially the voltage across the capacitor is 0 okay so if I try to plot V c versus time okay before t equal to 0 the voltage will be 0 so that is what I have assumed okay so this is because I say initial condition but I really mean 40 less than 0 okay on the capacitor now at the instant that at the instant that the V s steps from 0 to V p okay what happens to the capacitor will it change will the capacitor voltage change when I said t at t equal to 0 I mean at this instant will it suddenly change that's the question what do you think yeah so as Aarti answered the capacitor voltage cannot change suddenly now as I emphasized before this means that we are already assuming that the currents are finite okay so if the current is infinite the capacitor voltage can change suddenly but the current is finite and in this case we cannot have infinite current okay because if we do an infinite current through the capacitor it means that we also have infinite current through the resistor which in turn means an infinite voltage across the resistor okay so this capacitor voltage will not change will not jump at t equal to 0 okay now my question is what is the current through the capacitor at t equal to 0 what is the capacitor current at t equal to 0 so clearly it is V s by r or V p by r because the value of V s is V p so the capacitor current at t equal to 0 this is V p divided by r okay so that means that this current I see at t equal to 0 is V p by r now because the current is flowing through the capacitor its voltage tends to increase its voltage tends to change and from the direction of the current we know that V c is going to increase okay so what happens is it increases so I will show a little bit of increase like this now what is the slope of this increase going to be what will be the slope please try to find out because we know the current through the capacitor so we should know the slope of the voltage across the capacitor okay because the capacitor current is nothing but c times dv by dt dvc by dt so the current through the capacitor is basically related to the slope of the voltage across the capacitor okay so what is the slope of the capacitor voltage versus time so when I say slope of the capacitor voltage versus time obviously it should have a dimensions of volts over time volts per second something like that right so we know the capacitor current V p by r okay so we have to equate that we have to use that in the equation for the capacitor current and find out the slope of the capacitor voltage please try to calculate that so someone please try to answer this question so the current through the capacitor is V p by r which also in terms of the capacitor voltages c times dvc by dt okay so what I was looking for was dvc by dt the slope of the capacitor voltage okay so clearly that is equal to V p by rc okay so the slope at t equal to zero will be V p by rc okay so it increases a little bit I have shown it with this blue line over here now let's say after it increases a little bit let's say I come to this point what happens then to the current through the capacitor please try to answer this now initially at this point the current equals V p by r what will it be at the other point a little bit I mean for t a little bit more than zero the capacitor voltage increases so what happens to the current through the capacitor in this circuit somebody please attempt this current through the capacitor here what happens to the current through the capacitor so there is an answer that says slope less than one but keep in mind that here we are plotting voltage versus time so the slope is measured in volts per second okay so when we have a dimension quantity like that it doesn't make sense to say it is less than one okay so it has to be less than some quantity okay so another answer is that it is V p by r but is it true why is it V p by r what is the voltage across the resistor now the voltage across the resistor at this time instant what is it going to be is it V p is it less than V p is it more than V p exactly so the current through the capacitor is V p minus V c by r now when the capacitor voltage was zero the current was V p by r but when the capacitor voltage increases a little bit the current will decrease okay so the current here is smaller than the current there now what does that mean because the current is proportional to the slope of the capacitor voltage or the other way around the capacitor voltage the slope of the capacitor voltage is proportional to the current through it so the capacitor voltage will increase a little okay but after that the slope will reduce because the current has reduced now if you look at this point the voltage across the capacitor has increased further okay so that means the current through it is reduced further if the capacitor voltage increases the voltage across the resistor decreases because the capacitor voltage plus resistor voltage equals V p okay in this time in this range of time it equals V p which is a constant so as capacitor voltage increases the resistor voltage decreases so the slope reduces further okay so the slope will go on reducing okay now in reality it doesn't reduce in discrete steps like this continuously it will go on reducing because as soon as the capacitor voltage increases a little bit the slope will reduce then it increases a little further it will reduce and so on okay it turns out that the slope will go on reducing then finally it will asymptotically read some value okay it will read some value and after that the voltage does not change much okay so the slope becomes almost zero now what is this voltage what is the voltage it almost reaches what is the voltage at which it stops changing further please try to answer this clearly if the capacitor voltage is V p the voltage here is also V p and the resistance has zero volts across it so that means there is no current flowing through it okay so the capacitor voltage is V p and there is no current through the capacitor so that means that the capacitor voltage will not change at all okay so this would be equal to V p okay so I hope this intuitive description of how an RC circuit responds to a step is clear if not please ask your questions and I will clarify it further what happens is let's say you start from a discharged capacitor that is a capacitor which has no charge and in this circuit at t equal to zero there will be no infinite currents so the capacitor voltage will not change instantaneously but at t equal to zero because the input voltage increased to V p the current will increase okay so the current through the capacitor increases and sorry the current through the capacitor will be non-zero it will be V p by r so that means that the voltage across the capacitor increases as the voltage across the capacitor increases the voltage across the resistor reduces and that reduces the current and the slope of the capacitor voltage okay and this happens continuously the capacitor voltage will increase little by by little and the current through it will reduce little by little reducing the slope of increase. Finally, the slope of the capacitor voltage almost becomes zero and you can kind of say that the capacitor voltage reaches some value and that value is Vp. That is because the capacitor voltage will stop changing when the current through this is zero. If the current through this is zero, current through the resistor is zero, the voltage drop across the resistor is zero. So, that means that here also you have Vp. So, please think about this and ask me any questions you may have. The slope is Vp by R at k equal to zero. So, gradually reduces as Vc increases and Vc syntotically reaches Vp. What is meant by asymptotically is that it gets very very close to Vp, but does not quite get there. It gets there only at t equal to infinity. So, finally, when Vc equals Vp, current through the resistor and the capacitor is zero and Vc does not change anymore. So, any questions about this? Behaviour of an RC circuit for a step input, this kind of input which jumps from one value to another value is known as a step input. Now, let us go the other way around just for completing the calculation. So, let us say initially the capacitor voltage Vs is at Vp and the capacitor voltage Vc is also at Vp for t less than zero. So, imagine that we take the capacitor and charge it up to Vp. So, previously we charged it almost to Vp. So, we charge it up to Vp and then after some time we bring the input Vs down to zero. So, this is exactly the opposite of what we had earlier. Instead of going from zero to Vp, we go from Vp to zero. So, now if you look at the differential equation that governs this circuit, it is RC dVc by dt plus Vc equals Vs and this is nothing but zero for t greater than zero. So, we have a differential equation with zero on the right hand side. That is the differential equation that governs this particular circuit in this condition. So, again I put Vc versus time. So, like I said initially Vc is at a value Vp and also for convenience I will redraw the circuit with Vs equal to zero. With Vs equal to zero it is a short circuit and I have this C and R and the voltage across this is Vp at t equal to zero. Now, what will be the current through the capacitor? So, let us say I call this as Ic and because Vs equals zero I have shown it as a short circuit. So, if the capacitor current is flowing like this I have used the passive sign convention. What is the current Ic? So, there is an answer Vp by R, but please mind the signs because I have used passive sign convention for Ic. So, flowing from top to bottom that is right. First of all the voltage across the capacitor and the resistor they are the same in this circuit they are in parallel. So, this voltage with this polarity is Vp at t equal to zero which means that a current Vp by R flows in that direction. So, this Ic which is the current flowing from top to bottom in the capacitor is minus Vp by R. So, what happens to the capacitor voltage? After t equal to zero capacitor voltage we see for t greater than zero. What happens now? Will it increase or decrease or remain the same? So, that is right it is going to decrease. So, it will decrease and the slope