 Hi, I'm Zor. Welcome to a new Zor education. We continue talking about self-induction and this lecture is about a couple of very very simple problems related to self-induction. Now self-induction is actually a very practical issue primarily because when you are disconnect something your self-induction works against reducing the current, which means it's trying to increase the current and that's why we have sparks whenever you're turning off the electricity and in some cases self-induction might be really really large and it might actually damage certain electrical equipment. So we have to really know how to calculate the value of this self-induction. So I consider a very simple setting where we will just calculate this particular value of self-induction and then the second problem will be about the current, which is the result of self-induction. Okay, so here is the problem. The problem is the following setting. You have a wire loop. It's connected to switch and the battery, so plus minus. Now switch is on-off switch. Now our purpose is to basically calculate what happens with self-induction in this case and then the second problem will be about the current. Okay, so this is the setting. Now this lecture is part of the physics 14th course presented on Unisor.com. I suggest you to basically take a look at the website Unisor.com because the whole course is presented there. But there are interdependencies between lectures. In this lecture, for instance, I will be using something which I was presenting in the previous lecture. So I do suggest you to take the course. And there is a prerequisite course to physics 14. It's called Math 14. And it's on the same website. It's free. All courses are free. No strings attached. So I suggest you to basically take the whole course and maybe even both courses. Math 14th and physics 14th. All right, so let's talk about this particular problem and let's go into conditions, whatever we have. Well, first of all, we have this battery and let's assume that it produces certain voltage, U0. So that's original primary EMF, primary electromotive force. In this circuit. Now, under normal circumstances, the resistance of this circuit is R0. So if the switch is on, battery is working, current is working, everything is fine. Basically, U0 is your voltage. R0 is your resistance. And I0 is, according to the Ohm's law, that's your current. Now, that's when everything is settled. But I'm interested in the moment of turning the switch on and off. What happens in these cases? Well, we will consider a specific model. What does it mean that I'm turning the switch on and off in mathematical, for mathematical position? Well, when it's off, I can assume that resistance of the whole circuit is equal to, well, basically infinity. Right? There is no current. So I assume that whenever it's off, it's infinity. Whenever it's on, resistance is R0. So during the time I'm switching on, my resistance is changing from infinity to certain value R0. Now, let's assume that from purely mechanical standpoint, the contact, the time between no electricity and the residual electricity, it's a very small actually interval of time, obviously. This time interval is equal to t. So during the time t, my resistance is reduced from infinity to R0 whenever I am turning the switch on. Whenever I'm turning the switch off, resistance is turning backwards from R0 to infinity. Now, that means that actually resistance is a function of time. Right? Now, time is changing on interval from 0 to time t. So from t is equal to 0 to t is equal to capital T, my resistance is changing when I'm switching on from infinity to R0, when I'm switching off from R0 to infinity. Okay. That's, you know, we are trying to put some mathematics into this purely physical world. Well, it would be nice if I can express it as a formula, right? Because R of t is a function. So function should have a certain relatively preferably simple way. Well, I can come up with a very simple function which does exactly what I was talking about. Now, if it's on, that's R0 times t divided by t. If t is equal to 0, that's denominator, the whole thing is infinity. If t is equal to, and then as t is growing from 0 to capital T, my expression is, it's a hyperbola, right? It goes this way. It's diminishing from infinity to certain fixed value at lowercase t is equal to capital T. Now, that will be t divided by t1. So it will be 0, R0 at t equals to t, right? So that's a simple formula. Now, for off, there is practically the same simple formula. Again, if t is equal to 0, I have t divided by t, it's 1, so R0. As t is increasing to t, capital T, the denominator goes to 0, so the whole thing goes to infinity. So these two functions in my specific problem describe how my turning on and off actually is happening. It's basically changing a resistance of the whole circuit between infinity and R0. Now, is it the same how it really happens in practice? Well, I can tell you a very honest opinion, I have no idea. But it seems to be that at least approximately, and that's what physics actually is. We cannot really know what exactly is happening in nature, but we can approximate it, and our approximation are supposed to be reasonable. So I think it's a reasonable approximation. That's all I'm saying. Well, maybe it's not just this time of proportion. I mean, there are many functions which go from infinity to certain fixed value. Well, there is a logarithmic, for instance, function, or there are many others or tangent who knows. But anyway, for simplicity purposes, it's good enough. Okay, so I know how the resistance is changing in time. Well, if I know that, and I know my original, my primary voltage on the battery, I can come up with change of the current. Now, why do I need the current? Well, I need the current, how it's changed, because I have to know how the magnetic field intensity is changing, because intensity depends on the current. If current is changing, intensity is changing. If intensity is changing, the flux, magnetic flux going through this loop is changing, and that's exactly what's causing the self-induction. So as soon as magnetic flux is changing, it generates extra EMF, the induced EMF, self-induced EMF. And that's what we are interested in, right? So we need to know what is the induced EMF in this case. So again, to get the induced EMF, we have to know how the flux is changing. Magnetic flux depends on the intensity of magnetic field and the area of this loop. Well, area is constant, so magnetic field intensity is changing. How? Because of the current is changing. Why current is