is again given by the capacitor current. Minus Vp by R is the capacitor current C dVc by dt. So, dVc by dt is nothing but minus Vp by Rc. So, slope is minus Vp by Rc. So, there was a question here that was whether after charging a capacitor to some voltage can we use it as a voltage source? Yes we can in fact it is used like that in some cases, but of course you have to remember that a capacitor is not a voltage source. As you draw current from it its voltage is going to change. So, after some point the voltage changes so much that you cannot use it anymore. So, let us say you have you want a 5 volt source you can charge a capacitor to 5 volts and use it, but as you draw current from it the voltage will start decreasing and at some point it may go so below 5 volts so far below 5 volts that it may be useless. But in fact this is used if you want a source for a very short time where you are not drawing so much charge or so much current from the capacitor you can use that. In fact in many electronic gadgets when you want to change batteries during the time you take to change the battery there will be no source. So, some of those gadgets may use a very large capacitor. So, in the time that you take to change the battery the capacitor will be acting like a voltage source and supplying power, but of course it cannot do it indefinitely. So, if you leave it without batteries for a long time it will just discharge and die out. So, the slope is minus Vp by RC. Let us say we come to this point. So, what happens to the slope after that? What happens to the current through the capacitor and what happens to the slope of the voltage across the capacitor? The voltage across the capacitor is decreasing, but after it decreases a little bit what happens to the slope of the voltage? Now clearly the voltage across the resistor is also VC and as VC reduces the voltage across the resistor reduces which means that the current through it reduces and if the current reduces the slope of the voltage also reduces. So, it does that and then does that and that and that and so on. So, finally it turns out that it will asymptotically reach 0 because the input is 0 for T greater than 0 and if this capacitor voltage goes all the way to 0 then there will be no current through the resistor and that means that the capacitor voltage cannot change at all which in turn means that the capacitor has reached its steady state. So, this asymptotically goes to 0. Is this clear? Any questions about this? So, now we have looked at the behavior of the capacitor voltage for two cases when the input jumps from 0 to Vp or from Vp to 0. Qualitatively it is the same, it starts changing with a certain slope the capacitor voltage and the slope goes on reducing. The magnitude of the slope goes on reducing. So, it starts of steeply and then becomes less and less steep and finally asymptotically reaches some value. When you change the input from 0 to Vp that value is Vp. When you change the input from Vp to 0 that value is 0. This is fine. So, key things to remember are that are the values of the slopes and so on. So, this slope is minus Vp by Rc. So, it starts from Vp and if it went like a straight line with that slope it would cut the time axis at the equal to Rc. But of course it does not do that because the moment the voltage reduces a little the current also reduces. So, the slope reduces and so on. So, now what do you think this shape is? We have done it intuitively and graphically but we have to also be able to write a formula, an expression for Vc as a function of time. So, what is that function? Any guesses for what that is? So, let us see. Let me take the second case where the input Vs was 0 after t equal to 0. So, for t greater than 0 this is the model with Vs set to 0. Now, let us see what shape it is. Somebody answered parabolic. It turns out it is not a parabola. The differential equation governing this is Rc dVc by dt plus Vc equals 0. Now, I will rewrite this as dVc by dt is minus 1 by Rc times Vc. So, what this equation is saying is that the derivative dVc by dt is proportional to the function itself. So, the derivative is proportional to the function. So, what is the function that gives you a derivative which is essentially the same function? It may be a scaled version of that function, but it is the same function. So, for instance let me do it with some other variables. I could have something like this. If I have dx by dt equal to x, so in this case also the derivative is proportional to the function. In fact, I have made it exactly equal to the function. Now, you know how to differentiate functions. So, what is the function that you can recall that has the derivative which is the same function? Please try to answer this. Derivative is proportional to the function or maybe even equal to the function. What is the function that meets this criterion? So, if you want you can think of the table of derivatives that you would have studied when you studied calculus. So, think of a function whose derivative is the same function. So, clearly an exponential satisfies this because the derivative of the exponential is also an exponential. So, now, let us say I do not want the derivative to be equal to the function, but only proportional to the function. So, I also know that if I had a scaling factor here exponential at and I differentiate it, I get a times exponential at. So, clearly in this case if we see is as an exponential, this function will be satisfied. Also, I want a particular scaling factor. So, you see that this equation is of the same form as this. So, the derivative of something equals a constant times the same function. So, what is the function that will satisfy this equation? It is Vc of t of the form of exponential minus t by Rc. So, clearly now, I also can have a scaling factor here. So, let us say V0 because that will cancel out. If Vc of t is V0 exponential minus t by Rc, please give me the value of dVc by dt What is dVc by dt? If Vc is V0 exponential minus t by Rc, I use these expressions to find out exactly what goes inside the exponential. If it is exponential at, the multiplying factor here is a. I want the multiplying factor to be minus 1 by Rc. So, a has to be minus 1 by Rc. So, Vc of t is V0 exponential minus t by Rc. So, just differentiate this and tell me what the value is. Clearly this is minus V0 by Rc exponential minus t by Rc. So, you can see by inspection that this will satisfy this equation because the derivative is minus 1 by Rc. So, this part of it times the rest of it which is the function itself. So, the solution to Rc dVc by dt plus Vc equal to 0. The solution to this is Vc of t equals V0 exponential minus t by Rc and V0 can be anything as far as the differential equation is concerned. So, any value of V0, this differential equation will be satisfied. But in our circuit, we will need a particular value of V0. So, how will we find this? How will we find the value of V0? So, there is another answer which says V0 by Rc exponential minus t by Rc. I think you have omitted a minus sign it should be minus V0 by Rc exponential minus t by Rc. So, now we know that the shape of this blue curve here, the shape of this blue curve is of the form of V0 exponential minus t by Rc. So, this is this exponential describes the response of these types of circuits. And also we know that at t equal to 0, it starts from Vp. What is the value of this function at t equal to 0? The value of this function at t equal to 0 is if I substitute at t equal to 0 in this exponential, I will get V0. But I know from the circuit that it is going to be Vp. So, for this to correctly describe the blue curve here, this V0 has to be Vp. So, this curve is Vp exponential minus t by Rc. So, if initially the Kepler's curve had a voltage Vp and the input also had a voltage Vp, then the current through the capacitor, current through the resistor would be 0, current through the capacitor would be 0 and it would stay at Vp. But if Vp jumps, Vs jumps from Vp to 0, then the output will slowly discharge all the way to 0 towards 0. Now, we have also determined that function. The function is Vp exponential minus t by Rc. And we have intuitively determined the shape that it has some slope initially and then it goes on decreasing. And that is exactly what the exponential does. Please keep in mind that this exponential is the exponential of t, but with a minus sign inside. So, if we have an exponential with a plus sign, it will blow up. It will go on increasing with time. Here exponential of minus t by Rc, it will go on decreasing with time. So, any questions about this? The capacitor voltage across this will follow an exponential. I will redraw the input and output. Vs changes from Vp to 0, it jumps. So, this is Vs of t. And Vc of t, we will assume initially it was at Vp and it slowly goes to 0 like that. And this is Vc of t and it equals Vp exponential minus t by Rc. And the differential equation governing this is Rc dVc by dt plus Vc equals 0. And the solution is Vc of t is Vp exponential minus t by Rc. So, by now you should be comfortable with these results. If you have any questions at all, let me know. I will clarify that further. Any questions about this? Now, in this circuit, in this condition, the condition we are evaluating is with 0 input. So, if you look at this differential equation, it has only Vc, it does not have anything corresponding to the input. The input is 0. So, such a differential equation where the input or the right hand side is 0, I will always try to group the variables on the left hand side and the constant inputs on the right hand side. Here Vc is the variable and this kind of an equation which has 0 on the right hand side and only terms with Vc on the left hand side, this is known as a homogeneous equation. So, if you have taken some mathematics courses on differential equations, you will be familiar with this term. But in this course, whatever we need of differential equations, we will study them ourselves. And this comes because we have 0 input is known as the 0 input response. And this term which appears inside the exponential Rc, this is known as the time constant. It also appears here as coefficient of dVc by dt, but we have to be careful. The coefficient of Vc has to be 1. In that case, the coefficient of dVc by dt will be the time constant. So, it is called time constant because clearly Rc has dimensions of time. We have exponential minus t by something. So, this has to have dimensions of time so that we have exponential and inside that some dimensionless constant, dimensionless number. So, this is called time constant. Any questions about this? So, like I said before, the initial slope is Vp by exponential of Rc. Now, at t equals Rc, what is the value of Vc? What is the value of Vc at t equal to Rc? Please try to calculate this. Many of you were able to answer this. It is Vp times exponential of minus 1 or Vp divided by exponential of 1 or numerical value is 0.3679 times Vp. So, if I mark one time constant over here, this is t equals Rc. Maybe it is not to scale. I should redraw this a little better. So, that is what happens. And here the value is 0.3679 Vp. Now, if I go to two time constant, it will be 0.3679 square and so on. So, it will be approximately 0.1353 and it will rapidly reduce. Now, it turns out that after approximately 5 time constants or 4.6 time constants, it goes down to 0.01 Vp. t equals 4.6 Rc. We approximate this to 5 Rc. The point here is that it will never completely discharge to 0 because if you look at this function Vp exponential by Rc, it will never become equal to 0. It becomes equal to 0 only when t tends to infinity. So, in a finite time it never goes to 0 but it becomes arbitrarily small. Now, how small you want it to be depends entirely on the context. Sometimes you may want it to be 0.1 percent of the initial value that is 0.01 Vp. Sometimes you may want it to be 0.1 percent or even smaller. But this is a good number to remember. For the voltage to reduce to 1 percent of the initial value, you have to wait for 4.6 time constants. So, many times the time constant is denoted by the letter tau and for this circuit it is equal to Rc. And in 4.6 time constants, it reduces to 0.01 Vp. Any questions about this? So, when the input steps from some value to 0, we know how to solve it. We have the homogeneous equation and by solving the homogeneous equation, by solving the homogeneous equation we get the output to be an exponential and this is the 0 input response also meaning we have calculated this response for the period when the input is 0. So, now let us get back to the other case where we have R and C, VS and the differential equation governing this is Rc dVc by dt plus Vc equals VS. The right hand side is no longer 0. So, again I will not take arbitrary functions for VS, but I will take something that jumps from 0 to Vp at t equal to 0. So, this red stuff is VS of t. So, for t greater than 0, the right hand side of this is a constant equal to Vp. So, now we already intuitively evaluated that it starts with a slope of Vp by Rc and it will do that. It will do something like that. So, now we also have to find a function that corresponds to this. So, any idea of what this function might be? What might be this function? So, there are a couple of answers. One of the answers is Vp 1 minus exponential minus t by Rc that is correct. In fact, we will derive it properly. Also there is an answer saying positive exponential and not sure what is meant there. If you mean that the argument of the exponential will be positive that is not correct because if the argument of the exponential is positive, you will get a curve that is like this.  At t equal to infinity, it will become infinity. That is not the kind of function we have as we have determined from intuition. So, in terms of that we can very easily derive this because we know the result to the previous one when the right hand side is 0 for the homogeneous equation. In a very easy way, we can reduce this to the homogeneous equation. So, there are standard tricks and calculus that all of you are familiar with I think. So, let me redefine a Vc 1 which is Vc minus Vp. Now please tell me. So, Vp is a constant. So, keep this in mind. So, what will be dVc 1 by dt? What is this going to be? If Vc 1 is Vc minus Vp, what is dVc 1? What is dVc 1 by dt? So, please tell me what is dVc 1 by dt if Vc 1 is Vc minus Vp. So, clearly if Vp is a constant dVc 1 by dt is the same as dVc by dt. So, what I will do is, I will rewrite this equation in terms of the new variable Vc 1. So, first of all dVc by dt is the same as dVc 1 by dt and if you see this I have Vc minus Vp equal to 0 if I take Vp to the left side. So, that is nothing but Vc 1 equal to 0. So, here Vc 1 in terms of Vc 1, this is a homogeneous equation with the right hand side to be 0. Vc 1 is just a new variable I have defined. So, while solving problems you do this all the time. You define some variable in terms of other variables and substitute. In this case I have defined Vc 1 to be a variable Vc minus Vp. What I want to calculate for is Vc.  So, if I do that and write the differential equation in terms of Vc 1, I will get a homogeneous equation. So, Vc 1 would be of the form V naught exponential minus 3 by Rc. So, Vc if I turn around this now Vc is Vc 1 plus Vp. So, Vc will be of the form Vp plus V naught exponential minus 3 by Rc. So, this is the general form of the solution to this equation which has Vp on the right side. So, this is not a homogeneous equation and the solution to this will have Vp. It will also have some terms like exponential minus 3 by Rc which was part of the 0 input response. It has this constant V naught which we have to adjust based on the initial conditions. So, what is V naught and Vp? First of all Vp is the size of the step right Vp the input jumps from 0 to Vp. So, that is Vp and V naught is just a constant. So, previously here I said that Vc will be some V naught times exponential minus 3 by Rc that will satisfy the differential equation and we calculated the particular value of V naught by looking at the actual solution at t equal to 0. So, V naught happened to be Vp in the previous case ok. In this case again we have to evaluate the value of V naught we will do that ok. So, now we have to evaluate the value of V naught what we will do is I know that Vc of t is Vp plus V naught exponential minus 3 by Rc and if I substitute t equal to 0 what will I get? What will I get if I substitute t equal to 0 in this function Vc of t? So, clearly exponential of 0 is 1. So, the answer is Vp plus V naught ok. Now what is the value of the function? Actual value of the function at t equal to 0 it is 0 right that we know initially the capacitor has 0 volts and just after the input step the capacitor still has a value of 0 volts. So, this should be equal to 0 which means that V naught equals minus Vp ok it was clear. So, there will be arbitrary constants in the solution to a differential equation because the differential equation in the general form could be satisfied for any value of this constant. But by looking at the initial condition you will be able to find the particular value that satisfies it for that case ok. So, in this case this V naught equals minus Vp. So, the actual solution to Vc is Vp minus V naught exponential minus t by Rc or minus Vp sorry Vp minus Vp exponential minus t by Rc. Also frequently this is written in this form Vp 1 minus exponential minus t by Rc ok. So, if you look at the solution here at t equal to 0 both are Vp and this cancels out or inside this this exponential will be 1 at t equal to 0 and you have 0 and as t becomes infinity this exponential part becomes 0. So, Vc of t will become Vp that is what we see from our intuition and it starts from Vp I sorry starts from 0 and then exponentially approaches this Vp ok. Is this clear any questions? So, ok. So, we have looked at the Rc circuit it is governed by a differential equation. We have solved for it intuitively we know what it means ok that is very important actually. We of course, taken a special case of only constant inputs. When I say constant it jumps from one constant value to another constant value ok. Now, if it does that what happens to the capacitor voltage and we will see later that this happens to every variable in the circuit is it undergoes some change with some exponential argument and the argument inside will be minus t by Rc or minus t by some time constant ok. Now, we solved it for 0 input that is you have some initial condition and the input then jumps to 0. We also solved it for you have 0 initial condition and it jumps to some value ok. So, let me just tabulate the results. The circuit we are looking at is this Rc and the variable I am trying to calculate is this I took two kinds of inputs one where the input jumps from some Vp to 0 ok and in this case the capacitor voltage Vc will do something like that ok and the other case was where the input jumped from 0 to Vp and the output it turns out thus something like that ok. This is Vc of t Vc of t. So, this corresponds to basically a 0 input and in this case we have a 0 initial condition on the capacitor voltage ok. Now, the capacitor voltage is also known as a state a capacitor can hold some voltage. So, that is known as the state of the capacitor. So, this is also known as 0 initial state or 0 state ok. So, the differential equation governing the 0 input circuit is that the important thing here is that you have terms containing Vc and its derivatives and the right hand side is 0 ok. So, this is known as the homogeneous equation and in the other case the differential equation is Rc dVc by dt plus Vc equals Vs and we are talking about constant values of Vs or Vp ok. And the solution to the 0 input case it turned out was Vp exponential minus 3 by Rc ok and the solution to this case is Vp minus Vp exponential minus 3 by Rc ok. So, we will look at these expressions and classify them into different parts ok. It turns out that this part is called the force response this is the natural response the entire thing is the 0 state response in this particular case and this is the 0 input response. We will look at all of those things in the next lecture. So, if you have any questions now I will clarify them any questions about any of this Rc circuits and differential equations ok. I will see you on Thursday ok.