changing? Because the resistance is changing. So that's a very simple three or four steps which we have to make. Okay, so let's consider on, whenever we are turning on the switch, what happens? I know R of t. So I can say that current, which is equal to my voltage divided by R of t equals to what? It's equal to voltage R of t is this. This is variable. You see, it depends on t. At t is equal to zero, my current is equal to zero because the switch is not turned on, right? As t is increasing, my current also increases and at t is equal to capital T, my current is equal to nominal current. Whatever this particular circuit is basically developed for. Okay, now I of t current is changing. Well, now if there is a current in a loop, we know there is a magnetic field and you understand that magnetic field goes something like around it. Well, obviously the direction depends on the direction of current. There is a right hand rule, et cetera, doesn't really matter right now. So there is a magnetic field and inside this loop it goes perpendicularly to the surface of the board, right? Now, what is the value of the magnetic field intensity inside this thing? Now, in the center, we have already calculated in one of the previous lectures. Whenever we were talking about magnetism of current in a loop, I think that was the name of the lecture. We basically calculated it. It's a very simple integration problem and the result was that B of, now I put the dependency on the time because current depends on the time. It's equal to mu zero I of t divided by two r, right? Where r is radius. Now, I have calculated this in the center of the loop. Now let me make another assumption that if I will go to any other place, it will be the same. Well, it's a stretch to tell you the truth, but it's a reasonable stretch. I mean, we can go into math and basically try to calculate what exactly the magnetic field intensity at each point inside based on the laws which we were using before. And one of the main law which we will be using in this particular case is the dependency of, if this is the current and this is infinitesimal area, let's put it gs of the current, then the magnetic field intensity at this point depends basically on the current and on this angle. So we can obviously calculate it. Now, in case of the center, the current is always perpendicular to the radius to the radius to the center. So there is always perpendicularity here. So it's easy. If it's not in the center, it's not that easy. But again, we can go through calculations and we can show that it's maybe equal, maybe very close. In some ideal case, maybe it's even equal. I'm not sure it's really not very easy to do. And I prefer basically, okay, let's just assume that the magnetic field intensity at every point is the same. Without the proof, it's just an assumption. And quite frankly, I just don't know if it's a true assumption. It looks like at least approximately true. Maybe it's true 100%. I don't know. And I didn't go through exercise of integration and basically finding out exactly. But let's assume. Now, so this is magnetic field intensity at every point inside this loop. So it's basically at any moment of time, I can always calculate the magnetic field flux, which goes through this loop. Flux is a product of intensity and the area of the loop in this particular case. So my flux, which is also function of t is equal to my magnetic field intensity times area of the circle, pi r square. Okay, so that's equal to mu zero i of t, I will put this, which is u zero t divided by r zero t times true r and times pi r square. Well, r will, so it will be pi r. Am I right? Let me check. Pi mu zero u t r t. Okay, fine. That's what it is. Now, since I know the magnetic flux, I can basically determine very easily my induced EMF. If you remember induced EMF, I will use it u i for induced. It's equal to minus first derivative of the flux by time. So it's a rate of change of the flux. So if flux is changing, there is an induced EMF. If flux is not a function of t, if flux is constant, then there is no induced EMF. So whenever this switch is on after the time, after the time t, when it's really solid on, we have a direct current, which is constant, which is equal to this one. If it's a constant direct current, there is no dependency on time in the magnetic field intensity and there is no dependency on time here with magnetic flux and derivative would be zero, so it would be no induced EMF. So induced EMF is generated only during this relatively small interval of time when the switch is really turning on, not yet turned completely. So this miniscule amount of time, it's like 10 to minus whatever degree of second this thing is happening and then it doesn't really happen anymore. Well, in this case, since I have assumed a relatively simple dependency resistance by time, I have a relatively simple function. Dependency on t is basically a linear dependency and all I have to do is just to take the derivative from the linear function and this is definitely minus mu zero pi u zero r divided by two r zero t. I just changed the sequence and this is my induced EMF. Now the switch is turning on. My induced EMF is negative, it's reducing the original EMF. So whenever I'm trying to increase the current in the circuit, my induced EMF works against my primary one, it doesn't really let me to increase as fast as it probably would be. So this is my induced EMF for the case when I'm turning the switch on. Now let's go and turn to the switching of situation. What happens in this case? Well, in this case, r of t is slightly different, right? Now, if r of t is slightly different, my current would have, instead of this, it will have this. Now, my magnetic field intensity is this, my flux is this, so all I have to do, I have to change this to t minus t. And what happens with my derivative? Well, I have to multiply by derivative of the inner function, which is t minus t, and the derivative is minus one, so in this particular case I have plus. So the absolute value when I'm switching off is exactly the same as when I'm switching on, but it's positive, which means when I'm trying to reduce the current, my induced EMF is trying to bump it up. It prevents actually to reduce the current to zero, which means it's just, it prolongs this agony of electric current. Whenever it's basically dying, it's trying to prolong it a little bit more, put some more life in it. Question is how much life? We don't want that to be too much because then it would overload our electric equipment, whichever is on this circuit, right? And that would be