Outside of Africa, three people have died and five others injured after a light aircraft crashed inside the supermarket in the municipality of Tamexco in central Mexico. Emergency workers arrived at the scene where they assisted the five injured people, four of whom were shot below the supermarket when the aircraft crashed. Tamexco's authorities said the four people on board the aircraft took off from Acapulco in Guerrero State and headed to Puebla. Now three of them were killed in the accident and the other person has been treated. Now Morales Prosecutor's Office and Civil Aeronautics will investigate the causes that led to the accident according to local authorities. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


The White Canoe, and other verse by Alan Sullivan, read for LibriVox.org by Bruce Kachuk. THE WHITE CANOE There's a whisper of life in the grey dead trees, and a murmuring wash on the shore, and a breath of the south in the loitering breeze, to tell that a winter is o'er. While free at last from its fetters of ice, the river is clear and blue, and cries with a tremulous, quivering voice for the launch of the white canoe. O gently the ripples will kiss her side, and tenderly bear her on, for she is the wandering phantom bride of the river she rests upon. She is loved with a love that cannot forget a passion so strong and true, that never a billow has risen yet to peril the white canoe. So come when the moon is enthroned in the sky, and the echoes are sweet and low, and nature is full of the mystery that none but her children know. Come taste of the rest that the weary crave but is only revealed to a few. When there's trouble on shore, there's peace on the wave of float in the white canoe. Tonight, sweetheart, when all about me lay in shadow deep the wood, I felt my soul within me reel and sway, and pulse my sluggish blood, as when along a quiet, land-locked bay swells some resistless flood. My spirit leapt from out its earthly prism, higher and ever higher, until it reached those barriers alitions where the eternal fire creates one great, impassable division, tweaks us and our desire. Up till it left the regions of the night of sorrow and of fear, emerging into that soft mellow light that radiance, pure and clear, where love reigns all supreme and all is bright if only love be near. There, through sweet meadows on by brimming streams, wandered my soul at will and saw such forms as haunt our loveliest dreams and waking haunt us still. Voices like music, smiles like sunny beams, lost in a rippling reel. But ah, my soul saw one supremely fair, one form the most divine, one face and haloed all with golden hair, in beauty most benign, surpassing all the perfect beauty there. Heart of my heart, twas thine, my soul went forth, but all grew strange and dim, meadow and stream were gone. I heard a sound as of a far-off hymn, by night winds softly blown, then all around me seemed to sink and swim, and I am here, alone. End of poem. This recording is in the public domain. A Question By Alan Sullivan Read for LibriVox.org Pale moon, whose tranquil orb resplendent sails the ethereal main, thy curved prow forever braving the celestial gales, serene and slow. Myriads of stars that ever dot the blue great fault of heaven, eyes that keep eternal watch, unshaken, strong and true, yet never sleep. Ye southern zephyrs, redolent with balm of myrtle, orange and the rose, blowing from islands where the fronded palm in beauty grows. Wind of the north, whose trumpet voice can shake the shattering echoes of the cave, storm-born, blast-driven, thou whose breath doth make the mighty wave, perpetual fire, whose never-dying flame consumes the glowing heart of earth, until a wide destruction shall proclaim a second birth. Tell me, O mighty concourse, have ye seen in all this great infinity of worlds unborn and planets that have been a place for me? This recording is in the public domain. Confession, Creed and Prayer by Alan Sullivan read for LibriVox.org Silent around me, a cathedral dim, still throbbing with the echoes of a hymn, lifted its ghostly arches great and grim. Slowly the worshippers had filed away, untenanted the vacant cloisters lay, as even followed on the steps of day. But one remained who bent his reverent head where graven figures slumber with the dead, and spake with faltering accents, and he said, Light, more light, great Father, give me light. I cannot see my way, so dark the night my finite heart shrinks from the infinite. Anon the shadow lifts, my straining eyes one moment see that which before me lies. This fades, and newborn hope within me dies. I looked for sunshine, yet there cometh rain. My sweetest pleasure turneth into pain. I would sink back to nothingness again. Beliefs are but perpetual ideas. The gospel worketh only on my fears, in bitterness and sorrow, void of tears. In one God I believe, eternally omnipotent and present, and that he rules heaven and hell, the earth, the sky, the sea. As carnal life by carnal love is given, so life divine by love divine is proven, of which the fountain head is God in heaven, that since each body is the fleshly home of something better, something not our own, so God all faults but foulness will condone. For I believe impurity is sin against the spirit life that dwells within, creating Father and created men, that every soul is judged true and well according to the light that on it fell, no light, no judgment, strong light, heaven or hell. All this and more, great Father I have heard of thy dear son, my heart expectant stirred to welcome him, confessing I had heard. Nay, said humanity within me, nay, I cannot grasp this mystery, so stay until I comprehend and I obey. Thy word yet cannot, herein lies my grief, thy sons bake comfort to the dying thief, so speak to me and help mine unbelief. Hear the voice faltered, ceased, God, cannot be the mourn has dawned on him and not on me. Is this the publican, I, Pharisee? End of poem, this recording is in the public domain. To My Pipe by Alan Sullivan, read for LibriVox.org. Others their nectar from the goblet sip, I draw sweet psalus from thine amber lip, a feast of reason and a flow of soul, lurk in the perfumed vapors of thy bowl. Some scoff and say I err from nature's rules, tobacco's poison, but friend, some are fools. If times are hard, no comrade like to thee, if prosperous, there it the priest of jollity. Browned in my service, silver rimmed through age, thy smoldering fire, reflections heritage. When the day comes, old friend, and I'm dead broke, then just one puff will both go up in smoke. End of poem, this recording is in the public domain. Supposing by Alan Sullivan, read for LibriVox.org. He. Supposing that when we were wed, love, and two were reduced into one, that a hot-tempered word should be said, love, and thoughtlessly mischief be done, that you should be proud and offended, and I should be heartless and cold. Do you think that our peace would be ended? The tale of our happiness told. Supposing that children should come, love, it may be a girl and a boy, and my heart should go forth to the one love, the other your pride and your joy. Do you think that although so divided, yet we still, in our plans, could agree, and always the best be provided, for our dear ones by you and by me? Supposing that times were so bad, love, that ends couldn't possibly meet, and I should get weary and sad, love, while you were still hopeful and sweet. Do you think you'd in spirit and cheer me, and help me to weather the gale, that your presence would ever be near me, your courage not falter or fail? She.  No, the bond of affection would stay, dear, independent of pocket or purse. As a wife I would honor, obey, dear, and love you for better, for worse. you and me ever drifting spellbound silent down a shimmering track of light, while around the gloom was throbbing with the mystery of night. Mute our lips what need of speaking, but our hard cords were as tense as a bowstring stretched to breaking. Every look was eloquence, till my soul had burst its barriers, and I told you my desire, told of love, undying passion, strong as ocean, pure as fire. You nor moved, nor sighed, nor answered, pale your cheek was as your dress, but the golden lashes drooping gave response, and it was, yes, that was five long years ago, dear. Can you hear me as I speak? For again I see the lashes falling on a pallid cheek. Still and ah, so silent, sleeping, motionless, you take your rest. I've your pledge of love beside me, and your image in my breast.  See, she nestles to my shoulder. Darling, can you see us too? Lullaby. Sleep, little one, sleep. Safe and strong is thy father's arm. He will guard thee from every harm. Soothe thee with kisses, soft and warm. Sleep, little one, sleep. Sleep, little one, sleep. Close the lids on the wandering eyes, deep and blue as the summer skies. Far in the west the sunset dies. Sleep, little one, sleep. Sleep, little one, sleep. Thou art more than the world to me. O my life shall be spent for thee, till nature comes with her lullaby. Sleep, little one, sleep. Once long ago, a summer night in June, when earth lay still, beneath the raining moon, and never sound or rustle in the wood, save the dull thunder of a far-off flood, hurling itself in ruin to the deep, or a great gulf, I lay and strove to sleep. The stars were out, I watched with aching eye, their slow grand march across a cloudless sky. But rest came not. When suddenly I heard, far in the slumbering forest, one lone bird give three sweet calls, as if in pure delight to fling its soul in music through the night. Like a cool hand upon a fevered brow came that dear song, all fear had vanished now, steady my pulse, sunk in oblivion's arms, forgetful as a child of past alarms. Ye who have doubts, who is it has them not? Ye who have fears, and troubled anxious thought, when the storm lulls will, if ye list aright, hear a bird singing in your darkest night. Then I read, clear writ as in a book or chart, the vast futurity, with all its joy and grief, success and failure, love, hate, unbelief and faith, and that blind parting at the last, where at my soul recoiled, nor could it bear to muse on so much labour, better far not to have been, or else to be perchance like a dumb brute, existence without care or consciousness.  Thou, with the black stone stem, what of the past? Where are the cunning hands that fashioned thee? Where are the stern brown lips that placently drew comfort from thee, neath the towering mast of some old pine? Or patient to the last, toiled over thee? Perchance thou were to God, worshipped and feared by those whose feet trod the dim green aisles of that cathedral vast. But now thine incense rises, and I see the still north land, and hear the otter dive, the rapids calling, and the great trout leap, and smoking here at Seymouth like to me as if some dead hands touched the hands alive, in token of the fellowship we keep. End of poem. This recording is in the public domain. ADVONDI AMOR by Alan Sullivan, read for LibriVox.org. Silence again, sweetheart. The shadows grow. I watch the white stars climb into the sky. Hear the dull rapids softened lullaby in smothered thunder, brooding sweet and low. Catch in the east the pallid silver glow of a new moon that floating pure and clear in perfect promise of the fuller sphere dips this dim world in glory, mounting slow. Not always had the heavens such a charm. Last year the rapids were not half so sweet. The wind had not such rhythmic melody till love, love came, and fanned the cold heart warm, attuned to music chords still incomplete, and set the whole night whispering of thee. End of poem. This recording is in the public domain. A Song of Life by Alan Sullivan, read for LibriVox.org. It came through the fields of air. It came through the silent night, born low on a sigh of a western breeze, like the far-off voice of tumultuous seas in a tempest's waning might. I heard the wonderful song. It made its home in my breast. The music of all the world was there. It hushed all murmur of pain or care, a sum of infinite rest, ever more clear and pure, ever more strong and sweet till some kindred chord in the outer air in response to the melody throbbing there sang come to my restless feet. I heard the mysterious call. I rose and followed it straight, or many a mount, through many a dale, past blazing meadow and shady veil, to the sunset's rosy at gate, and never a halt or stop till the song I could scarcely hear. It had sunk to an echo, faint and dim, of some melodious wonderful hymn, so I knew that the end was near, lower and fainter yet, and more imperceptible still, as I journeyed on, but I climbed one day, with courage that faltered so steep the way, the crest of a long, long hill. There, far as the eye could scan, was not but the fathomless deep, while down at the crags great base the waves crept in and out of the blind black caves and whispered ever of sleep. I looked at my hair, twas white, my hands were bony and long. The years of my life had vanished and fled, though they seemed but days that had quickly sped in pursuit of that fugitive song. Then out of the ocean's heart came swelling a grand refrain, and threw it their pulsed and angelic voice. Now we're immortal, rejoice, thou hast come to thy rest again. The song that stole into thy breast was the song of an earthly love. It was but an echo, faint yet true, of that mightier song that is peeling through the musical halls above.  So out of the void a sound from the vast dim space a breath that fanned the flickering flame of life till it flared went out and ended the strife, I slept, and the sleep was death. And a poem. This recording is in the public domain. Voices by Alan Sullivan, read for LibriVox.org. My heart within me stirred with a nameless trouble and dread of evil that should be tied, and a voice in my bosom said, What pause from this weary toiling? What end to this endless strife? The day bringeth not but labour, and death follows hard upon life. Ever I see the false one triumphing over the true, the foul out once the fair, the many oppressed by the few. Answer me, mortal master, after the battle is fought, six feet of earth for a couch may have a stone, then what? How could I answer my heart when suddenly in my breast there fell a hush as of a wind sinking at eve to rest? The voice within me was stilled, and I felt its murmuring cease, for somewhere out of infinity an angel had whispered, Peace. End of poem. This recording is in the public domain. Fifty years hence. By Alan Sullivan, read for LibriVox.org. Again, twas night and on the wave the moon in silver lay, vanished had all the petty cares and troubles of the day, no sound in all the wide expanse, no rustle in the wood, save when some evening zephyr stirred in whispers on the flood, breathless and motionless she stood, unquestionably dumb. Twas as a world were waiting there, waiting for God to come. Then back through long dead years her heart winged its reflective flight to ponder childhood's days again, to muse unpassed delight.  Old voices spake, old scenes recurred, old friendship lived once more. Serene the skies, no fear, no care, no tempest and no storm, wild birds and sunshine in the air, and south winds sweet and warm. Ah, perfect youth, ah, perfect life, free as a cloud above, ah, fount when spring the purest hopes, whence flows the purest love. For if ambitions wildest dreams, success should crown, in truth the cups she holds were tasteless still beside the wine of youth. All silent now, ah, for the power again those tales to tell, to wake afresh those sleeping chords that memory loves so well. But echoing clear and low those notes that song we still may hear. For faintly yet its music floats in old age atmosphere. And a poem, this recording is in the public domain. Farewell to the White Canoe by Alan Sullivan, read for LibriVox.org. The summer is dead, for the air is chill, and winter is nigh again. The maples ablaze on each ruddy hill are dripping with crimson rain. Black dusk comes hard on the steps of day, the breath of the south that blue has turned to the north and bids me say, farewell to the White Canoe. How wildly she leapt at each measured stroke and mounted the curling swell, how the white foam hung at her boughs like smoke when the great waves rose and fell. No terror for her could attempt to find no wrath in a frowning sky. Her birth was the union of sea and wind, her life is a mystery. She swam like a ghost through the ghostly night that bowed but to her as queen. She sped like a wreath in the silver light or a spirit of things unseen. As a leaf in the autumn she sank to sleep by babbling ripples caressed and lay in the arms of the cradling deep on the river's responsive breast. The summer is dead, and alas no more may we wander alone and free. By still deep pools and the shadowy shore and the rapids soft lullaby. Farewell, farewell to the peace that lies in that solitude deep and blue, an answering voice from the great stream size. Farewell to the White Canoe. End of poem. This recording is in the public domain. End of The White Canoe and Other Verse by Aaron Sullivan.