the subject of my second problem. So the first problem is done basically. We have this expression as the magnetic, as the induced EMF produced by switching off and on. If on, it's negative, if it's off, it's positive. Now I have to use this, all these calculations somehow, for basically determining how strong that induced EMF actually is. Now my purpose is not to overload the circuit. So my second problem is the following. I would like to know, now by the way, you have to really understand that if T is small, the smaller T is, let's put this way, the bigger EMF, induced EMF, is generated. So the shorter my time when the switch is turning off, and off is actually more important than on, because on we are just reducing the current, but on we are increasing the current. So that's dangerous. So if my T is very small, then this thing might be very big. So the more abruptly, the faster I'm switching off, the greater EMF is induced EMF is really developed in the circuit. So that's why it's very important, especially in cases when you have really large loads and big voltage, etc., etc., big voltage, you see, if voltage is very big, that might actually be a problem. So it's very important to reduce this current slowly. So it's not just abruptly cut off the electricity. It would be nice if you would gradually do this. That's why on the very large installations, whenever you have switches, large, I mean, in terms of current it consumes, whenever you have these switches, switches are not really like electric switch you have in the room. They are really very, slowly changing the current down to zero when you're switching off to prevent sudden growth in the induced EMF. And that's why whenever there is some kind of a breakage in electrical equipment on big installations, that's dangerous for all the electrical equipment because the current might actually have a spike. And a spike might kill actually the electric equipment. So my purpose is, what is the minimum T when it's really safe? Now what does it mean safe? Well, that's just my definition of safe. So I would like I of T to be less than or equal to 1.1 times my nominal T. So nominal T is what the circuit is basically developed for. I don't want more than 10% increase. So it's 1.1 multiplied by this nominal. I don't want I of T to exceed this value. So question is, what is my minimum T, minimum time during which I am switching to satisfy this condition? Okay. So basically we have to find out the T when this is equal to 1.1 of I zero. Not this one, I'm sorry, that's the electromagnetic force. So we need the total voltage. This is the total voltage, right? The original plus induced. And this is the constant by the way. It doesn't depend on time. This induced EMF depends only on these parameters which are constant. So this is my total voltage. If I will divide it by resistance, I will get my real current in the in the circuit, right? Okay. So what I have to do is I have to basically make sure that this is so I have to basically resolve this U zero plus U I divided by R of T. It's equal to 1.1 times U zero divided by R zero. So I have to put whatever the values are and resolve it for T, for capital T. Right? So instead of R of T I have to put R zero C here. Well, in this particular case I'm interested only, okay, let's think about it this way. Since my total voltage induced plus original is constant, when the current will be the maximum, when the resistance will be the minimum, right? Now when we're switching off, my resistance is growing from basically original value R zero to infinity. So when it's minimum, when in the very beginning, at the time T is equal to zero, at the time T equals to zero, my resistance is smaller. That's why my current will be the bigger. So instead of divided this, divided this to R of T, it's better to divided by R of zero because R of zero is always less than R of T. We're switching off. So in the very beginning, R zero is the maximum value and then it's increasing. So my maximum, my spike is in the very beginning. So I basically have to divide it by R of zero, which is equal to R zero. If T is equal to zero, that would be R zero. So this is an equation. So let's just substitute proper U i. So we have U zero plus U i is this one, mu pi mu zero. Pi U zero R divided by two R zero T. Okay. Now I can safely reduce both sides equals to one point one U zero. Now this is one point one, which means I can subtract one of them and have zero point one, right? And this is very simply to resolve by T. T is equal to mu zero pi U zero is also cancelling out, right? R divided by about zero point one goes here and T goes there. So it's zero point two R zero. I think that's where it is. Let me check. Yes, correct. So this is the minimum value during which my switch should work if I don't want to exceed 10% of the original current. So it depends basically only on the initial resistance of the circuit and the radius of this loop. Well, that's what it is. This is a T minimum. It should be at least this or greater. If it's greater, it would be slower process, less EMF would be induced. Now as far as what exactly this value is, now you know that every lecture on Unisor.com has comments, notes, whatever. It's like a textbook basically. Whatever I'm saying right now is presented in a textual format. And I ended up with some calculation for some practical numbers. And for those practical numbers, I mean real practical numbers, I've got T minimum approximately 10 to the minus eight of a second, which is a very, very small value. So that's the minimum value. Now, most of the switches which manufactured, you know, for regular purposes are not as fast. Now, at least the smaller ones maybe really are something on this level, but they don't really generate big spikes. And the bigger ones are designed specifically to do it slowly. So it's definitely time is greater and that prevents destructive spikes in the current. So if you would, if you read the notes for this lecture, which I always recommend you to do, you will see what kind of a practical values I substitute for these numbers. Mu is a very small actually, I think. And these are just got something, whatever seemed to be reasonable to me. Alright, so that's it for today. Thank you very much. I recommend you to, again, recommend you to take the whole course and even start with a mass for teens. But this lecture, read the notes, it's very interesting to read again, whatever you have, you know, listened to once and pay attention to these numbers, the calculation they're kind of educational. Alright, thank you very much and good luck.