Good evening aspirants, I have an important announcement to make. The third batch of the pre-storming test series is about to commence. Like the previous two batches, this batch will also have 66 tests. The first test is about to commence on 20th November. I urge the aspirants to use this test series to boost your brilliant score. With this news, let us get into the Hindu News Analysis by Shankarayesh Academy for the date 15th November 2022. Here are the list of news articles we will be discussing today. Now without wasting time, let us start our discussion. Let us start today's discussion with this editorial article. This editorial article highlights a symbiotic relationship of mutualism. Yes, here the article is talking about the US Treasury Secretary Janet Allen's visit to India last week. And it also highlights how both the countries can be benefited by using a single tool called trade. Although in this article discussion, we will see some of the important points mentioned in the editorial. Firstly, you have to know that the US Treasury Secretary's visit to India highlighted the renewed focus in the United States on strengthening economic ties with India. This was confirmed when Ms. Allen particularly said that the US and India is planning to strengthen supply chains in the world where certain countries are using trade as a geopolitical weapon. If you cannot understand this, let me explain it with an example. What happened when Russia invaded Ukraine? Russia weaponized its supply of natural gas to Europe, right? That is, after economic sanctions were imposed by the West on Russia, Russia cut off gas supplies to Europe indefinitely. This created huge demand supply gap in Europe. So this is how Russia weaponized its trade with Europe. The second example is in relation to China. We know that China is a dominant player in the manufacture and selling of solar panels. This over-concentration of solar panel production in China is an issue. Because this factor is giving China a power to cause global supply chain disruption and interruption. To counter this only, the economic ties between US and India is important. Because US is going to use India to create alternative supply chain to counter China. By creating an alternative supply chain, the dominance of China in the global market can be countered. This is the first point mentioned in the editorial. Moving on to the second point. The second point mentioned in the editorial article is about India-US economic ties. What happened during the Trump administration? The Trump administration removed India from the generalized system of preferences. This affected Indian exporters and this act strained the US-India economic ties. So through this visit, US is trying to amend the US-India economic ties. This is about the second point. See both these points can be used in your main sensor. The first point can be used when a question is asked about the weaponization of global supply chain. And the second point can be used when a question is asked about the US-India economic relations. So that's all regarding this discussion. In this discussion, we saw two main points mentioned in the editorial article. With this, let us conclude this discussion and take up the next news article. Let us take up this news article for our next discussion. Yesterday, in the Supreme Court, the Union government asked for more time to clarify its stand on the validity of the Places of Worship Act. The government said that detailed consultations are needed at a particular level regarding the act. So the government needs more time. And the court ordered the center to file its affidavit by December 12. This is about the news article. In this context, let us learn about the major provisions of the Places of Worship Act and the issues surrounding it. The Places of Worship Act was enacted by the parliament in 1991 in the backdrop of the Ram Mandir agitation. The act basically prohibits the conversion of any place of worship to any other purpose. The act also provides for the maintenance of the religious character of any place of worship as it existed on the day of our independence, that is, 15 August 1947. To put it simply, the act says that the religious character of the place of worship should be maintained as it existed on the day of 15 August 1947. Now coming to the major provisions of the act. First let us take section 3. Section 3 of the act bars the conversion of a place of worship. It prohibits the conversion in full or section of a religious place of worship into a place of worship of different religion or a different denomination of the same religion. For example, different sect in the same religion. This is in regards to section 3. Now moving on to section 4. Here let us first take class 1 of section 4. Class 1 declares that the religious character of a place of worship shall continue to be the same as it existed on 15 August 1947. Now moving on to class 2. It says that all appeals, suits or other proceedings pending before any court, tribunal or other authority with respect to conversion of the religious character of a place of worship shall end on the commencement of the act. And also fresh appeals won't be allowed to be filed. This means that this provision takes away the powers of the court or tribunal or any other authority in adjudicating the cases related to the conversion of the place of worship. Now moving on to class 3. This section 3 provides for some exception. The first exception is provided to any place of worship which is an ancient and historical monument or an archaeological site covered by the ancient monuments and archaeological sites and remains act 1958. The second exception is provided to any dispute that has been settled by the parties before the commencement of the act. The third exception is the conversion that took place by consent or agreement before the commencement of the act. And finally exception is also provided to any suit that has been finally settled or disposed of by the court or a tribunal or any authority before the commencement of the act. This is all about the section 4 of Places of Worship Act 1991. Moving on to section 5. This section says that nothing contained in this act shall be the place or place of worship commonly known as Ram Janmabhoomi Babri Masjid which is situated in Ayodhya in the state of Uttar Pradesh. Section 5 also says that the act does not apply to any suit, appeal or other proceedings pending before any court, tribunal or other authority relating to the Ram Janmabhoomi. And finally there is section 6. Section 6 provides for punishment. It says that whoever contravenes the provisions of section 3 shall be punishable with imprisonment for a term which may extend to 3 years and shall also be liable to 5. See these are some of the important provisions of the Places of Worship Act 1991. Having seen the important provisions, now let us see the important issues associated with the act. Let us take up the first issue. The first issue is regarding section 4 of the act. We saw that section 4 of the act bars the judicial review. And we know that judicial review is one of the basic structure of the Indian constitution. So this is an important issue. Moving on to the second issue. See this act violates the principle of secularism. How? Because we know that this act was created in the backdrop of the Ram Mandir agitation. So the act was created by focusing only on the problem of one particular religion. So due to this, some people are worried that this act is against the principle of secularism. This is the second issue. Moving on to the final issue. The third and the final issue is regarding the cut-off date. The cut-off date of 15th August 1947 is completely arbitrary and it is also one of the most contented provisions of the Places of Worship Act 1991. Because there are no proper records of the worship places. So this particular cut-off date is contributing to the chaos that is currently existing in our country. So these three are the major issues associated with the Places of Worship Act. So the government needs to review the act to address the issues that is prevalent in our country. So that's all regarding this discussion. See in this discussion we saw the major provisions of the Places of Worship Act 1991. Along with that we also saw the major issues associated with the act. With this let us conclude this discussion and take up the next news article. Take a look at this text and context article. It deals with the G20 summit which is going to be held in Bali. The article talks about various things like the origin of G20, the leaders attending the current G20 summit and also about the agenda of the current Bali summit. This is the essence of the article given here. In this context let us learn about the Bali summit and also about India's presidency takeover of G20 which is going to occur in December this year. Now before getting into the discussion I have highlighted here the syllabus regarding this discussion. You can go through it. With this let us start our discussion. Firstly we will see about the origin of G20. G20 as an intergovernmental organization was established as a result of a G7 finance ministers meet in the year 1999. See G20 was born in the backdrop of the Asian financial crisis that gripped much of East Asia and Southeast Asia beginning in July 1997. It raised fears of a worldwide economic meltdown. Why did the countries fear that the Asian financial crisis will lead to global economic meltdown? This is due to globalization. All the world economies are linked to one another right. So if one country is affected rest of the world nations will also suffer the consequences of economic slowdown. This is why the world nations feared that the Asian financial crisis will turn into a global economic meltdown. However there was a rapid recovery in the 1998-99 period and due to this the worries of the global economic meltdown subsided. As a continuation of the G7 finance ministers meeting the inaugural meeting of G20 was held on 15th December 1999 in Berlin and so the G20 grouping was born. Again in 2008 another financial crisis happened. This is nothing but the US subprime crisis. At that period also the world nations feared that this will turn into a global economic meltdown. To prevent the US subprime crisis in turning into a global meltdown coordinated action from the world nations was required. So the then US President George W. Bush invited the leaders of the G20 member countries to coordinate actions in order to respond to the crisis. This led to the start of the actual G20 leaders summit. Since 2010 the G20 meeting is held annually and the host country changes every year. Now let us see few terms related to G20. First is SHERPA. SHERPA is a term used to denote G20 leaders personal representatives. SHERPA's duty is to usually oversee negotiation over the course of the year discussing agenda items for the summit and coordination of the work of the G20. Here note that India's G20 SHERPA for the Bali Summit is Avitabh Khan. SHERPA's can recall Avitabh Khan earlier served as the CEO of Nithiyog. Now let us take up the next term which is G20 TROICA. TROICA refers to the top grouping within the G20 that consists of the current, previous and the incoming presidencies. Here note that the present G20 TROICA are Italy, Indonesia and India. The TROICA will change every year. SHERPA is to ensure consistency and continuity in G20's agenda. The next TROICA will be Indonesia, India and Brazil. This is all about the background information of G20. Now coming to the member states of the G20. G20 is a forum comprising of 19 countries with some of the world's largest economies as well as the regional block of the European Union. I have displayed here the member countries of G20 forum. You can go through it. Here note that Spain is invited as a permanent guest. Other than this, there will be special invitees to the meeting. Special invitees include leaders of other countries which are not part of G20 and other regional organizations. Here note that G20 does not have a permanent secretariat or a headquarters. This is all about the background information of the G20 grouping. Now moving on, we will see the agenda of the Bali summit. The motto of the Bali summit is, recover together, recover stronger. The Bali summit is going to focus on food and energy security with special emphasis on health partnership for global infrastructure and investment and digital transformation. Apart from this, the discussions relating to climate change will also take place. Note that Russian President won't be participating in the meeting. The summit in Bali will be attended by leaders and also by the heads of several international agencies like the UN, IMF, ASEAN and the African Union. Another important thing you have to note here is that there will be bilateral meetings of the world leaders which will also be happening in the sidelines of the summit. This is all about the G20 summit which is going to be held in Bali. Now coming to the next G20 summit which is going to be preceded over by India. As per the article agenda for the year ahead under India's G20 presidency will be focused on the global south the problems it is facing due to geopolitical tensions, food and fuel shortages. Here global south denotes the grouping of developing countries. This is the basic information provided in the article about India's presidency of the G20 summit. With this we have come to the end of the discussion. In this discussion we saw about the origin of the G20 summit. We also saw about two terms. The terms are Sherpa and Troika and we also saw about the Bali summit and India's presidency next year. With this let us conclude this discussion and take up the next news article. Look at this news article. Yesterday at the United Nations climate change conference that is at COP27 Egypt, India announced its long term low emission development strategy. This strategy is aiming to limit greenhouse gas emissions and strive for low emissions. India also highlighted that these steps were in line with India's five decade journey to net zero or being carbon neutral by 2070. This is the crux of the article given here. In this context in our discussion today let us learn about long term low emission development strategy. To understand better let us start with the Paris Agreement. Paris Agreement is a legally binding international treaty on climate change. This you all know. It was adopted by 196 parties of UNFCC at the 2015 United Nations Climate Change Conference that is COP21 in Paris. The goal of the Paris Agreement is to limit global warming well below 2 degree Celsius and preferably below 3 degree Celsius. This limitation is fixed in comparison to pre-industrial levels. And this goal can only be achieved by reducing greenhouse gas emissions substantially. It is the 2015 Paris Agreement which invited the parties of the UNFCC to formulate long term low greenhouse gas emission development strategies and to submit them by 2020. But what happened in 2020? COVID-19 pandemic happened right? So due to the COVID pandemic time was extended. And the COP26 held in Glasgow last year required the parties to submit the strategies by COP27. So yesterday due to this only India announced its long term low emission development strategies. Through the long term low emission development strategies the countries must explain how they will transition their economies. And they will also have to explain how this will help them in achieving their emission reduction targets. The strategies that the countries formulate must be based on each countries various responsibilities and capabilities. For example a country like UK will have higher responsibility and higher capability because of its contribution to long term CO2 emission. But for example a country like Fiji will have that much responsibility for climate change and it also has low capabilities. So it will have some limited strategy. This is about long term low emission development strategy. Now let us see about the salient features of India's strategies one by one. First is the strategy on energy sector. India said that it will focus on the rational utilization of natural resources with due regard to energy security. India added that the transition from fossil fuels will also be undertaken and the transition will be done in a sustainable and an inclusive manner. Here what India is trying to say is it will balance both the transition from fossil fuels and the energy security of India. This is the first feature of India's strategy. The second is the strategy on transport sector. India said steps will be taken to drive the low carbon development of the transport sector. This is to be done by increasing the use of biofuels, mainly ethanol blending in petrol and then increasing electric vehicle penetration and increasing the use of green hydrogen fuel. The third is the strategy on industrial sector. India said that the low carbon development transition in the industrial sector would be done without affecting the energy security, energy access and employment. Here the focus will be more on improving energy efficiency. Here the energy efficiency is achieved by using four strategy. First is the perform achieve and trade scheme. The second is the national hydrogen mission 2021. The third is the electrification of the process and activities. Finally, India will focus on enhancing material efficiency and recycling. Through this, India will strategize its transformation in the industrial sector. The fourth strategy is on the forest cover. The strategy says that India has a strong record of enhancing forest and tree cover in the last three decades. This is alongside the high economic growth achieved by India. As of 2016, India's forest and tree cover are having the capacity of observing 15% of its CO2 emissions. As a part of this strategy India highlighted that India is on track to fulfilling its commitment of 2.5 to 3 billion tons of additional carbon sequestration in forest and tree cover by 2030. In the fourth strategy India is saying that India is taking steps to increase the forest cover to absorb CO2. These are the four strategy announced in India's long term low emission development strategy. That's all regarding this discussion. With this, let us conclude this discussion and take up the next news article. Take a look at this editorial article. The article itself begins with a quote, In our dreams, wind will blow into women's hairs. In our dream children will not be forced to learn ideologies of the Middle Ages. In our dreams, wind will blow into women's hairs. In our dreams, children will not be forced to learn ideologies of the Middle Ages. In our dreams, no one will attack girls' school. No one will shoot at them from big A. These are the lines set by an Iranian-Canadian activist at a rally organized to support the protests in Iran. Yes, this article talks about the ongoing protest in Iran in which protesters are calling for an end to the compulsory headscarf and other religious laws. What led to this protest? We will see about this in our discussion today. Before that, I have highlighted the syllabus regarding this discussion. You can go through it. Let us begin with the question, What led to this protest? The protest began when Masha Amini, a 22-year-old went into coma and died after arrested by the Morality Police in Tehran on September 13th. The Morality Police detained her as she was wearing and she was detained for hours to be re-educated on proper dress codes. So, who are these Morality Police and what are the hijab rules in Iran? We will see about them now. In Iran, walking unveiled or without wearing a hijab is a punishable crime under the Islamic Hijab Rules. Since the country's 1979 Islamic Revolution, Iranian law requires all women, regardless of their nationality or religious belief, to wear a hijab that covers the head and the neck while concealing the hair. So, what happens if a woman does not follow this dress code? That is where the role of the moral or the Morality Police comes in. The Morality Police are known as Gash-e-Irshad, which means Islamic Guidance Patrol. The Morality Police use white vans with green stripes and are often stationed in areas where pedestrians or young people are gathered. The force employs men and women. Often, their mere presence prompts many women to rearrange their headscarves to comply with regulations. The officers enforce the country's dress code only by issuing a verbal warning. But some people are occasionally detained. If someone is detained, they will be brought to a guidance center where they are re-educated on proper dress code for hours. They are then made to sign documents pledging not to repeat the offense. They will be released after signing the documents. So, like the usual process, Masha Amini was interrupted by the Morality Police. But the issue here is that according to the eyewitnesses, she was beaten up and then taken to the notorious Ozara Detention Center. She was later admitted in a hospital where she fell into a coma and eventually passed away. After her death on September 16th, Iranians took to the streets across the country calling for justice for Amini.  So, this is what is happening in Iran. I hope you all remember about the recent Karnataka Hijab issue. Even though they are two situations far from each other, it happens to involve the same piece of cloth that is the hijab. Here you might think that it is ironic to compare an Islamic state like Iran with a secular state like India. But the similarity between the agitations is that both the women in Iran and India are fighting against the control over their autonomy of their body. And they are fighting for their freedom of choice. The difference is also the same. For example, in India, women who want to wear their hijabs are fighting against a system that is forcing them to take their hijabs off. But in Iran, women are battling against regime that is forcing them to wear the hijab. The protest in Karnataka is against the legislation of anti-Muslim prejudice and patriarchal attitudes. While the protest in Iran is about the patriarchal setup and against how religion is used as a pretext to enforce injustice towards. So, this is about the Iranian hijabla issue and the comparison between the protest in Iran and the protest in Karnataka. So, since already we are talking about Iran, let us also see in this discussion about the India-Iran relationship. First, let us start with the strategic importance of Iran for India. Firstly, Iran is important for India because of its location. Iran is located at a strategic and crucial geographical location between Persian Gulf and the Caspian Sea. And this is why Iran is very important for India. Secondly, Iran is important to India as it provides an alternative route of connectivity to Afghanistan and Central Asian Republics in the absence of permission for India to use the land route through Pakistan. For this only, India has been constructing the Chabar port in Iran and also the Chabar-Zaranj Railway project which connects Chabar port to Afghanistan. In addition to this, Iran is an important link in the international north south Khaidor which will enhance India's connectivity with the Central Asian countries. This is the second strategic importance of Iran for India. Finally, Iran is important for India because Iran has one of the largest deposit of crude oil and natural gas in the world. And since India is a developing energy-hungry country, a strong relationship with Iran will help India address its crude oil and natural gas requirements. See, these three are the major strategic importance of Iran for India. See, although Iran and India has stable and friendly relations, due to various geopolitical issues, the relationship between India and Iran is deteriorating in the present times. So, here now we will see the issues in the India-Iranian place. First is Iran's proximity to China. See, Iran is part of China's ambitious built-in road initiative. So, this might come in conflict with India's interest in Iran. The second issue is US sanctions on Iran. See, earlier, India used to import lot of crude oil and natural gas from Iran. But after US sanctions, India has slowly reduced its crude oil imports from Iran and substituted it with oil from Iraq. India has also stalled its funding to the Chabar port. So, this is pushing Iran towards China. And this is the second major issue in the India-Iranian ties. And the last major issue is regarding the recent grouping that is the I2U2 grouping. The grouping consists of Israel, India, UAE and USA. This grouping is titled as anti-Iranian according to the officials in Iran because Israel, the United Arab Emirates and the US are adversaries of Iran. So, Iran is viewing India skeptically in regards to the I2U2 grouping. So, these three are the major issues in the India-Iranian ties. So, that's all regarding this discussion. In this discussion, we first saw about the Hijab protest in Iran. Then we saw about the comparison between the Hijab protest in Iran and the Hijab protest in Karnataka. Then we saw about the strategic importance of Iran for India. And finally we saw about the issues in the India-Iranian relationship. So, with this, let us conclude this discussion and take up the next news article. Look at this article. Yesterday, Iran has launched a wave of cross-border missile and drone strikes against Kurdish opposition groups based in northern Iraq. The local authorities in northern Iraq have reported that one person has died and several others were wounded. This is about the news article. In this context, let us learn about the Kurds and the issues between Iran and the Kurds. Who are the Kurds? The Kurds are one of the indigenous people of the Mesopotamian plains and the highlands in the southeastern Turkey, northeastern Syria, northern Iraq, northwestern Iran and southwestern Armenia. Know that the majority of the Kurds are Sunni Muslims and there are some Kurds who follow other religious beliefs also. The Kurds form a distinctive community. Even though they have no standard dialect, they are united through race, culture and language. Now we will see the problems faced by the Kurds. In the early 20th century, many Kurds began to raise their voice for the creation of their own homeland, which was generally referred to as Kurdistan. But their voice for the creation of the country of Kurdistan was never addressed. To understand this, we have to go back in time. We have to go back till the World War I. In World War I, the Ottoman Empire was defeated by the Western Allies, which comprised of France, Britain, Russia and the United States. During the World War I, the Kurds helped the Western Allies. So, the victorious Western Allies made an provision for a Kurdish state in the 1920s Treaty of Severus. The Treaty of Severus was signed between the Allies of the World War I and the Ottoman Empire. In that only, the provision of the Kurdish state was made. But such hopes of the Kurdish state was destroyed three years later. In 1923, the Treaty of Lausanne was signed. This Treaty of Lausanne set the boundaries of modern Turkey and this treaty had no provisions about the Kurdish state. So, this left the Kurds only with the minority status in the respective countries like Iran, Iraq, Turkey and Syria. So, from that time onwards, the Kurds have been continuously straining to set up an independent state. This is the basic about the Kurds, the issue faced by them and the country of Kurdistan. Now, coming to the issue between the Iran and Kurds. The Kurds are presently inhabited in western Iran region and this region shares border with parts of Iraq and Turkey and in Iraq and Turkey also, Kurds are present. As we saw earlier, majority of the Kurds are Sunni Muslims and we know that Iran is dominated by Shia Muslims. So, naturally the difference in ideology resulted in conflict. The Kurdish separatism in Iran or the Kurdish-Iranian conflict is an ongoing, long-running separatist dispute between the Kurdish opposition in western Iran and the government of Iran. The Kurdish separatist movement in Iran first started with the establishment of the Republic of Mahabad during the 1946 Iranian crisis. The Republic of Mahabad or the Republic of Kurdistan was a short-lived, Kurdish self-governing, under-organized state in the present-day Iran, which stretched over Iran's border with Turkey and Iraq. The Republic of Mahabad was established with the help of Soviet Union but the Republic lasted for one year. Due to the pressure from western powers, the Soviet Union pulled out from Iran. So, the Iranian government took back the control of the region and this fueled the continuous agitation by the Kurds for a separate state. Then, during the 1969 revolution, there was open conflict between the Shia revolutionaries of Iran and the Sunni Muslims of the Kurdish party of Iranian Kurdistan. And the fight for independence is an ongoing process. So, the present conflict between Iran and the Kurdish people is due to this long history of conflict between the Sunni Muslims of Kurds and the Shia Muslims of Iran. See, this is an ongoing process which will be addressed only if a state of Kurdistan is created in which the Kurdish people are in majority. So, that's all regarding this discussion. In this discussion, we saw about the Kurds where they are present, the issues faced by them and the current conflict between Iran and the Kurdish people. With this, we have come to the end of the news article discussion session. Now, let us take up the practice problems questions. We have three practice problems questions today. Let us see them one by one. Let us take up the first question. This is a two statement question asking us to find the correct statements. And this question is asked in reference to G20. Let us take up the first statement. The term Sherpa denotes a country's representative to the G20 summit. See, this statement is wrong. A country's representative to the G20 summit is nothing but the Prime Minister or the President of the country. So, in India's case, our country's representative to the G20 summit is our Prime Minister, Mr. Modi. But Sherpa denotes the representative of our Prime Minister, that is Amitabh Khan. So, the first statement is wrong. Moving on to the second statement, the term G20 Troika refers to the current and the next two presidencies of G20. See, this statement is also wrong. Because in our discussion, we saw that Troika represent the previous, the current and the next presidency of G20. So, since both the statements are incorrect, the correct answer here is option D, neither one nor two. Moving on to the second question, let me read out the question, which of the following best defines the term LT-LEDs, which is often mentioned in news. See, this is a very easy question because in our news article discussion, we saw that LT-LEDs represent long-term, low-emission development strategies. So, the correct answer here is option B. LT-LEDs is a strategy aiming to limit greenhouse gas emissions that has to be formulated by the parties of UN FCC. So, once again, the correct answer here is option B. Moving on to the last question, this is a quiz question for you today. Interested aspirants can comment the answer for this question in the comment section. The main questions based on today's discussion are displayed here. Interested aspirants can write the answers for this question and post it in the comment section. If you like today's video, like, comment and share it with your friends. 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